Preface

During the past century, the impact of mathematics on humanity has been more tremendous than ever since Galileo's agonizing fight against the old establishment and the revolution which physics experienced after Newton's subsequent synthesis.
At the beginning of the last century, mathematical ideas and techniques were spread to theoretical and applied physics by the influence of two of the greatest mathematicians of all times, D. Hilbert and H. Poincar6, being then at the zenith of their careers. Their ability to establish very deep at first glance often hidden connections between a priori separated branches of science convinced physicists to adopt and work with the most powerful existing mathematical tools. Whereas the 20th century really was the century of physics, mathematics enjoyed a well deserved reputation from its very beginning, so facilitating the huge impact it had subsequently on humanity. This reputation has been crucial for the tremendous development of science and technology. Although mathematics supported the development of weapons of mass destruction, it simultaneously promoted the advancement of computers and high technology, without which the substantial improvement of the living conditions humanity as a whole has experienced, could not have been realized. In no previous time the world has seen such a spectacular growth of scientific knowledge as during the last century, with mathematics playing a central role in most scientific and technological successes. But, not surprisingly, the extraordinary development of mathematics and science politics created an impressive number of difficulties which are to be solved in order to facilitate further real advance in mathematics. These concerns are not new as is witnessed by the following excerpts from a speech of R. Courant, given in 1962 in G6ttingen on the occasion of Hilbert's hundredth birthday: i
Although mathematics has played an important role for more than two thousand years, it is still subject to changes of fashion and, above all, to departures from tradition. In the present era of over-active industrialization of science, propaganda, and the explosive manipulation of the social and personal basis of science, I believe that we find ourselves in such a period of danger. In our time of mass media, the call for reform, as a result of propaganda, can just as easily lead to a narrowing and choking as to a liberating of i We follow the translation of C. Reid in Hilbert (Springer, New York, 1996, p. 220).

ix

J. Ferrera, J. L@ez-G6mez, E R. Ruiz del Portal

mathematical knowledge. That applies, not only to research in the universities, but also to the instruction in the schools. The danger is that the combined forces so press in the direction of abstraction that only that side of the great Hilbertian tradition is carried on.
Living mathematics rests on the fluctuation between the antithetical powers of intuition and logic, the individuality of 'grounded' problems and the generality of farreaching abstractions. We ourselves must prevent the development being forced to only one pole of the life-giving antithesis.
Mathematics must be cherished and strengthened as a unified, vital branch in the broad river of science; it dares not trickle away in the sand.
Hilbert has shown us through his impressive example that such dangers are easily preventable, that there is no gap between pure and applied mathematics, and that between mathematics and science as a whole a fruitful community can be established. I am therefore convinced that Hilbert's contagious optimism even today retains its vitality for mathematics, which will succeed only through the spirit of Hilbert.
Indeed, one of the main difficulties we are faced with is to treat and to filter the huge amount of information at our disposal, originating in the explosive growth of human knowledge.
What is the best way to deal with such an abundance of information? How to filter the new results in order to ascertain that the most novel and interesting discoveries and relations between the different areas of science and, in particular, within and with mathematics, do find the place they deserve? Certainly, as in former times, sometimes it may be difficult to publish really new results lying outside of the fashionable main stream research.
This is an indication for a severe fragmentation of mathematics and the community of mathematicians m , as well as for the failure of the media to scrutinize and propagate mathematical discoveries. As envisioned by R. Courant, science politics and influential groups may severely affect the quality of mathematics as well.
Not tackling these serious problems might lead to an increasing waste of enthusiasm and energy of a large number of highly specialized researchers who, in practice, stay almost completely ignorant of important mathematical progress, due to the strong fragmentation of the field. Undoubtedly, the enormous amount of information can add to this fragmentation, whose real cost in practical terms seems very difficult to estimate. Quite surprisingly, although mathematical methods for scrutinizing quality are, a priori, stronger than those used in other scientific disciplines, like physics, chemistry and biology, the fragmentation of mathematics is substantially higher than that of these disciplines. Why? Could it be possible that our stronger scrutiny methods facilitate tendencies of established networks to propagate almost exclusively main stream work of their associated groups instead of ensuring independence and quality of mathematical production? Certainly a challenging question to contemplate!
Another serious difficulty lies in the high number of mathematicians still having a genuine 19th century idea of mathematics. Actually, instead of encouraging and supporting the outset to new mathematical frontiers and real innovation, most of the world leading mathematical associations tend to reward almost exclusively the ability of mathematicians to solve open problems coming from the past occasionally highly marginal ones m ,

Preface

xi

instead of looking ahead and encouraging experts to face the new challenges of the globalized world in which we are living.
It is a significant irony that the list of Hilbert's Problems did not predict what Hilbert would work on by himself during the subsequent two decades, as there is no word about functional analysis or integral equations, and, most surprisingly, there is almost nothing on topology, which Poincar6 had founded and which L. E. J. Brouwer revolutionized a decade after Hilbert's celebrated speech at Paris.
In his book The Structure of Scientific Revolutions (Chicago University Press, 1962),
Thomas S. Kuhn advocated the existence of two different types of science: normal science and revolutionary science. In any scientific discipline, normal science follows a number of well established rules, concepts and methodologies, accepted by all researchers in the field. Occasionally, within normal science there arise unexpected discoveries inconsistent with the established paradigms. These discoveries generate high tension that increases in intensity until a scientific revolution is accepted to emerge beneath the surface of such an unorthodox finding. Then a new paradigm naturally arises under which all experts start to do normal science again. Normal science expands within existing paradigms, while revolutionary science sees a necessity to update the old established paradigms. Of course, too strong scrutiny within a strongly fragmented scientific discipline is far from facilitating the diffusion of unorthodox thinking, and so, is hampering progress.
Although these thoughts might be rather contentious, besides being irrelevant for the advance of mathematics, they were at the roots of this book - - at the very beginning a rather fuzzy project. We decided to collect a series of papers covering wide areas of mathematics that have advanced along separate paths around the common theme approximation theory, in an attempt to bring together those fragmented areas, and, simultaneously, to facilitate the development of further connections between them. As a result, this book collects ten mathematical essays around approximation theory, understood in a broad sense, within the context of one of those mathematical fields that has enjoyed a most remarkable growth during the past century: Nonlinear Analysis. Since the pioneering results of H. Poincar6 and L. E. J. Brouwer, Nonlinear Analysis defines itself as that part of mathematics dealing with the existence, structure, and multiplicity of solutions of equations in general topological spaces, Banach spaces in particular. Its main impetus comes from the fact that a huge number of mathematical problems - - pure and applied ones admit such an abstract formulation. The mathematical analysis of the general properties of the set of solutions to abstract equations, that are preserved by continuous deformations, was an important impetus for the development of Topology, and, simultaneously, stimulated the extraordinary development that Functional Analysis linear and non linear experienced during the past century.
Undoubtedly, the classification of the resulting local algebraic manifolds motivated the creation of Algebraic Geometry. The implementation of iterative schemes and continuation methods to compute solution manifolds is on the basis of modem Numerical Analysis. The study of the sensitivity of the states with respect to initial values and parameters triggered the theory of Dynamical Systems. The determination of the correct functional spaces to

xii

J. Ferrera, J. L6pez-G6mez, F. R. Ruiz del Portal

analyze Partial Differential Equations was a milestone in the development of Functional
Analysis since the pioneering results of Hilbert and Banach.
Combining topological, analytical, algebraic and geometrical tools has proved to be imperative for finding the hidden structures of the solution sets in broad classes of nonlinear differential equations. Aside from the enormous interest of nonlinear differential equations per se, and for modelling a wide variety of real world problems, these equations have always provided an extraordinary testing ground for abstract mathematical techniques. Simultaneously, new mathematical tools have been generated by their study and novel ideas emerged that extrapolate to and have an impact on other areas of mathematics, economics and science. This methodology has tremendously facilitated the rise of scientific revolutions since Galileo and Newton, passing through the synthesis of Poincar6, Volterra and
Hilbert, until today.
Linear results are the basis upon which the local nonlinear theory is constructed, while topology provides us with the most universal global properties of nonlinear problems. In complete agreement with I. Stewart in The Problems of Mathematics (Oxford Univ. Press,
1987, p. 156), we corroborate that:
The current success of topology ... owes little to Poincar6's abilities at crystal-gazing, but an enormous amount to his mathematical imagination and good taste. Topology is a success precisely because it forgot the details of those original problems, and instead concentrated on their deeper structure. This deep structure occurs in any scheme of mathematical investigation that makes use of continuity. In consequence, topology touches on almost everything. Of course, the extent to which this fact helps depends on the question you want to answer.
In the papers collected in this volume the authors discuss central problems of their respective research fields which are closely related to approximation theory, understood in the broadest possible sense, connecting topology, analysis, and applications to nonlinear differential equations. These jewelry pieces of mathematical intra-history will be a delight for many forthcoming generations of mathematicians, who will occupy themselves with these fascinating and important subjects. Some authors kindly outlined, as requested by the editors, the path that led to some of their most celebrated results, as well as the personal circumstances of their discoveries.
Before concluding this preface, we express our deepest gratitude to all who have generously contributed to the production of this volume, which could not have taken form without their friendship and unselfish efforts. Our warmest acknowledgements to all of them! Madrid, January 2005.
The Editors

Ten Mathematical Essays on Approximation in Analysis and Topology
J. Ferrera, J. L6pez-G6mez, F. R. Ruiz del Portal, Editors
(~) 2005 Elsevier B.V. All rights reserved

Maximum Principles and Principal Eigenvalues
H. A m a n n
Institut ffir Mathematik, Universit~it Z/irich,
Winterthurerstr. 190, CH-8057 Zfirich, Switzerland

Abstract

In this paper well-known maximum principles are extended to second order cooperative linear elliptic systems with cooperative boundary conditions in strong, weak, and very weak settings. In addition, interrelations between maximum principles andprincipal eigenvalues are studied in detail, as well as continuity properties of principal eigenvalues under domain perturbations.
Key words: Maximum principles, principal eigenvalues, cooperative systems, cooperative boundary conditions, weak and very weak solutions, domain perturbations

1. Introduction

It is the main purpose of this paper to study maximum principles for linear second order cooperative elliptic systems under general linear first order cooperative boundary conditions. We are particularly interested in weak settings, in view of applications to nonlinear systems in situations where higher regularity either cannot be expected or does not constitute a convenient frame to deal with such problems.
Maximum principles for cooperative systems have already been discussed by several authors under various assumptions (of. [20], [22], [32], [36], [43], [56], [60], [63], [71],
[77]). However, in all these references, with the exception of [63], the case of Dirichlet boundary conditions is studied only. Furthermore, in almost all cases maximum principles in the strong sense are considered, that is, for C 2 functions, or, at least, for Wq2 functions where q is sufficiently large.
It is well-known that maximum principles are of great importance for the study of existence and qualitative properties of nonlinear equations. For example, one of the most

2

H. Amann

useful techniques in the theory of second order scalar elliptic (and parabolic) boundary value problems, the method of sub- and supersolutions, is based on maximum principles
(cf. [1], [62], [64], [67]). This is true for systems as well, as has already been observed in [1, Sections 5 and 10] and has since been worked out by several authors under various hypotheses (cf. [59], [62], [66], and the references therein). However, in all those papers either Dirichlet conditions are considered only or, if Neumann boundary conditions are studied at all, it is assumed that either the boundary conditions decouple, a rather particular situation (e.g., [40], [41 ]), or that very strong regularity conditions are satisfied (e.g.,
[62]). It is one of the advantages of our work that our maximum principles allow, among other things, comparison theorems for semilinear problems with nonlinear boundary conditions, the latter depending on all components of the unknown vector function, in a weak setting. The validity of maximum principles is closely related to the existence of a principal eigenvalue, that is, of a least real eigenvalue determining the position of the smallest closed right half plane containing the spectrum. This eigenvalue plays a predominant r61e in the qualitative study of nonlinear boundary value problems via bifurcation theory and in the method of sub- and supersolutions (cf. [37], [51 ], [53], [54], [57], [58], and the references therein). Consequently, we investigate in some detail questions of existence and continuous dependence on the data of the principal eigenvalue.
It should be noted that our results on maximum principles in weak settings are new, even in the scalar case. The same is true for our continuity results for the principal eigenvalue, since we allow perturbations of the Robin boundary as well.
To give a flavor of the content of this paper we describe now some of our results in a simple setting. Here we restrict ourselves to a 2 • 2 system with the diagonal Laplace operator as principal part. The general case is studied in the main body of this work.
Throughout this paper 9t is a C 2 domain in ~n, where n _> 1, with a nonempty compact boundary F. We denote by u " - ( u l , . . . , u n) the outer unit normal on F.
However, to illustrate some of the main results by means of prototypical examples, we assume throughout the rest of this introduction that f~ is bounded.
Let u be a superharmonic distribution in f~, which means u C D'(f~),

{ - A ~ , u} _> 0

for all ~ C D(f~) with ~ > 0.

(1)

Then it is known that u is a regular distribution, in other words: u E Ll,loc(f~). If, moreover, for some point-wise defined representative ~ of u, lira inf ~(x) _> 0 xCf~ for all y E F,

(2)

x-+y

then u _> 0, that is, u(x) >_ 0 for a.a. x E ~ (e.g., [30, Propositions II.4.20 and II.4.21]).
It is clear that from (1) alone nothing can be said about the boundary behavior of u E
Ll,loc(f~) since every test function ~ E 7)(f~) vanishes near F. Thus (1), without the additional information of (2), does not imply that u _> 0.

Maximum Principles and Principal Eigenvalues
The situation is different if we require the validity of the inequalities in (1) for a larger class of test functions and a little more regularity for u. For this, given q C (1, oc), we put w~,~(a, R) . - { ~ e w ~ ( a ,

where

1
-+
q

1

~) ; ~

- o },

-1

and 7 is the trace operator. We also denote by _ O.

(3)

_ Ofor all v C l/Vq2,,.~(a,IR) with v >_ 0
Very weak maximum principles are of importance in nonlinear problems involving low regularity data, for example (e.g., [ 12]). The maximum principles studied below are valid for cooperative systems also. To illustrate this we consider the model system (A, B) on f~ defined as follows: we put u := (u 1, u 2) and assume that there are two decompositions ofF: r - r 1 u r~ - r~ u r~ r 1 n r l - r~ n r~ - ~, such that P~ and F~ are open, hence closed, submanifolds of F. Then we define
, A u - (,Alu, A2u) by ,Alu

.-

_ / k u 1 -+- a11u 1 q- a12u 2,

A2u .-- _ A u 2 _+-a21u I q_ a22u 2, and by
I

on Fo ,
1

~31~ .__ ~ ~i

[ OuU1 + bllu I -}- b12u2 and t"
~2 u .-- ~ U2

on F~ onF~, [ O,,u 2 + b21u 1 + b22u2

on F12,

where we assume that a r* C L ~ ( a , JR),

br* C C I - ( F , I ~ ) ,

r , s e {1,2},

with the hypothesis of 'cooperativity'" a 12 < 0,

a 21 < 0,

As usual, C 1- means 'Lipschitz continuous'.

b12 < 0,

b21 < 0 .

4

H. Amann

[10]

[alla12] [bllb12]

Denoting by X" the characteristic function of F[ and introducing matrix notation,
X'-

0

X2

'

a'-

a 21 a 22

b'-

'

b21 b22

'

we can rewrite this system in the concise from
,,dU - - - - A U

nu att~

(4)

Bu = x(O~,u + bu) + (1 - X)u.
Of course, the boundary operator is to be understood in the sense of traces.
We also define the formally adjoint problem (A lI, Bll) by

.A~v := - A v + a-rv,
Bllv "- x(O,v + bn-v) + (1 - X)v, where aT- is the transposed of a, etc.
We endow all spaces of functions with their natural point- and component-wise defined order. First we consider the eigenvalue problem

A u = Au in 9t,

Bu = 0 on F.

(5)

It will be shown that every eigenfunction of (A, B), that is, of (5), associated with any eigenvalue A is regular in the sense that it belongs to
2
%;_(a,c~).-

["1 Wq~(a,c~). l 0

imply u > 0.

f

Theorem 1. (,,4, ~) satisfies the very weak maximum principle/ffAo (~4, ~) > 0.

(6)

Maximum Principles and Principal Eigenvalues
Since the principal eigenvalue of the Dirichlet Laplacean is positive, Theorem 1 is an extension of (3) to system (4).
As already mentioned, this theorem is new, even in the well-studied scalar case (where obvious analogues of the theorems of this introduction are valid). Indeed, to the best of our knowledge the very weak maximum principle has not been observed so far. Also note that there is no restriction on q, besides 1 < q < ~ .
In order to guarantee that the principle eigenvalue is the only one with a positive eigenfunction we have to impose an additional condition. For this, the pair (a, b), more precisely:
(a, xbx), is said to be irreducible if b12 [V~ N F~ - 0 implies a 12 # 0 and b211F ] n F~ - 0 implies a 21 # 0, putting b12[~ := b21]o :--0.

Note that these conditions can be rewritten as a 12 ~ 0

if

~1512X2 __ 0,

a 21 ~ 0

if

x2b21x1 __ 0.

[a110]

For example, the pair (a, b) with a m

a 21 a 22

b'

[b116121
0

b22

is irreducible if a 21 -r 0 and b121F ~ N F 2 -r 0. Then the following improvement over the mere existence of a principal eigenvalue with a positive eigenfunction is valid.
Theorem 2. Let (a, b) be irreducible. Then A0 (.A, 13) is a simple eigenvalue of (.A, 13) and

the only one with a positive eigenfunction.
We refer to the main body of this paper for a precise definition of the simplicity of an eigenvalue of (.4,/3) and for further properties of A0(.A,/3) and the associated eigenfunction.
If (a, b) is irreducible then we obtain another useful characterization of the positivity of the principal eigenvalue. For this we say that u is a very weak strict supersolution for
(A, B), provided

u C Lq(f~, ~2) for some q C (1, ~ ) and
(A~v, u) >_ 0 for all v C Wq2,,~ with v > 0, with a strict inequality sign for at least one v.

6

H. Amann

It follows from Green's formula that u is a very weak strict supersolution for (A,/3) if u C Wq2(~, IR2) for some q E (1, ~ ) a n d (Au,15v) > 0, meaning, ofcourse, that Au > 0 in ~, /3u _> 0 on F, and (Au, 13u) :fi (0, 0).

Theorem 3. Let (a, b) be irreducible. Then (~4,13) satisfies the very weak maximum principle iff there exists a positive very weak strict supersolution for (,A, 13).
Theorems 2 and 3 (and their more general versions presented in Sections 6 and 7) generalize considerably the results of [56] and [71]. Indeed, besides of the fact that those authors consider only Dirichlet boundary conditions, our regularity hypotheses are substantially weaker than theirs. In particular, in Theorem 3 we are dealing with very weak supersolutions only.
In the following section we formulate the hypotheses used throughout most of this paper and give a precise formulation of the differential operators under consideration. In Section 3 we fix some general notations and describe the boundary spaces for our systems.
Our main results very weak, weak, and strong maximum principles and their interrelations as well as monotonicity and continuity properties of the principal eigenvalue are contained in Sections 4-11, where only the more elementary proofs are given. The somewhat deeper statements as well as additional results are proved in Sections 15-18.
In Section 12 we collect some functional analytical tools, and in Section 13 we recall the version of the maximum principle for scalar equations from which we derive our results for systems. Section 14 contains the fundamental solvability results for nonhomogeneous problems inthe strong, weak, and very weak setting.
For all these results we impose enough regularity on the coefficients of the differential operators to guarantee that the assertions are independent of q C (1, or In Section 19 we present weak maximum principles in W 1, assuming minimal q-dependent regularity only.
They lead to comparison theorems for semilinear elliptic boundary value problems which are of importance in the study of such problems in situations where strong solutions do not exist. We also show that the various realizations of cooperative elliptic systems generate positive analytic semigroups. These results have important implications for parabolic problems.
Since this paper is already rather long we refrain from giving details.

2. Elliptic boundary value problems
In this section we give precise formulations of the elliptic problems under consideration and state the hypotheses used throughout the following, unless explicitly stated otherwise.
We assume that
2yen

• .- N\

{0}.

Maximum Principles and Principal Eigenvalues
The space of real N x N matrices, a linear subspace of all diagonal matrices,

[arS], is denoted by R NxN ' and ~ x x N
""diag

a - diag

7 is the

[al,..., aN].

We always use the summation convention with respect to j and k belonging to { 1 , . . . , n}.
We also assume that
9

ajk -- akj E B U C I ( Q , I"'~diag )' 1 0 then (jl, 13) is inverse positive. Conversely, if (,,4,15) is inverse positive and A is surjective then Ao > 0.
(3) The semigroup HA is positive.
(4) Ifcr(A) r ;g then Ao E o(A).
The proof of this theorem is given in Section 16.

Remarks 7.
(a) Although our regularity assumptions guarantee that (15) holds for every q E (1, oo), we do not know whether Ao is independent of q. This would follow from the spectral invariance of elliptic operators. However, the known results (see [13], [31], [45],
[50], [70]) do not seem to apply to the present situation. It is also not known whether cr(A) ~ ~, in general.
(b) Suppose that only the weaker assumption (16) is satisfied. Then Theorem 6 remains valid, provided we omit assertion (i) in (1).
V1
We emphasize the fact that the results of this section are true under the mere assumption that ft has a compact boundary. We are not aware of any related theorem valid for the case of exterior domains.

5. Nonhomogeneous problems
Of course, the validity of a maximum principle has implications on the solvability of nonhomogeneous elliptic boundary value problems. This is made precise in the present section. We put
W - 2 .__ ( % 3 , ) t q,B B~

'

W-i

q,B

: = (W~,(l_x),.) ' )!
._ W-1 q,(1--X)- ),

with respect to the duality pairings (., .) naturally induced by (10). It follows that

Wq ,

Lq

w q,B

w q,B "
-:

(19)

We endow Wq-k for k E { 1 2} with the natural dual order whose positive cone is the dual
,13
-1
-2
of the cone ( Wqk, ~ ) + . Then Wq,~ and Wq,B are O B S s and each one of the injection maps in (19) is positive.
Next we consider the nonhomogeneous boundary value problem
,Au=finft,

Bu=9onF.

A (strong) Wq2 solution is a u C Wq2 with

(.Au, Bu) = (f, 9)

in Lq • 0 2 W q .

(20)

Maximum Principles and Principal Eigenvalues

13

By a (weak) Wq1 solution we mean a u r Wq1 satisfying

a(v, u) -- (v, f)
(1 -

X)Tu

-

(1 - x)g

for V E W ~ , ( I _ x ) , y , on F.

J

(21)

Lastly, u is said to be a (very weak) Lq solution of (20) if u r Lq and
(M:v, u) - (v, f) + (Ouv, (X - 1)9)r + (Tv, x g ) r

(22)

for v C Wq2,,t~.
The following theorem gives a further characterization of the positivity of Ao.
T h e o r e m 8. The following are equivalent:
(i) Ao > 0.

(ii) Problem (20) has for each ( f , 9) E (Lq x OWq2)+ a unique nonnegative W 2 solution.
(iii) Problem (20) has for each (f, 9) r (Wq,-~ • OWqa)+ a unique nonnegative Wq1 solution. (iv) Problem (20) has for each (/, g) C ( q,t3 x tion. owo)+ a unique nonnegative nq

solu-

The proof of this theorem is also given in Section 16.
R e m a r k 9. If we presuppose only the weaker hypothesis (16) then Theorem 8 remains valid if assertion (iv) is omitted.
K]

6. The principal eigenvalue

Throughout this section we suppose that f~ is bounded. Then we can considerably improve on the results of the preceding section. For this we put

w:_:=Nw:. lO for all

xEf~

Maximum Principles and Principal Eigenvalues

17

implies the existence of the maximum principle for (,4,/3) (cf. [63, Section II.5]). In the usual weak H1 setting and with weak regularity assumptions it has also been shown in [23] that the existence of a positive H I supersolution characterizes the weak maximum principle for the Dirichlet problem. The fact that (i) and (iv) of Theorem 13 as well as the existence of a positive strict classical supersolution w, such that wr(x) > 0 for all x C 9t and 1 < r < N, are equivalent has first been observed in [56] in the case of Dirichlet problems for cooperative elliptic systems satisfying the strong irreducibility condition explained in the preceding section. In the scalar case, L6pez-G6mez [52] could then relax the hypotheses on a classical strict supersolution w by requiring w(y) >_ 0 for V in F, keeping the assumption w(x) > 0 for x C ft. This result has been extended to the case of a general boundary operator in [ 11, Theorem 2.4], where positive strict W 2 supersolutions with q > n are being considered. The proof in [11] relies on [3, Theorem 6.1] which, in turn, is a consequence of the Protter-Weinberger result cited above and a construction of a strict Wq2 supersolution w for q > n satisfying w(x) > 0 for all x C ~ (see [3,
Lemma 5.1 ]), The latter construction is somewhat involved and complicated (but see Remark 36(a)). This prompted L6pez-G6mez [55] to give a simpler proof of Theorem 2.4 in [ 11 ] in the framework of C 2+~ solutions by extending a version of the maximum principle due to Walter [76]. It should be remarked that in none of those results (,4,/3) is required to possess divergence form.
The fact that a strict positive supersolution in the class W~ (f~, I~N ) N C ( ~ , I~N ) implies
2
the maximum principle and the existence of a unique positive eigenfunction is the main theorem in [71 ] for the Dirichlet problem of irreducible cooperative systems (also see [20]).
Our Theorem 13 is much more general since it applies to systems with general coupled boundary conditions and replaces Wq2 supersolutions for q >_ n by the much weaker concept of Lq supersolutions.
The importance of the results of this section is seen from the theorems in Sections 8-11.
Furthermore, it should be noted that the results of Theorems 13 and 15 suffice to apply the abstract techniques of [ 1] to irreducible cooperative elliptic systems. By this way one obtains extensions of the existence, multiplicity, and bifurcation results contained in [ 1]. For example, one can extend the three solutions theorem [ 1, Theorem 14.2] to such systems, etc. We leave the details to interested readers.
It should also be remarked that, thanks to Theorems 12 and 15 (and their proofs), it is not difficult to extend the anti-maximum principle of C16ment and Peletier [26] (also see [72]) to cooperative irreducible systems of the form (,4,/3). Details are also left to the readers.

8. Monotonicity of the principal eigenvalue
Throughout this section 9t is again bounded and only the weaker hypothesis (16) is

imposed.
We discuss monotonicity properties of the principal eigenvalue with respect to variations

18

H. Amann

of a, b, f~, and the boundary conditions.
First we suppose that 8 E L~(ft,1R NxN) and b C C I - ( F , I ~ NxN) are cooperative.
Then we define (A,/3 ) by replacing a and b in (8) and (9) by ~ and b, respectively.

Theorem 16. Suppose that either N - 1 or (a, b) and (~d,b ) are irreducible. If

(a, XbX) < (~, XbX)

(27)

then

~o(A,~) < Ao(~,~).
Proof Let u0 be a positive eigenfunction of (A,/3) to the eigenvalue

Ao := Ao(A, t~).
Then

( , 4 - ~o)~o - (~4- ~o)uo + ( ~ - a)~,o - ( ~ - a)~o

inf~

and
/3no --/3Uo + X ( b - b) XTuo - X ( b - b) X"7'uo on V.
Hence it follows from (27) and the strong positivity of uo, guaranteed by Theorem 12, that uo is a positive strict Wq2 supersolution for ( . 4 - Ao, B). Thus

0 < ~o(d- ~o,~)

-

Ao(d,~)

-

~o

by Theorem 13 and Remark 14.

1--1

The next theorem shows, in particular, that the principle eigenvalue decreases strictly if a Dirichlet boundary condition is replaced for at least one component of u by a Neumann boundary condition on at least one component of F. To make this precise we suppose that

C C(F, II~N• ) is a boundary identification map. Then we put
"'~diag
~

.-

~ ( o , ~ + b~) + (1 - ~ ) ~ .

Note that X < ;~ means that F~) D F~) for 1 _< r _< N where at least one of the inclusions is proper.

Theorem 17. Suppose that either N - 1 or (a, xbx) is irreducible. Also suppose that
X < ;~ and

( ~ - x)b A 0

on P.

Thus ao is a strict W1 supersolution for (A - ~o,/3). Hence

o < ~o (.4 - ~o, ~) - ~o(~4, t3) - ~o by Theorem 13 and Remark 14.

VI

Note that (28) implies that brr _< 0 on each component o f F on which a Dirichlet boundary condition is replaced by a Robin one.
For example, it follows from Theorem 17 that, given that a is irreducible if N > 1, the principal eigenvalues for the pure Neumann, the general Robin, and the pure Dirichlet condition satisfy
~o(~4, O~) < ~o(~4, B) < ~o(~4,'y), provided br~ _< 0 for 1 _< r _< N with at least one strict inequality sign. Also note that, thanks to the cooperativity assumption, the second inequality in this chain is consistent with Theorem 16.
Our next theorem shows that the principle eigenvalue increases if the domain shrinks and if on each boundary component being moved inside the original boundary condition is replaced by a Dirichlet one.
Theorem 18. Let f~* be a proper C 2 subdomain of f~ with boundary F*. Denote by E the union of all components of F* having a nonempty intersection with f~, and put
E' . - F* \ E.
~NxN
~.
Define a boundary identification map X* C C(F*, "~diag )for by x*lr~--

0,

x*lz'

-

xlZ'

and put

(ta;k],

,a:),a*)-

( a l , ,an),a) la*

H. Amann

20

and
A*u "- Au, for u E Wq2 (f~*, I~N ). Then

13*u "- x*(Ou, u + bu) + (1 - X*)u

o(A, s,

<

o(W, t3*, w )

provided (a*, x*bx*) is irreducible if N > 1.
Proof Observe that the irreducibility of (a*, x*bx*) implies the one of (a, xbx). Also note that the strong positivity of a positive eigenfunction u of (.A,/3) to the eigenvalue
Ao " - A0(.A, B, f~) implies that u ] E > 0. Hence u is a positive strict W2 supersolution for (.A* - A0,/3*, 9t*). Now the assertion follows once more from Theorem 13 and
Remark 14.
[-I
The theorems of this section, as well as their proofs, are more or less straightforward extensions and sharpenings of corresponding results established in the scalar case (i.e.,
N - 1) by L6pez-G6mez [52] for Dirichlet and by Cano-Casanova and L6pez-G6mez [21 ] for general boundary conditions (also see [37] and, for the Dirichlet problem in a weak setting, [24]). In the particular case of Theorem 16 where X - 0 and ~ - 0, that is, for
Dirichlet problems, the monotonicity of Ao as a function of a has also been shown in [56], given the much more restrictive assumption that a, ~ E C ( ~ , ~ U • N) and satisfy ~ >__ a and ~rs (x0) > a rs (xo) for some x0 r 9t and all r, s r { 1 , . . . , N}.
The weak Dirichlet problem in W.1 for scalar equations has attracted a lot of interest. In
2,"y
this case general perturbation theorems are due to Arendt and coauthors [14], [17], Stollmann [69] and, in particular, Daners [28], [29], who has perhaps the most general results.
For the weak Robin boundary value problem in the scalar case we refer to [27].

9. Continuity of the principal eigenvalue
We suppose again that f~ is bounded and put
E 1 (~-~) .__ C 1 ( ~ , ]"'~diag
~N•

n X L ~ (f~ ' II~N• N ) n X Zoo ( ~ , I~N x N )
"~diag
X

CI-(F,~ N•

x C( FT~ •
,lN

)"

Given
9

-

x) c

we define (~4(a),/3(a)) on f~ by (8) and (9). T h e n we denote by g'(f~) the set of all a C E 1 (f~) such that X is a boundary characterization map for Q and (.,4(a),/3(a)) is a cooperative elliptic boundary value problem on ~ such that (a, b) is irreducible if N > 1.
Let f~ be a bounded C 2 domain in ~n and let ~ 9 ~ --+ ~ be a C 2 diffeomorphism. Then
~o " - ~ ] F is a C 2 diffeomorphism from F onto 0~. Given
9- ([ ajk], ( a l , . . . , ~n), a, b, ~) E E 1 (~'~),

Maximum Principles and Principal Eigenvalues

21

we put

an) ~*a ~0 b, ~OX) ~ E1 (f~)

([~* ajk], (~* al,

~

where, for a function f on f~, the pull back of f by ~ is defined by
,-,.,

qs* f "-- f o ~s.
,.,.,

,-,.,

Note that 9~*~ E g (f~)if ~ C g ( ~ ) .
Let ( ~ i ) be a sequence of bounded C 2 domains in IRn . Then it is said to be C 1 converging towards f~ if there exist orientation preserving C 2 diffeomorphisms ~;i " f~ --+ f~i such that 9~i --+ idg in C 1 (~, R N ). Such a sequence (~i) is said to be a representation sequence for (f~i).
Sequences of bounded C 2 domains being C 1 convergent towards f~ are often obtained by deformations of 9t be means of sequences of transformation groups for f~. This is true, in particular, if the transformation group is generated by a C 2 vector field.
Example 19. Suppose that 9t is bounded and f E C 2 (~, IR'~ ). For each x C ~ let ~(r, x) be the solution at time r of the initial value problem
- f((),

((0) - x.

(29)

Then there exist r - < 0 < r + such that
C C 2 ( ( r - , r +) x ~, IR~),

(30)

and ~ - "- ~ ( r , - ) is for each r E ( r - , 7-+) an orientation preserving C 2 diffeomorphism from ~ onto ~ - " - y)~-(~). Thus Ft~- " - r is a C 2 domain in Rn and, given any sequence (rj) in ( r - , r +) with rj -+ O, the sequence (f~,-,) is C 1 converging (in fact:
C 2 converging) towards f~, and (cp,-j) is a representation sequence for (f~-~).

Proof The theory of ordinary differential equations (e.g., [6, Section 10]) implies these assertions. Ul

Now we fix r, s, t C [1, oc] satisfying

(n,n/2, n - 1 )
-

n>_3,

e (2, oc] x (1, oc] x (1, oo]

if

n-2,

(1, 1, 1)

(r,,,t)

if

if

n-l,

(31)

and put
E,

,

(a) .-

NxN
,"'~diag )

>< Lr(~ ' ~N•
"'~diag ) n • L s

]i~N• N )

• Lt(F, ~N•

• C(F, ~NxN..,~diag
)"

Note that E 1 (f~) and Er, s,t (f~) are Banach spaces satisfying E 1 (f~) ~-+ Er,s,t (f~).

22

H. Amann

After these preparations we can formulate the following general continuity theorem for the principle eigenvalue, whose proof is given in Section 18. Note that we allow not only all coefficients to vary but the domain as well.
Theorem 20. Let (31) be satisfied Suppose that (f~i) is a sequence o f bounded C 2 domains C 1 converging towards f~, and let (qPi) be a representation sequence for (f~i). Also suppose that a E E(f~) and ai C ca(f~i) such that (~*ai) converges in Er,s,t(f~) towards a. Then
Ao(~4(ai),13(ai), fti) --+ Ao(.A(a),/3(a),ft)

as i --+ cx~.

Furthermore, if u, resp. ui, is the unique positive eigenfunetion o f (~4(a),/3(a)), resp.
( M ( a i ) , B ( a i ) ) , o f W~ norm 1 then ~ ; t t i ---+ ~t in W 1.

It should be remarked that ~)" ui -+ u in W1, provided r, s, and t are replaced by suitably chosen numbers ~, r/, and ~ depending on q (cf. Theorem 47).
Example 21. Suppose that 9t is bounded and f E C 1 (~, R n). Let ~ be the flow defined by (29) and fix T- < 0 < 7-+ such that (30) is true. Let V be an open neighborhood of

U

~,. x {,-,-}

r-- 0. Then
Ao(.A, B) - ,sup inf a(v, u) ' ce (v,u)

(41)

the infimum being taken with respect to all nonzero v C (Wql,,(l_x).y) +.
Proof Suppose that
< Ao := Ao(.A,/3).

Then, as in the proof of Theorem 23, we see that
~0(A-

~,t~) > 0

and there exists a positive strict Wq2 supersolution u for (.A - A,/3). Hence u is strictly positive by Theorem 15. From Green's formula we deduce that u is a positive strict W 1 supersolution for (.4 - A, B). Consequently,

a(v, u ) >

u),

vc

+,

H. Amann

26

and ( 1 - X ) T u _> 0. Since (v, u> > 0 for each v > 0 by the strict positivity ofu, we see that
A<

a(v, u)

inf

- ve(Wl ,(I--x)~ )+\{o} qt (v,u> "

This implies Ao "- Ao(A, ~) _< A*, with A* denoting the right-hand side of(41).
Suppose that c "- (A* - A)/2 > 0. Then there exists u E Q satisfying fl(V,?_t) >

()~*

-

E)(V,2t),

V e (Wql,,(l_x),7

)+\{0}.

Hence u is a positive strict W1 supersolution for (A - A* + c,/3). By invoking once more
Theorem 13 we arrive at a contradiction as in the final part of the proof of Theorem 23. E]

I I. C o n c a v i t y o f the p r i n c i p a l e i g e n v a l u e

Suppose again that f~ is bounded and only assumption (16) is satisfied
In this section we give an application of the minimax characterization of Theorem 25 to the study of the behavior of Ao (.A,/3) as a function of (a, b). For this we first note that, if
N _> 2, the set of all cooperative pairs (a,/3) is a convex cone in the algebra
(L~(f~,E) • C I - ( F , I ~ ) ) N x N .

(42)

Given (aj, bj) in (42) for j - 0, 1, we put
O_ 0 f o r A > c o .
We denote by s+ ( - A ) the infimum of all such co.
The next two theorems are basically known and included for easy reference only.

H. Amann

28

Theorem 28.
(a) The following are equivalent:
(i) A is resolvent positive.

(ii) (A + A) -1 _> Ofor A > s ( - A ) , that is, s + ( - A ) 1 satisfying IIUA(t)IIs
~_ M e ~t for t _> O. Hence the assertion follows from well-known results in semigroup theory
(eg., [25, Proposition 7.1 ]).
(b) This follows from part (b) of the proof of Proposition 3.11.2 in [15] (by observing that it is valid without the standing hypotheses of those authors that the positive cone is normal and generating ).
[]
The positive cone E + is said to be generating if E - E + - E +. It is normal if each order interval

[x,y].-{z~E; x -Ao such that (A + A) -1 is irreducible.
(ii) ()~ + A) -1 is for each )~ > -)~o strongly irreducible.
(iii) blA is strongly irreducible.
Proof It follows from [65, App. 3.1 ] that the preceding definition of a positive irreducible bounded linear operator is equivalent to the one used in [25, Section 7.1 ]. Hence the assertion is implied by [25, Proposition 7.6 and Corollary 7.8].
UI

Suppose that
Eo ~ E - l ,

A-1 C 7-/(Eo, E_I),

A-1 D Ao "- A.

(46)

Then we denote for j C { - 1 , 0} by Nj ()~) the algebraic eigenspace of the eigenvalue A of Aj.
Lemma 32. Let (46) be satisfied Then
Crp( A _ I ) = Crp(Ao).
Furthermore,

X_l()k) = Xo(~),

/~ C ap(Ao).

Proof (i)It is clear that Crp(Ao) c Crp(A_l) and No()~) c N-1(s

for ~ C Crp(Ao).

(ii) Suppose that there exist )~ E C and x, y C Eo satisfying
A _ l x = )~x - y.

(47)

Fix co E p ( - A o ) M p ( - A - 1 ) . Then (47) is equivalent to x - (w + A_1)-1 ((w +/~)x - y).

(48)

From A_I D Ao it follows that
(co -k- A_I) -1 D (co nt- Ao) -1 .
Thus we deduce from (c~ + A)x - y C Eo and (48), thanks to (co + A0)-I (Eo) C El, that x C E1 and, consequently, Aox = A x - y.

H. Amann

30

(iii) Suppose that A C C, m C IN, and x E ker[(A- A_l)m+l].
Then there exist x o , . . . , Xm C Eo satisfying xo - x and

A - l Xk -- Axk -- Xk-+-l,

0 ~__ ~ ~__ m ,

where Xm+l "- O. Thus we deduce from (ii) by backwards induction that Xk C E 1 for
0 < k < m and that xo E k e r [ ( A - Ao)m+l].
This implies N_I(A) C No(A).

[-]

Next we prove a perturbation theorem for resolvent positive generators of analytic semigroups.

Proposition 33. Suppose that 0 C (0, 1) and (., ")o is an interpolation functor of exponent O. Put Eo "- (Eo, E1 ) o. I f B E f_.(Eo, Eo) then
A - B E 7-[(E1, Eo).

I f A is resolvent positive and B >_ 0 then A - B is also resolvent positive.
Proof Since A C 7-/(E1, Eo) there exist positive constants M and co such that
IAI1-j I[(A + A)-IIIC(Eo,Ej) co,

j - 0 , 1.

Thus, by interpolation, there exists M1 such that
ReA >co.

II(A + A)-II[C(Eo,Eo) < M1/IAI 1-~ ,
Hence we can find col _> co such that
IIB(A + A)-lllc(Eo) < 1/2,
It follows that 1 - B(A + A) -1 E s

ReA > (.01.

has an inverse on Eo, bounded by 2, and o (49)

j=0 in f-,(Eo). Hence
[Re A > COl] C p ( - A + B) and (A + A - B ) - 1 - - (/~ + A) -1 (1 - B(A + A ) - I ) -1,

R e a > col,

(50)

so that
II(A + A -

B)-~llC(Eo,~) < 2M1/IAIa-J,

ReA > Wl,

j-

0,1.

This proves that A - B C "H(E1, Eo). Furthermore, i f B _> 0 and A is resolvent positive, we deduce from (49) and (50) that (A + A - B) -1 _> 0 for A > col.
[-]

Maximum Principles and Principal Eigenvalues

31

It should be remarked that the first part of the assertion is well-known (eg., (I.2.2.2) and
Theorem I. 1.3.1 in [8]).
The next result shows that the set of resolvent positive operators in 7-I (El, Eo) is closed.

Proposition 34. Let (Aj) be a sequence in 7-/(El, Eo) converging in s
A. I f each Aj is resolvent positive then A is resolvent positive as well.

Eo) towards

Proof It follows from [8, Corollary 1.1.3.2] that there exist t~ > 1 and w > 0 such that
[Re z > CO]belongs to p ( - A ) N p ( - A j ) and
II(A + A)-IlI~(Eo,E1) + IAI II(A + Aj)-III~(Eo) _< ~,

ReA > w,

j e N.

Hence we infer from
(A + Aj) -1 - (A + A) -1 - (A + A j ) - I ( A -

Aj)(A + A) -1 ,

Re A > CO,

that

II(A +

Aj) -1

-

(A q- A)-IIIc(Eo) ~ IA1-1 ~2 IIA- Ajllc(-1,~o),

ReA > co.

Thus, in particular,

(A + A j ) - l x --+ (A + A ) - l x in E0 for A > C and x E Eo. Now the assertion is a consequence of the closedness of the
O
positive cone.
[]

13. The strong maximum principle for the scalar case
1
In this section we suppose that N - 1 and set Fj " - F j for j - 0 , 1 , of course. Then we put
AOU := --ajkOjOkU + ajOju + au, where ajk - a k j , a j , a C L ~ , with [ajk(X)] C I~n• x C f~. We also put

being positive definite for a.a.

/

~0u "-- ~ u

/

Ozu + bu

on Fo, onFi, where/3 is an outward pointing nowhere tangent C 1 vector field on F1, and b is a C 1- function on F1. Clearly, in this case the derivative 0Z is used in the definition of strong positivity.
The following theorem is the basis for the proofs of the following sections. It slightly improves [3, Theorem 6.1 ]. Its importance stems from the fact that there is no sign restriction for b.

Theorem 35. There exists COo C ~ such that (.Ao + co,/30) satisfies for w > COothe strong maximum principle.

H. Amann

32

Proof If f~ is bounded, this is a reformulation of Theorem 6.1 in [3] (where the statement is incomplete since the condition u(y) = 0 for the validity of0~u(y) < 0 is missing). Since

w~ ~ Co (fi),
~

q > n,

the same proof applies if it is only supposed that F is compact, provided Lemma 5.1 in [3] is valid. But this follows easily from the proof of the much more general Theorem B.3 in [5].
7-1
Remarks 36.

(a) The proof of [3, Lemma 5.1 ] is somewhat complicated and perhaps not too transparent. By restricting the arguments leading to Theorem B.3 in [5] to the relevant cases k - 0 and k - 1, one gets a simpler and more lucid demonstration.
(b) Suppose that (a, b) _> 0. Then COo < 0. Furthermore, (,A0, B0) satisfies the strong maximum principle unless ft is bounded, F = Fa, and (a, b) = (0, 0).

Proof This is a consequence of the classical maximum principle.

7-1

14. Strong and weak solutions

We return to the case of a general N E N x and the hypotheses of Section 2. We set

W q,/3 "- Lq and define linear operators
~
Ak-2 E s

k-2
' Wq,B )

k C {0,1},

by
(V, A-1 u) := a(v, u),

(V, U)C W~,(I_x), ), X WI(I_x),7

(51)

and

(v, A_2u) "- (A~v, u),

(v, u) e Wq2,,t~ x Lq,

(52)

respectively. A k is called W -k realization of (A,/3) We also put Ao "- A and denote by N_j(A) the algebraic eigenspace of A C ap(A_j) for j C {0, 1, 2}.
-

q,B

9

Theorem 37. For j C { 1, 2}

(i) A j - 2 C qr-[(wJB, W~B2) "
(ii) A-2 D A-1 D Ao,"

(iii) cr(A_j) -- cr(Ao) and crp(A_j) - Crp(Ao).
If A C ap(Ao) then N_j(A) = No(A).

Proof Fix co > s ( - A o ) and let [(E~,Bc~) ; a C I~] be the interpolation extrapolation scale generated by (Eo, Bo) := (Lq, co + Ao) and the complex interpolation functors [., "]0,
0 < 0 < 1. (We refer to [8, Chapter V] for the general interpolation extrapolation theory,

Maximum Principles and Principal Eigenvalues

33

and to [7, Section 6] for a summary of the main results.) Then (cf. Theorems 7.1 and 8.3 in [7] and observe that they remain valid if it is only assumed that F is compact) we find that
E-j~2

9 W q,l~ '
-j

B-j~2 -A

--J '

where - means 'equal except for equivalent norms'. Hence (i), (ii), and the equality of a ( A _ j ) and or(A0) follow from (15) and the general interpolation extrapolation theory
(cf. Theorems V. 1.4.6 and V.2.1.3 as well as Corollary V.2.1.4 in [8]). The remaining part of (iii) is now a consequence of (i), (ii), and Lemma 32.
[~
R e m a r k 38. Suppose that only the weaker assumption (16) is satisfied9 Then Theorem 37 remains valid for j = 1 and with A-2 being omitted in (ii).
Proof Of course, the interpolation extrapolation scale generated by (Lq, • q- Ao) is still well-defined. However, since in this case the dual of A is not explicitly known, the space
E-1 cannot be identified in terms of a known space of distributions. But it is not difficult to see that

E-1/2

9

W-1 q,/3 is still true.

I--1

The next theorem concerns the solvability of the nonhomogeneous problem (20) and the parameter dependent boundary value problem
(A+A)u=finf~,

/3u=gonF.

(53)

Theorem 39.
(i) Every strong Wq2 solution o f (20) is a weak Wq1 solution, and each weak Wq1 solution is a very weak L q solution. j-2 (ii) Suppose thatA C p ( - A ) , j E {0 , 1,2}, a n d ( f , 9 ) C W q,t~ x OWqj. Then (53) has a unique Wqj solution.
-1
(iii) If (f, 9) C W q,t~ x OW1 then every Lq solution o f (20) is a Wq1 solution. Similarly, if (f, 9) C Lq x OWq2 then every W 1 solution is a Wq2 solution.
Proof (i) is an easy consequence of Green's formulas.

(ii) First suppose that j = 2. From [5, Theorem B.3] we know that there exists

c(OWq satisfying 13n2q; - q; for qD C cgWq Set w "- n 2 9 . Then u is a Wq2 solution of (53) iff
2.
v := u - w satisfies
(A+A)v=hinf~,
/3v=0onF, where h :- f-

(A Jr- .A)w C Lq,

that is, iff (A + A)v = h in Lq. This proves the assertion i f j = 2.

H. Amann

34
Suppose that j -

1. The trace operator is a continuous retraction from the space W 1

onto W 1-1/q ( r ) , that is, there exists

C f--,(wl-1/q(r), W g) with 7T/~ - ~; for ~ E Wq1-1/q (F) (e.g., [5, (B.21)-(B.23)]). Set w " - 7~(1 - X)9-Then u is a W 1 solution of (53) iff v " - u - w c W 1 satisfies
A(~, v) + a(~, v) - (~, f ) - A(~, w> - a(~, w),

~ C W ~ , ( I _ x ) . ),

(54)

and
(1 -

X)?v =

0,

(55)

thanks to ~2 _ ~. Since a is a continuous bilinear form on Wql,,(1_x)7 x W1, the right-hand side of (54) defines an element h in W q,B " Thus (54) and (55) are equivalent to
-1
y E W . q,(1--X)~'
1

()~ + A - 1 ) v - h.

Now the assertion for j = 1 follows from o ( A _ l ) - o(A).
Finally, assume that j = 0. Note that
(1 - ;~)0, E s

Wql,-1/q' ( r o ) ) .

Hence
(0u) , (1

-

, W -q,t3)"
2

X) C s

Similarly, and, consequently,
, W -2
F r o m this and (22) w e infer that u is an Lq solution of(53) iff

(A + A_2)u = h, where -2 h " - f + (0•)'(X - 1)g + ~'xg C W q,B"
Thus o-(A_2) = (7(A) implies the assertion in this case also.
-1
(iii) Suppose that (f, g) C W q,B x OW1 and u is an Lq solution of (20). Then there exists w > 0 such that
)~ : = w + ) ~ E p ( - A ) .
Hence (20) is equivalent to
(A~o + .A)u = f,~ in f~,

Bu = g on F,

(56)

where
-1
f~ "- f + wu C W q,13"
Thus (ii) implies that (56) has a unique W 1 solution v. From (i) we infer that v is an
Lq solution of (56). Since it is unique, by (ii), it follows that u - v, that is, u C W q,B" This
1
proves the first assertion. The second one follows by similar arguments.
E]

Maximum Principles and Principal Eigenvalues

35

R e m a r k 40. If we presuppose only condition (16) then Theorem 39 is valid with any reference to Lq solutions being omitted.

Proof This follows from the above proof and Remark 38.

71

15. Resolvent positivity
The next theorem is the basis of all the following positivity results.

Theorem 41.

-1
-2
(i) W q,6 and W q,B are OBSs and the natural injection maps (19) are positive.
(ii) (WqJ, + is dense in (wqk,t3)+ f o r - - 2 < k < j < 2. ts) (iii) A_j are resolventpositive for j C {0, 1, 2}.

Proof (1) First we assume that a E Lc~(Q, I~NxN )
"~diag

and

b E C I - ( F , I~NxN )"
"'~diag

Then Theorem 35 applies to the boundary value problem (A ~, B ~) for 1 < r < N. Hence there exists wo > 0 such that

(A + A o ) - l v >_ O,

A > aJo,

v c T)+,

(57)

where 7? . - 7?(ft, 1R is the space of all smooth IRN valued functions with compact
N)
support in f~. Since (A + A0) -1 C s and 7?+ is dense in L + it follows that (57) is true for all v E L +. Thus A0 is resolvent positive.
The same arguments show that A~, the Lq, realization of (A t~, B~), is also resolvent positive. Note that Lq is generating since Lq, is a Banach lattice. Thus, fixing A > s ( - A ~)
+
and setting
P " - (A + AI~)-ILq ,
+
it follows from Theorem 28 that
PC

(W~,~)+

and

W~,t~ ~ - P - P .

Hence (Wq2,,B~)+ is generating, thus total. Now (cf. the proof of Theorem 37) Theorems V.1.5.12, V.2.3.2, V.2.7.2, and Corollary V.2.7.3 in [8] imply that (i)-(iii) are true in this case.
(2) We consider the general case. First we observe that W q,~ and Wq,B are independent
1
-1 of a and b. Hence it follows from step (1) that they are OBSs, the injection maps
Wql,B ~ Lq ~-+ W - 1 q,t~ are positive, (Wq,~) + is dense in L +, and the latter cone is dense in (Wq,-~)+
1

H. Amann

36

Recall (25) and (26). Define (A A , B A) by replacing a and b in the definition of (,4,/3) by a A and b A, respectively. Then step (1) implies that AA1 the Wq,B realization of
-1
(f[zx ' BA), is well-defined, belongs to qr~(Wq,B, W q,t3), and is resolvent positive.
1
-1
--

'

Fix s C (l/q, 1) and put
Wq: B

"--

; (1

Wq:(I_x),), "--" { V E W

-

X)3"v

-

0 }.

Then [7, Theorem 7.2] implies that
W q,B 9 ( W q,B' W.q,B)(l+s)/2,q
S
-1
1

where (-, ")0,p are the real interpolation functors for 0 < 0 < 1 and I < p _< ec.
By the trace theorem,

% . - 7lWq~,U ~ s

(58)

Lq(F1)),

and
I

~W-1

e

q B),

(59)

where
"71 "- 3'[Wql,,(l_x).y C s
Consequently, setting

B u "- a~ + 3"~xb~ it follows from (a~ ~ E L ~ ( ~ , ~ N•

u C Wq~B,

• L~(F,I~NxN),

Wq: u ~ Lq r

W q,B '
-1

(60)

and (58) and (59)that B C E(Wq~,, , W q,B)- Furthermore, B _> 0 thanks to the posi-1 tivity of (a ~ b~ of the trace operators % and 3'1, and of the injection maps (60), and thanks to the fact that L + and (Wq,-~)+ are the dual cones of the positive cones of Lq, and Wql,,(l_x).r, respectively. Note that

(v, (A~_I - B)u} - a(v, u),

(v, u ) C W~,(l_x). r • wl(l_~).y,

so that
A_I - AAI_ - B.
Hence Proposition 33 guarantees that A_I is resolvent positive.
From Theorem 37 we infer that
(A -~- A_2) -1 D (A -~- A _ I ) -1 D (A -k- Ao) -1,

A > s(-Ao).

Thus Ao is resolvent positive as well. The same arguments apply to the boundary value problem (.A~, B~) and guarantee that A~o is resolvent positive. Thus we see, as in step (1), that ( W+q,• is a proper cone, that is, W-2q,B is an OBS. Now the remaining assertions
-2)
follow by the arguments of step (1).
UI
R e m a r k 42. Suppose that only the weaker assumption (16) is satisfied. Then Theorem 41 remains true if all assertions involving Wq,B and A-2 are omitted.
-2

Maximum Principles and Principal Eigenvalues
Proof This follows from Remark 38.

37
D

16. Proofs of the weak maximum principles

Now it is not difficult to prove the theorems presented in Sections 4 and 5.

Proof of Theorem 6

(a) Theorem 41 (iii) guarantees that A is resolvent positive. Thus (3) and (4) follow from Theorems 28 and 29, since Lq is a Banach lattice.

(b) If (,4,/3) is inverse positive on W2 then A is inverse positive9 Thus, if A is surjective then Ao > 0 by Corollary 30.
From (52) we infer that (.A, B) satisfies the very weak maximum principle iff A-2 is inverse positive. Suppose that Ao > 0. Then A is inverse positive by Corollary 30. Since
Ao - A0(-A_2) by Theorem 37(iii), it follows that

0 C p(A_2).
Also suppose that u C Lq and A_2u > O. Then Theorem 4 l(ii) guarantees the existence of a sequence (fk) in L + converging in Wq-2 towards f "- A_2u " Hence the sequence (uk) '
,B
where u} - - (A_2) - 1 f k , converges in Lq towards u - (A_ 2) - 1 f . Since A_2 D A by
Theorem 37(ii), it follows that ( A _ 2 ) - I D A -1 Thus u} E W q2B and
,
9

Auk -- f k >_ O.
Consequently, Uk >_ 0 by the inverse positivity of A. Hence Theorem 41(i)implies u > 0.
This shows that A-2 is inverse positive and proves (2).
(c) Suppose that u r W1 satisfies assumption (18). Then we define f r (Wq,-~) + by
y) . - a(v, u),

e

We also set g " - (1 - X)Tu r (Wql-1/q(v0)) + ~

(OW~ +.

By Greens's formula,

(A~v, u> - a(v, u) + ((X - 1)Ouv, "Yu}r
= (v, f> + 0 for v c (Wq2,,t~)+. Thus u > 0 if (i) is true. Hence (i) implies (ii).
Suppose that u G W2 satisfies (Au, Bu) > O. Then
(f, g ) ' - ( A u ,

Bu)r

(nq • OW:) + ~ (Wq,-~ x OWql) +

by (13). From Green's formula we infer that

a(v, u) - (v, Au) + (XTv, Bu)r - (v, f ) + (~/v, Xg)r > 0 for v r (W~,(l_x).y)+. Thus u r Wq2 ~-+ W1 satisfies (18) so that u > 0 if (ii) is true.
This shows that (ii) implies (iii).

H. Amann

38

Now suppose that A is surjective and (A,/3) is inverse positive on Wq2. Then A is inverse positive. Hence Ao > 0 by (2), so that
0 C p(A-2) = p(A).
From the second part of (b) we know that A_2 is inverse positive. Thus (iii) implies (i).
This proves (1).
U]

Proof of Remark 7(b) Replace in the second part of step (b) of the preceding proof the very weak maximum principle by the weak one and A_2 by A-1. Then it follows that the inverse positivity of A implies the one of A_I. Hence from (51) and (18) we deduce that
A0 > 0 implies that (A,/3) is inverse positive.
Similarly, by replacing in the beginning of the last paragraph of the preceding proof A_2 by A_I, we see that (iii) of Theorem 6 implies (ii).
UI

Proof of Theorem 8.

It is an easy consequence of Theorems 6 and 39 that

(i) ~ (iv) ::v (iii) ~ (ii).
Suppose that (ii) is true. Then it is obvious that (.A,/3) is inverse positive. Given f C Lq, it follows from (ii) that there exists u + E Wq,B satisfying Au • - f+ where f + (resp, f - )
2
is the positive (resp. negative) part of f. Thus, setting u "- u + - u - C Wq2,B,we see that
Au = f. Hence A is surjective. Hence we infer from the second part of Theorem 6(2) that
Ao > 0. Thus (ii) implies (i).
U]
It is obvious from this proof that Theorem 8 remains valid if only the weaker assumption (16) is satisfied, provided assertion (iv) is omitted.

17. B o u n d e d D o m a i n s
In this section we prove the theorems presented in Sections 6 and 7. We begin with a simple bootstrapping result.
Proposition 43. Let ft be bounded and suppose that only condition (16)/s satisfied. Suppo e that C
(53). If q < ; <
( f , 9) C
• O W l t e, C

Proof Put r := p A n q / ( n - 2q) ifq < n/2, and r := p otherwise. Then u C Lr by
Sobolev's embedding theorem. Fix co := cot C a ( - A ( ~ ) ) , where A(~) is the L~ realization of (.A,/3). Then (53) is equivalent to
(co+.A)u=f~in~,

Bu = g on F,

where

f~ := (co- A)u + f C L~ and 9 c OW~2, thanks to the boundedness of f~ (and the compactness of F). Hence we deduce from Theorem 39(ii) that u C W2. I f r < p we repeat this argument to arrive after finitely many steps at the assertion.
V1

Maximum Principles and Principal Eigenvalues

39

Proof of Theorem 10 Since 9t is bounded, embedding (14) is compact. Hence A has a compact resolvent and the assertions concerning c~(A) follow from the general theory of linear operators with compact resolvent (e.g., [48]).
Suppose that )~ C (2,

V E Lc~- "=

N

Lp,

l_ No] it follows that Re p > 0 if A c a(A).
Put
(1 - X ) T u - 0 }.

E "-- { ~ e C 1 ( ~ ) ;

Then E is an ordered Banach space whose positive cone has nonempty interior. Indeed, every strongly positive u belongs to i n t ( E +). Also set
" - (w + A ) - I [E.

T'-T~o

Then the compactness of the embedding W q,• '--+ E implies that T is a compact endo2 morphism of E. Thus we infer from Theorem 4 l, Proposition 44, and E r Lq that T is strongly positive, that is,
T ( E + \ {0}) C i n t ( E + ) .
Consequently, the Krein-Rutman theorem (cf. [1, Theorem 3.2]) implies that the spectral radius r " - r~, of T is positive and a simple eigenvalue with a positive eigenvector. Moreover, it is the only eigenvalue of T with a positive eigenvector. Clearly, u C E satisfies r u - T u iff u E Wq,B and u satisfies (63) with # := r. But this is equivalent to the fact
2
that u is an eigenfunction of A to the eigenvalue
Ao "-- - w +

1/r.

Thus a ( A ) ~ 0 and Ao is an eigenva|ue of A with a positive eigenfunction Uo. From (63) we also deduce that Uo is a positive strict supersolution of (w + .A, B). Hence Uo is strongly positive by Proposition 44.
Suppose that there are u o , . . . , Um C Wq2,t3 satisfying
Uk+l,

0 _< k < m,

r u k -- T u k -- r T U k + l ,

0 < k < m.

AUk -- A.Uk

--

where Um+l " - O. Then

Thus, if m > 1, it follows from

Turn

--

rum

that

r u m - 1 -- T u r n - 1 - r2Um 9

(64)

Since r is a simple eigenvalue of T there exists a E ~ such that Um - aUo. The KreinRutman theorem guarantees also that there exists an eigenvector g) of the dual T ' E s to the eigenvalue r satisfying (~, v) > 0 for v C E + \ {0}. By applying the functional g) to (64) it follows that r 2 a ( ~ , Uo) - ( r ~ - T ' ~ , ~tm-1) -- 0.

Hence a -- 0. Thus we find by backwards induction that Uk -- 0 for 1 _< k _< m, which shows that
NA (Ao) -- ker(A. - A).

Maximum Principles and Principal Eigenvalues

41

Hence the equivalence of the eigenvalue problem for A with (63) and the simplicity of r imply that A~ is a simple eigenvalue of A.
Assume that A C p ( - A ) M Ii~ with A r w. Note that (A + A)u equivalent to
( 1
)
1

w- A

T u-

co- A Tf"

f for f C E is
(65)

Suppose that co > A. Then (65) has for each f > 0 precisely then a positive solution if
1/(co - A) > r (cf. [1, Theorem 3.2(iv)]), that is, if A > co - 1/r - -Ao. Hence, if
(A + A) -1 > 0 and co > A then it follows that A > -Ao. Clearly, given any A E K we can fix co > A such that the above arguments apply. (Note that A~ is independent ofw although
T - T~, and r - r~ depend on this choice.) This shows that -Ao - s+(-A). Hence
Theorem 29 implies - A ~ - s ( - A ) - -Ao.
Lastly, suppose that A C a ( - A ) \ {A0} satisfies Re A - Ao. Then it follows from [39,
Theorem 2.4] (also see [16, Corollary C-III.2.12] or [25, Theorem 8.14]) that
Ao + i k I m A C a(A) for k C Z. But this contradicts the fact that or(A) is contained in a symmetric sector around the real axis with an angle of opening less than 7r, as follows from A c 7-l(Wq2,u,Lq). Thus
Ao is the only eigenvalue of A with ReA - A0.
K]

Proof of Theorem 15 Thanks to Proposition 44 it suffices to show that every positive strict
Wq2 supersolution is strictly positive.
Fix w > Ao and put
K "-- (w + A) -1 E ~(Lq).
Then K >_ 0. By repeated application of Proposition 43 we deduce from Proposition 44 that there exists m C N such that K ju is strongly positive whenever j > m and u > 0.
Consequently, given # > r(K),
OO

K(/_t -- K ) - l u -- E l t - J K J l t j=l is for each u C (L +) \ {0} a quasi-interior point of L +.
Set A "-- w - 1 / # and note that

r(K)

-

1/(~ +

Ao) implies A > -Ao. Furthermore,

( A + A ) -1 - # K ( p - K )

-1.

Hence (A + A) -1 is strongly irreducible, thus irreducible. Now Theorem 31 implies that
(A + A ) - l u is for each A > -Ao and each u C (L +) \ {0} a quasi-interior point of L +, hence strictly positive.
Let u be a positive strict Wq2 supersolution for (A, 13). Set

(f, g) . - (Au, B~) > o.

42

H. A mann

Fix co > (-Ao) V 0 and put f,~ " - cou + f. Then f~ > 0 and the above considerations show that v "-- (w + A ) - l f c o

is a strictly positive element of Lq. Since 9 E (OW2) + and
)~o(co + A) - co + )~o(A) > 0, it follows from Theorem 8 that there exists a unique w r (Wd2)+ satisfying
(co + A ) w - 0 in ~,

B w - g on F.

Since u - v + w we see that u is strictly positive.

[--1

Suppose that Ao > 0. Then 0 C p ( A ) so that A is surjective.
Hence it follows from Theorem 6(1) and (2) that (.4,/3) is inverse positive and that this is equivalent to (ii) and (iii). The inverse positivity of (A,/3) and Proposition 44 imply that
(.A, 13) satisfies the strong maximum principle. From this we deduce that

P r o o f o f Theorem 13

(i) ::v (ii) ::v (iii) :::> (iv).
(iv)=v(i)

Suppose that Ao _< 0 and let uo be a positive eigenfunction of (,4, B). Then
AUo - Aouo < O in l2,

Buo - O on F.

2
Hence uo C W e r
{0} and the strong maximum principle imply - u o > 0, which is impossible. Thus )~0 > 0.
(i)::V(v) Every positive eigenfunction to the eigenvalue )~o is a positive strict Wq2 supersolution, hence a positive strict Lq supersolution.
(v)::v(i) Recall that A ~ - A ~, where A is considered as an unbounded operator in Lq.
Hence o-(A ~) - o-(A). Note that (.A ~,/3 9) satisfies condition (7) also and the irreducibility of (a, b) implies the one of (a T, b-r). Thus Ao is also the principal eigenvalue of (A ~,/3 ~)
2
and it has a strongly positive eigenfunction ~o r W ~ _ .
Let u be a positive strict Lq supersolution for (.A,/3). Fix co > -)~o and put
9- (co + A ) - l u .

Then ~ C (Wq2,t3)+ by Theorem 4 l(iii). From A - 2 D A and
(co + A_2) -1 D (co -k- A) -1 we deduce that f "- A~-

A_2(~ + A-2)-lu

- (co + A - 2 ) -~ ( A _ 2 u ) > O,

where the last inequality sign is also a consequence of Theorem 41 (iii) and Theorem 37(iii).
Hence ~ is a positive strict W 2 supersolution for (fl,,/3), and f belongs to L + \ {0}. Thus the strict positivity of ~o implies
0 < (~o, f) - (~o, A~) - (A'~o, ~) - Ao (~o, ~) and (~o, u) > 0. Hence )~o > 0.

[-7

43

Maximum Principles and Principal Eigenvalues
Proof of Remark 14 by A_I.

It suffices to replace in the last paragraph of the preceding proof A_ 2

Proof of Theorem 11 Thanks to Theorem 12 we can assume that N > 1 and (a, b) is reducible. Thus we can also assume that [(a ~ )~b~ has a block triangular structure of the form (24). If the first diagonal block is either one-dimensional or irreducible then we can apply Theorem 12 to the reduced system obtained by setting u M + I , . . . , u N equal to 0.
This guarantees the existence of a real eigenvalue of (,4, B) with a positive eigenfunction.
If M > 2 and the first diagonal block is reducible we can repeat this argument to arrive at the existence of at least one real eigenvalue of (A,/3) with a positive eigenvector. Thus or(A,/3) -r O and Ao is an eigenvalue of (A, B).

Fix co > -Ao. Then (co + A) -1 is a positive compact endomorphism of Lq, and
1 / (co + Ao) is its spectral radius. Hence the Krein-Rutman theorem (e.g., [ 1, Theorem 3.1 ]) guarantees that (co + A) -1 has a positive eigenfunction uo to the eigenvalue 1/(w + Ao).
Thus uo is a positive eigenfunction of A to the eigenvalue Ao.
Finally, if (,4,13) is inverse positive then A is injective. If A is not surjective then 0 C
~(A). Thus ker(A) r {0} since ~(A) = ~p(A). This being impossible, A is surjective and the last assertion follows from Theorem 6(2).
[]

18.

Domain

perturbations

In this section we prove Theorems 20 and 22. For this we need some preparation.
We fix ~, ~7, ~ C [1, ec] satisfying
1 ( (L~ + L ~ ) ( a , ~ N x N )

x LC(r, ~:N•

X c(r,

Given
Og "-- ([ajk] N x N ,

(al,...,an),a,b,~)

E E(,r/,~(a),

l~NxN"'~di)"g a H. Amann

44 we put

a(a)(v, u) := (Ojv, ajkOku) + (v, ajOju + au> + (7v, xb'yu>r for (v, u ) C Wq1, x W1 .
For Banach spaces E and F we write s
F; 1R) for the Banach space of all continuous bilinear maps E x F -+ 11{,endowed with its usual norm.
Given b E Z;(E, F; I~), let Bb be the unique linear operator i n / ; ( F , E') satisfying

(Bbf, e)E = b(e, f),

(e, f) C E x F.

It follows that the map

s

F;

-+

C(F,E'),

b

Bb,

(69)

is a linear isometry.
Lemma 45. The map

~{,r/,~ (~'~) ---} ~(Wq 1,, %1 ; ]~),

a

a(a),

is well-defined, linear, and continuous.
Vq

Proof This follows from Sobolev embeddings and the trace theorem.

Suppose now that f~ is bounded. Let ~ be a bounded C 2 domain in IRn with boundary and trace operator 3'- Also suppose that X; is a boundary identification map for ~. We set m X'-~

and

Y'-~2

and denote by ( x l , . . . , Xrz) and (yl,..., yn) the standard Euclidean coordinates of X and Y, respectively. Then X and Y are compact oriented n-dimensional Riemannian
C 2 manifolds with boundary and the standard Euclidean metric

(" ]") x "- dx j | dx j

and

(.].)r'-dy

j | dy j,

respectively.
Also suppose that ~ 9 X --+ Y is an orientation preserving C 2 diffeomorphism satisfying ~ ; V - X. Then

~* C s

~ (~, IRN ),

%J,(1-x)~')'

j C { - 1 , 0 , 1},

p C (1, oo),

with inverse
Indeed, this is easily verified i f j C {0, 1}, and follows by duality i f j - - 1 .

(70)

M a x i m u m Principles a n d Principal Eigenvalues

45

Put
( I )M

(71)

(dy j | dy j) - 9jk d x y | dx k

where
(0cP I 0 ~ )
9j

'-

?- xJ

l O.

Then
(i) (A,/3) satisfies the weak maximum principle in Wq1"

(96)

H. Amann

54
(ii) The boundary value problem

~ 4 u - f in f~,

Bu-

g on F

-1 has for each ( f , 9) C Wq,t~ • OWqla unique Wqlsolutionzt, and u~_O if ( f , g ) ~_0.

Proof (i) follows from Theorems 28 and 50.
(ii) Assumption (96) guarantees that 0 C p ( A - 1 ) . Thus the proof of case j = 1 of
Theorem 39(ii) applies to give the unique solvability. The last part of the assertion is a consequence of (i).
[-1
Remarks 52.

(a) Note that Theorem 28(b) guarantees that A 0 ( - A _ I ) belongs to a ( A _ l ) if the latter set is not empty. However, we do not know whether this is true, in general, even if the domain is bounded (in which case A_ 1 has a compact resolvent) and even if N - 1.
(b) In the weak setting studied above it is natural to consider operators of the form
A u . - --O~(ajkOku + ~ju) + ajOj + au,
~N • U where dj 9 f~ -"9" ]"'~diag satisfy appropriate regularity assumptions9 It is not difficult to determine these optimal conditions and to show that Theorem 50 and its corollary hold in this case also. Note that the corresponding boundary operator is now formally - - given by

B~u "- x(Ouu + u j . ~dju + bu) + (1 - 5)u.
We leave the details to the interested reader.

D

The scalar case (N = 1 has been studied by many authors (see [23], [24], [38], [49],
[61 ], [68], [73]-[75], and the references therein). However, in all those papers only the case q = 2 is considered. In that situation one can, of course, weaken the regularity conditions on F considerably, and it suffices to assume that the ajk a r e only bounded and measurable
(in fact, Trudinger [73]-[75] considers even the case of nonuniformly elliptic equations).
It is well-known that this is no longer true if q ~ 2. We do not know of any work dealing with weak maximum principles in a W1 setting, except for [7, Theorem 8.7], where the resolvent positivity of A_x is proved if N = 1 and the lower order coefficients satisfy stronger regularity assumptions.
Now we can easily derive a comparison theorem for semilinear elliptic boundary value problems. For this we recall that, given a a-finite measure space (X, m) and Banach spaces
E and F , a function f : X • E --+ F is said to be a Carath6odory function if

f (x, .) : E -+ F is continuous for m-a.a, x E X, and f(.,~) : X - - + F

Maximum Principles and Principal Eigenvalues

55

is m-measurable for each ~ E E. We denote by Car(X • E, F) the set of all such functions.
We assume that
9

f C Car(~ • ~ N RN), 91 C Car(V x RN, ~u).

9

r, s C (1, cx~)with r f(x,~)inf~,

(98)

Bu - g(x, u) on V

B~ > y(x, ~) on F

if ~ c W1 and in the weak sense, that is, a(v, ~) >_ _ (1 - X)go

for V e (W~,(I_x),7) -~-, on F.

(99)

J

If both inequalities in (99) are reversed then ~ is said to be a W~ subsolution of (98).
Theorem $3. Let (95) and (97) be satisfied and suppose that
A0(A-1) > 0.
I f ~ is a W 1 subsolution and ~ is a Wq1 supersolution of (98) such that

(F(~), a(~)) _< (r(~), a(~)) then ~ 1 Property A1 implies Ap. Assume that X has
Property A1. Let a n 9 X -+ l l ( X ) be a sequence of functions satisfying the conditions
(1)-(2) from the definition of A1. Then

yCX

and a z (y) _> 0 for all z, y C X. We define
~

bn(y) - a'~(y)-~.
Then ]]bnllp - 1. The condition (l) is satisfied automatically. We check the condition (2).
In view of the obvious inequality

tp / m + ( 1 - t )

p/m 1, t C [0, 1], we have that la - bl p < la p - bPl for all a, b _> O. Hence

I[b~z - b~,ll~ - E

Ib~(x) - b~(x)lP < E

xCX

This implies the condition (2).

xCX

Ib~(x)P - b~,(x)Pl -Ila~ - a~ll~.

On Some Approximation Problems in Topology

71

Then we show that Ap, p >_ 1, implies Am for m >_ p. Assume that X has Property Ap.
Let a n 9 X --+ I v ( X ) be a sequence of functions satisfying the conditions (1)-(2) from the definition of A1. Then
E (an (y))p -- 1 yEX n and a z (y) >_ 0 for all z, y C X. We define bz (y) - az (y) ~ . n ~
Then ]lb~iim - 1. The condition (1) is satisfied automatically. We check the condition (2).
In view of the inequality t p/m + (1 - t) p/m _> 1 for p _< m, t C [0, 1], we have that
]a - b]p/m >_ ]ap/m - bp/m ] for all a, b _> 0. Hence
Ilbz - b ~ , l l ~ ~

Ibz~(X) - b ~ ( x ) l m -

~

xCX

~_. I(a~(x)) p/m - (a~(x))P/ml

m

xCX

_ 1 for all points in X. Let c be a constant from the definition of discrete geodesic metric space applied to X. Then a coarse dilatation of a map f : X --4 Y is bounded from above by

dy(f(x),f(x')).

sup dx(x,x') 2.
Since dim X _< n - 3, we still have the inequality dim(X • Y) < n.
By Proposition 47 the Approximation Conjecture holds for Y. Hence we can approximate
9 " Y --+ ~'~ arbitrary closely by 9' " Y --+ Rn with (n - 3)-simple 9' (Y) and with
DIM(g'(Y)) - DIM(Y).

By Theorem 30 and the Negligibility Criterion 9 ' ( Y ) is X-negligible. Therefore we can take an arbitrary close approximation f ' of f with f'(x) This ends the proof.

n 9'(Y) - o.

IN

R e m a r k 49. The Casson finger move allows to prove the Simple Approximation Conjecture for n - 4.

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25. Gromov, M., Spaces and questions, GAFA Special Vol. I (2000), 118-161.
26. Gromov, M., Random walk in random groups, GAFA 13 (2003), 73-146.
27. Higson, N., Bivariant K-theory and the Novikov conjecture, GAFA 10 (2000), 563-581.
28. Higson, N., and Roe, J., On the coarse Baum-Connes conjecture, in Novikov
Conjectures, Index Theorems and Rigidity, Vol. 1, 2, pages 227-254, London Math.
Soc. Lecture Note Ser. 226, Cambridge Univ. Press, Cambridge, 1995.
29. Higson, N., and Roe, J., Amenable action and the Novikov conjecture, J. Reine Angew.
Math. 519 (2000), 143-153.
30. Lubotzky, A., Discrete Groups, Expanding Graphs and Invariant Measures,
Birkhauser, Basel-Boston-Berlin, 1994.

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A. N. Dranishnikov

31. Matou~ek, J., On embedding expanders imo /p-spaces, Israel J. Math. 102 (1997),
189-197.
32. Mitchener, P., Coarse homology theories, AGT 1 (2001), 271-297.
33. Roe, J., Coarse Cohomology and Index Theory for Complete Riemannian Manifolds,
Memoirs Amer. Math. Soc. 497, 1993.
34. Roe, J., Index Theory, Coarse Geometry, and Topology of Manifolds, CBMS Regional
Conference Series in Mathematics 90, 1996.
35. Shchepin, E.V., Arithmetic of dimension theory, Russian Math. Surveys 53 (1998),
975-1069.
36. Skandalis, G., Tu, J.L., and Yu, G., Coarse Baum-Connes conjecture and groupoids,
Topology 41 (2002), 807-834.
37. Spiez, S., On pairs ofcompacta with dim(X x Y) < dimX + d i m Y , Fund. Math. 135
(1990), 213-222.
38. Spiez, S., and Toruflczyk, H., Moving compacta in It~ apart, Top. Appl. 41 (1991), m 193-204.
39. Stanko, M.A., Approximation of compacta in E n in codimension greater than two
(Russian), Mat. Sb. 90 (1973), 625-636.
40. Tu, J.L., Remarks on Yu's property A for discrete metric spaces and groups, Bull. Soc.
Math. de France 129 (2001 ), 115-139.
41. Walsh, J.J., Dimension, cohomological dimension, and cell-like mappings, in Lecture
Notes in Math. 870, pages 105-118, Springer, 1981.
42. Yu, G., The coarse Baum-Connes conjecture for groups which admit a uniform embedding into Hilbert space, Inv. Math. 139 (2000), 201-240.
43. Yu, G., The Novikov conjecture for groups with finite asymptotic dimension, Annals of Math. 147 (1998), 325-355.

Ten Mathematical Essayson Approximationin Analysisand Topology
J. Ferrera, J. L6pez-G6mez, F. R. Ruiz del Portal, Editors
(~) 2005 Elsevier B.V.All rights reserved

95

Eigenvalues and Perturbed Domains
J. K. Hale
School of Mathematics, Georgia Institute of Technology,
Atlanta, Georgia 30332, USA

Abstract
For elliptic partial differential equations on a bounded domain, we present a survey o f some results on the dependence of eigenvalues and eigenfunctions on smooth and nonsmooth perturbations o f the domain.

Key words." elliptic operators, eigenvalues, eigenfunctions, domain variation

1. Introduction

In the study of the global dynamics of certain types of partial differential equations, the stability properties of equilibria play a very important role. These properties often are closely related to the eigenvalues and eigenfuntions of a linear partial differential equation given by the linear variation from an equilibrium. If the partial differential equation is defined on a bounded domain, then one must investigate the dependence of the eigenvalues and eigenfunctions on the boundary conditions and perturbations of the domain. The purpose of these notes is to survey some of the results dealing with this latter problem for second order elliptic operators.
The first situation deals with regular perturbations; that is, the boundary of the original domain and the perturbed domain are Ck-close for some k _> 1. By a change of coordinates onto the original domain, the regularity properties of eigenvalues and eigenfunctions are reduced to the study of the dependence of these quantities on variations in coefficients in the equation and in the boundary conditions. A differential calculus with respect to the domain is needed to discuss nonlinear problems; for example, bifurcation theory, generic hyperbolicity and transversality of stable and unstable manifolds with respect to the domain, maximization of functions over a domain with fixed volume, etc. Problems of this type are discussed in Section 1.

J. K. Hale

96

If the domain is irregular (that is, not regular), then the definition of eigenvalues and eigenfunctions for some boundary conditions is a nontrivial task. For example, if the domain is irregular and the boundary conditions are Dirichlet, then one must give first a precise definition of Dirichlet boundary conditions as well as what is meant by eigenvalues. This is discussed in Section 2 for the first eigenvalue and eigenfunction and it is stated that a maximum principle holds if the first eigenvalue is positive. This definition depends upon the domain and then it becomes important to give a topology on the domains in order to know the maximum principle remains true under small perturbations in this topology. It also is necessary to do the same thing for the complete spectrum of the operator.
In Section 3, for the Laplacian with Neumann boundary conditions, we present some results on the dependence of eigenvalues and eigenfunctions on exterior irregular perturbations of the domain, including perturbations near points on the boundary of the domain, dumbbell shaped domains, thin domains and more general perturbations.

2. Regular domain perturbations
Let f~o C I~'* be a bounded domain and let/Ck(flo) be the collection of all regions which are Ck-diffeomorphic to rio. We introduce a topology by defining a sub-basis of the neighborhoods of a given fl as
{h is a small C ~ (f~, Ii~ ) - neighborhood of the inclusion in : f~ C R n }. n When [[h-inl[ck is small, h is a Ck-imbedding off~ into ~n; that is, a Ck-diffeomorphism to its range h(f~). Micheletti [42] has shown that this topology is metrizable and that
K~k(f~o) may be considered a separable complete metric space. We say that f~ E K~k(f~0) is a Ck-regular perturbations (or, sometimes, simply a regular perturbation) of a given domain f~o if h is a small perturbation in C k (9t0, II~ ) of the inclusion in o. n Courant and Hilbert [15] studied the effect of regular perturbations of the domain on the eigenvalues and eigenfunctions of boundary value problems for PDE. For example, if f~r C Cr(f~o) is a continuous family of domains converging to f~o as e -+ 0 and Ak(gt~), k _> 1, denotes the ordered set of eigenvalues of the Laplacian - A Be with some boundary conditons BC and {~k,~ } is a set of normalized eigenvectors, they proved that the eigen~o values Ak (f~) and eigenfunctions {~k,~ } converge to those of--ABc. The proof consisted of constructing the family of diffeomorphisms h~ which map 9t~ onto f~0 and reduce the problem to the study of a family of operators L~ on ~0.
This result is very interesting, but it is desirable to have more information about the eigenvalues and eigenfunctions. For example, if the eigenvalues and eigenfunctions are smooth functions of e, what is the Taylor series in ~? Other important problems arise which are concerned with the determination of those domains which are critical values of some function of the domain such as maximization of torsional rigidity with fixed area of the domain, minimization of the principle eigenvalue of the Laplacian with Dirichlet boundary conditions over domains with fixed volume, etc. To discuss such questions we need to have a differential calculus of boundary perturbations. Many people have been concerned about

Eigenvalues and Perturbed Domains

97

questions of this type. We are going to present the approach of Henry [29], [30], [31] and refer the reader to [31 ] for extensive references.
If F 9 (f~o) --+ Y, Y a Banach space, then we can define the smoothness of F at f~o in the following way. For any f~ C/~k (f~o) which is close to Fro, there is an h C Ck(flo, ]~n) which is close to the inclusion i~ o such that
-- h(~o).
Therefore,
-

-

(r

o

We say that F is C" (resp. C ~ ) (resp. analytic) if the map h ~ F o h is C k (resp. C ~ )
(resp. analytic). In this sense, problems of perturbation of the boundary (or, of the domain of definition) of a boundary value problem is reduced to differential calculus in Banach spaces. Consider a non-linear formal differential operator

Fa (u)(x) - f (x, L u ( x ) ) ,

x

ft,

where L is a constant coefficient linear differential operator depending upon

u,

Uxj, l < j < n,

Uxjxk, l 0in f~, (L + p)(p _< 0in 9t}.

(11)

The following very interesting results have been proved by Berestycki, Nirenberg and
Varadhan [ 12].
Theorem 1. RMP holds for L if and o n l y / f A ( L , Ft) > 0.
Theorem 2. If A (L, Ft) > O, then there is a positive constant

A - A(Ft, co, Co, b, A(L, Ft)) such that, for any f C Ln(f~), there is a unique solution v of
Lv-f

in Ft,

v - O(u ~M)

in 0~,

(12)

and
(13)
Both of these results depend upon knowing that A(L, f~) > 0. If we know that this condition is satisfied for Ft, how do we characterize the class of perturbations of f~ for

Eigenvalues and Perturbed Domains

105

which it will still be true? Arrieta [5] has introduced a complete metric space of equivalence classes of bounded open sets in which ,k(L, f~), as well as the solution of(12), is continuous in f~. We now describe this result.
Let
O-

{f~ C B1 C If{ 9 f~ is open}, n where B1 is the unit ball with center zero. For f~ C |

if

r a m -- {x C 0f~" 3{x (j) } C f~,x (j) -+ X, uaM(x (j)) -~ 0 a s j --+ Oc}, then the set f~,M -- f~ \ r a m

is open. We say that f~l, f~2 C O are equivalent relative to the operator M and Dirichlet boundary conditions, ~-~1 "~M ~'-~2, if f~,M _ f~M. With this equivalence relation, following Arrieta [5], we define
(~M -- O / ~ M

and the metric a~= 9 o M • ~)M
(al,~2)

>R t---} dLMo~(~l,a2) -- ]luf~IM --

2Lf~2MI[L~176

Arrieta [5] shows that (Ore, d M ) is a complete metric space and also proves the following result. Theorem 3. If/k (L, ~) is defined as in (11), then/k (L, f~) and the corresponding eigenfunction are continuous in the metric d~o~. If )~(L, Q) > 0, then so is the unique solution of (12).
This result shows that, if the conditions of Theorems 1 and 2 hold for a given domain fl0, then they hold for an open neighborhood of f~o in the space ( o M dM ).
3.2. Operators in divergence form
Several important questions arise with respect to the above metric imposed on the domains.
(1) Is it possible to show that the equivalence relation ~ M is independent of M for M in some class?
(2) In the definition of the metric, is it possible to replace L ~ ( B 1 ) by LP(B1) or
H1 (B1)?
(3) Is there a class of operators for which the metric for M and M* are equivalent if they belong to this class?
(4) In these metrics for M in some class of operators, is it possible to obtain continuity of all of the spectrum?

J. K. Hale

106

Arrieta [6] shows that the answers to these questions are mostly affirmative in the class o f operators which can be described in divergence form. To describe the results, we need some notation. For a fixed constant u > 0, let

D

- { L - Ei,j Ox~(aiJOxJ) + Zi biOx' + c, a ij C C 0'1 (]t~n ), bi c C L~(It~ n), 1 < i j < n

n a~J~j >_~,l~[2 }

i,j=l

Do - { L e / ) ' c - O } ,
79o0 - { L E 79o " bi - O , 1 _< i _< n}

Proposition 4. The equivalence relation "~M is independent of the operator M r 7)0," that is, ~ * M __ ~ * M *

for every M, M* C 79o.
From Proposition 4, we can define f~, _ f~,M

and

O_~)M

for any M r 790. From now on, when an open set 9t is considered, we can suppose that
9t - f~* since the properties o f an operator L r Do are the same on [~ and [~*. As we did for the metric dLM~, we can define the metrics dMp, 1 _< p < ec, and dM1 on 6).
The metric d M is strictly weaker that the metric dM~. On the other hand, as noted by
Arrieta (1997), even though the space ((2), d M ) is complete, the space ((2), dMrp) is not complete for any 1 _< p < ~ . As compensation, we have the following

Proposition 5. For any M C Doo, the metrics dr 1 O, is a sequence in 0 and define L k to be the operator L with
Dirichlet boundary conditions on f~ k. I f

Eigenvalues and Perturbed Domains

107

lim d ~ (f/k, 12o) -- 0,

k--+oc

then the following statements are true:

(i) For any C1-Jordan curve F in the complex plane such that F D cr(Lo) = ;g, there exists a k0 = k0(F) such that F A a ( L k ) = ;g for k >_ ko. Moreover, if Pr,Lk is the spectral projection over the part o f the spectrum inside F, then lim k-+oo

liNt ' L~

- Pr,LoIICCL2(B1) H]CB1)) -- O.

(ii) I f R(A, Lk ) is the resolvent o f Lk, then lim I]R(A Lk) -- R(A LO)IIC(L2(B1),H~(B1)) -- 0 and the convergence is uniform in any compact F C p(Lo ).

Since the Laplace operator A is the simplest second order elliptic differential operator, it is natural to define a canonical metric d2 by d2(f/1, f/2)

-

II ?-tf~lA -

Uf~2/X]]L2(B1)-

With this notation, Arrieta [6] obtains the following interesting result.
Theorem 7. Suppose that ftk, k > _O, is a sequence o f domains in ~) and let u k - u n~A.
_
For any L E 7), let L k be the operator L k with Dirichlet boundary conditions acting on f~k. For the following statements."
(i) d2(f/a, f~o) --+ 0 as k -+ oc,
(ii) The spectrum o f L k aprroaches the spectrum o f Lo and the spectral projections o f
Lk approach the spectral projections o f Lo in Z;(L 2 (B1), H I (B1)) as k --+ oc, we have (i) implies (ii). Moreover, if L is self-adjoint, then both statements are equivalent.

Micheletti [42]-[45] has given results about the convergence of the spectrum of operators in the case of regular perturbations of the domain in the Courant metric. The Courant metric is stronger than the d2 metric and therefore we can have convergence of the spectrum for more general domains.
Most of the results in the literature related to the behavior of the spectrum of an operator when the domain is perturbed put the emphasis on geometric conditions on the perturbations of the domain to guarantee the continuity of the spectrum (see the previous and the next section). For Dirichlet boundary condtions, the conditions in Theorem 7 are different from the conditions being imposed on the convergence properties of solutions of the simplest nontrivial elliptic equation Au = 1 in the perturbed domains.
It is clear that it would be interesting to characterize, in some more analytic way, large classes of domains for which the condition (i) in Theorem 7 is satisfied. It would also be interesting to see if some similar theory is valid for other types of boundary conditions.

108

J. K. Hale

4. Neumann conditions and irregular perturbations
If the perturbed domain depends upon a parameter c in a metric space containing zero, then a family of domains f~, is said to be an irregular perturbation of the domain f~0 if the measure of Ft, \ f~o approaches zero as c --+ O. The set of irregular perturbations contains but is more general that the set of regular perturbation of f~0 as defined in Section
1. For example, the domain f~, could be a perturbation of f~o which introduces an irregular bump at a point on the boundary of f~o. Another example could be a dumbbell shaped domain for which the connecting bar degenerates to a curve as c -+ O. A domain f~, C It{
'~
which degenerates to a domain f~o c IRm with m < n (thin domain) also is an irregular perturbation. In this section, we study the properties of eigenvalues and eigenfunctions of elliptic operators with Neumann boundary conditions as a function of external irregular perturbations of a bounded domain.
Problems of this type have independent interest and also play an important role in the dynamics of nonlinear equations. For example, if the nonlinear system is gradient, then the compact global attractor (that is, the maximal compact invariant set which attracts bounded sets uniformly) consists of the union of the unstable sets of the equilibrium. Knowing convergence properties of the eigenfunctions and eigenfunctions with respect to the domain leads, without too much difficulty, to results on the upper semicontinuity of attractors at the limit domain for parabolic equations. For hyperbolic equations, the upper semicontinuity is more difficult to prove. In some cases (for example, the variational case), it is easier to show upper semicontinuity directly. If each equilibrium is hyperbolic, then one can deduce continuity properties of the unstable manifolds and, as a consequence, deduce that the compact global attractors are Hausdorff continuous at the limit domain. We do not discuss this problem and refer the reader to Hale and Raugel [25], [26], Raugel [55], Arrieta [7].
In this section, we concentrate on Neumann boundary conditions for these types of perturbations. However, we begin with a few remarks about other types of boundary conditions.
If we assume Dirichlet boundary conditions, then it is possible to prove very general results. In fact, Babu~ka and Vyborny [9] proved that the eigenvalues and eigenfunctions converge for a general 2m-order elliptic operator with Dirichlet boundary conditions when the domains 9t, satisfy the following conditions:
(i) For all compact sets/x" C f~o, there exists c(/x') C (O, co)such that A" C f~, for

c (0,
(ii) For each open set U with 9to C U, there exists c(U) C (0, co) such that f~, C U for

(0,
Other references dealing with these problems for Dirichlet boundary conditions are
Courant and Hilbert [ 15], Dancer [ 17], [ 18], [ 19], Daners [21 ], L6pez-G6mez [41 ].
We will not discuss Robin boundary conditions and only mention that some references

Eigenvalues and Perturbed Domains

109

for this case are Dancer and Daners [20], Daners [21 ], Ozawa [47], Ozawa and Roppongi
[48], Stummel [59], Ward, Henshaw and Keller [62], Ward and Keller [63], [64]. Results related to convergence of eigenvalues and eigenvectors are more closely related to the
Dirichlet problem than to the Neumann problem.
It was shown by an example in Courant and Hilbert [15] that the eigenvalues of the
Laplacian with Neumann boundary conditions may not be continuous if the perturbation of the domain is irregular. In the last few years, Neumann problems have received considerable attention by Arrieta [2], [3], [4], [5], [6], Arrieta, Hale and Han [8], Beale [10], Brown,
Hislop and Martinez [13], Chavel and Feldman [14], Ciuperca [16], Hale and Raugel [25],
[26], Hale and Vegas [27], Hempel, Seco and Simon [28], Hislop and Martinez [32], Jimbo
[33], [34], [35], [36], Jimbo and Morita [37], Lobo-Hidalgo and Sanchez-Palencia [40],
Rauch and Taylor [54], Raugel [55], Vegas [61], as well as others contained in the references of the above papers.
4.1. Perturbations near boundary points
It is instructive to begin with an example of Courant and Hilbert [ 15]. Let
- {(x, y ) -

Ix1 <

yl <

of area cr2/4 with center (0, 0). For any e > 0, ~- > 0, let
- {(x, v )

0 < x <

Ivl <

and define

~~e,7

--

~-~1 [--J

(/~e,w -~- (1/2, 0)) t_J (gt~ + (1/2 + C, 0)).

For T -- c4, the domain ~,~4 can be viewed as a C~ perturbation. Let

of ~0, but not a C 1-

A~ 9 D ( A ~ ) C L 2 ( ~ , ~ 4 ) - + L2(t~,~4),

For all e > 0, 0 is an eigenvalue of A~. I f ) ~ are the ordered eigenvalues of A~, then )~ > 0 for e > 0. It is shown in Courant and Hilbert [ 15] that )~ --+ 0 as e --+ 0. The eigenvalues exhibit a singular behavior at e - 0 in the sense that the second eigenvalue for e > 0 is not close to the second eigenvalue for e - 0.
For some cases, a C~ does not yield singular behavior of the eigenvalues of A~. In the example of Courant and Hilbert [ 15], if ~- - eZ and/3 is too small, this will be the case. A more trivial example can be obtained by eliminating the retangular square of size e from the perturbation.
Arrieta, Hale and Han [8] have given a complete description of the behavior of the eigenvalues and eigenfunctions of the Laplacian with Neumann boundary conditions for a general class of perturbations including the example above of Courant and Hilbert [ 15]. As we will see, the singular behavior of the eigenvalues relies on the way in which the original

110

J. K. Hale

domain is perturbed as well as the relative sizes of the domains used as perturbation, but the shape of the perturbation is of no importance.
We now give a precise definition of the domain considered in Arrieta, Hale and Han [8].
Let fro, D1 be bounded, connected smooth domains such that
(H. 1) There exist positive constants c~,/3 such that
{(X, y) E ]l~ X R n-1 " Ixl < O~,

- {(x, y).

lyl

< 9} n

-~

~o

< x < o, lyl < 9 } ,

{(X, y) E ~ X ]l~n-1 " 0 < X < 2OL, lYl < 9 } n D ,

={(x, y)

o~ < x < 2o~, lYl < 9},

((0, a) x (-,3, ,3)) n (f'/o U D1) - Z,
{0} x (-,3,/3)C OQo,

{o~} x (-/3,/3)C OD,,

(H.2) (~o n / ) 1 -- Z .
(H.3) For any connected set
-t~l C { ( x , y ) E ~x X ]~x - 1 " 0 ~ X ~ OL, lYl < /~}, n the set f~o U D1 U R1 is a bounded connected smooth domain in If{n . Also, if F 1 =
OQo n OR1, then R1 n 1-'~ 5r Z .
The set (R1 \ FI) n D1 is a bounded connected domain with smooth boundary except probably at some points of FI. Let r / > 0 be a constant which will be fixed later. For e > 0 small, let
R ~ , , - {(ex, e'Y)" (x, Y ) e R1},
D e - {(ex, ey)" (x, y)E 01 }.

(14)

There is an eo > 0 such that, for each e. E (0, co), we have

fionb~-0 and _~,, u O~ c {(x, y). o _< x < ~, lyl _< 9}.
The set
~ - ~0 U R~,n U D~ is a bounded open connected smooth domain.
R e m a r k 8. The fact that 0f~o is a piece of a hyperplane near (0, 0) is merely technical.
It is shown in Arrieta, Hale and Han [8] how to attach the perturbation near a point for arbitrary smooth domains f~o and so all of the results below will be valid.

Eigenvalues and Perturbed Domains

111

For each fixed c1 r (0, co), the domain 9t~, for c close to el, is a Cl-perturbation of 9t~ 1 .
Although this is not true at e - 0, we do have
#(~r

as e--+0,

where # is Lebesgue measure. Let us also introduce the set Sz by the relation
S,y -- { ( x , y )

E ]1~ X I~ n - 1

9 X 2 Jr-]y]2 ~ 9`2} A ~='~0.

There is a ~/o such that, for 0 < "7 < 9`o, we have
S,y c { ( x , y ) c R • IRn-1 9 -c~ < x < 0, ]y[ < / 3 } .
For 0 _< e < co, we denote by
___

<

a set of orthonormal eigenvectors corresponding to the ordered set of eigenvalues
{A~, 1 _< m < ce} of the Laplacian on fl~ with Neumann boundary conditions.
The following result regarding the second eigenvalue and eigenfunction is due to Arrieta,
Hale and Han [8].

Theorem 9. Let

~

-

Qo U R~,, 7 U D~

with R~,,, D~ defined by (14). For TI > (n + 1 ) / ( n - 1), the following conditions hold." lim A~ - 0,

e---+0

lira Ilw~ ]Hl(~o) -- 0,

e--+0

~-+olim
[[w~}lH2(no,,) -- 0 lira - 1,

e---+O

~-+olimff(D~) ( /D ~;)~ -- 1.
1

Furthermore, if Qo is a C~-domain, then, for any integer ~ >__i and any "7 C (0, 9`o), lim c-+O

II

ll

(ao\S

)

-

0.

Therefore, for any 9' C (0, 9`o), the function w~ together with all derivatives up to order ( converge to zero pointwise in Qo and uniformly in Qo \ $7 as c -+ O.
The limit properties of the remainder of the eigenvalues and eigenfunctons is given in the following result.

J. K. Hale

112
Theorem 10. Let

~e f~o tO Re,, tO De with Re,o, De defined by (14). For rl > (n + 1) / (n - 1), the following conditions hold."
-

lira Am - A ~

e--+0

for m >__3.

The corresponding eigenvectors can be chosen so that, for any sequence o f positive numbers {ek, 1 < k < c~} with ek --+ 0 as k --+ c~, there is a subsequence {6k, 1 3, we have lim k-+cx~

I1~

-

0
Wm-lllnl(f~o)

-

O,

-

lim I[~llHl(R~kuO~k ) -- O.

k--+cx~

Furthermore, if f~o is a CC~-domain, then, for any integer g >_ 1 and any 3' C (0, 3'o),
6~
0 lim Ilwm
Ogm_ 111He(Fto\S.r) 0
-

-

-

k--+cx~

Therefore, for any 3" E (0, 3"o), the function co~, m >_ 3, together with all derivatives up to order g converge to win_ pointwise in f2o and uniformly in -~o \ S,y as e --+ O.
It is worth making a few remarks about these results. If we ignore the set Re,~ and consider the eigenvalue problem on f~o tO De, then there is no singular behavior in the eigenvalues. This is due to the fact that the only eigenvalue on the domain De that remains bounded as e -+ 0 is the eigenvalue zero. Theorems 9 and 10 assert that the double eigenvalue zero on the disconnected domain Fro U De becomes two simple eigenvalues, zero and A~ with A~ --+ 0 as ~ --+ 0 and the other eigenvalues converge to the eigenvalues of f~o as e --+ 0 provided that they remain bounded. Of course, this is under the restriction that r/ > (n + 1 ) / ( n - 1). If r/is too small, then the eigenvalue problem on f~e may not correspond so well to the one on the disconnected domain Fro tO De.
R e m a r k 11. The above result could have been stated in terms of spectral projections and then it would not be necessary to make a choice for the eigenfunctions.
R e m a r k 12. Mixed boundary conditions as well as perturbations at a finite number of points also are discussed in Arrieta, Hale and Han [8].
4.2. Dumbbell shaped domains
Let us now turn to the disucussion of dumbbell shaped domains. Jimbo [33], [34], [35] seems to have been the first to discuss this problem in some generality for some special smooth domains in IR2 . For example, suppose that
-

u

u

is a smooth, connected domain in 11~ for which f~o f2o R, are disjoint, f~o aoR are
2
L, R,
L,
smooth connected domains joined by a rectangular channel
R, = L x

(0, e),

L=[0,1].

113

Eigenvalues and Perturbed Domains

Jimbo pointed out that the relevant limit problem should consist of the following three eigenvalue problems:
Au -- #u in a g U f~L, uxz = # u

in L,

O u / O n -- 0 in 0fro U fro
R
L,

u=0

(15)
(16)

in OL.

We order the eigenvalues of the problems (15), (16) as
, 1o _ ~o _ o > . o _ . o _>...,

and let ~po, ~po,... be a corresponding set of normalized eigenfunctions. He proved the convergence of the eigenvalues and eigenfunctions on ft~ to those of (15), (16) as e --+ 0.
Arrieta, Hale and Han [8] considered a more general type of dumbbell shaped domain for which the connecting channel R~ could have a boundary which may not even be connected.
Allowing this complicated type of channel is the main difference between this situation and the one considered by Jimbo [35] and Hale and Vegas [27].
We now give a precise definition of the perturbed domain. Let f~o f~o be bounded
L, R connected smooth domains such that
(H.4) There exist positive constants c~, fl, 7 such that
{ ( x , y) e R • R ~-1 9 - ~

< x < ~, lyl < 9 } n ao~

= {(x, y).

-~

< x < o, lyl < 9 }

{(X, y) e I[{ X ]~n-1 . 0 < X < "~ nt- OZ,
= {(x, y).

{yl

n ~g
~, I~1 <

< 9}

~ < x < z +

91

( n . 5 ) rio n (~g - z .
L

(H.6) For any connected set
R1 C { ( x , y ) e ]1~ • l~n-1 " 0 ~ X ~ 7, lYl < ~},

the set f~L U f~o U R1 is a bounded connected smooth domain in I~n .
R
For e > 0 small, if we let
Ro - { ( x , ~ v )

(x, y) e R~ }

(17)

and define a~ -

no u R~ u a0~,
~

then f~ is a bounded open connected smooth domain.
Remark 13. As noted in Remark 8, the fact that 0f~o and 0f~o are pieces of a hyperplane
L
R near (0, 0) is merely technical.
Let

{#i-o > ~ > ~ >...}

114

J. K. Hale

be the ordered set of eigenvalues of the Laplacian with Neumann boundary conditions on f~ and let
~-22~ " ' "

be a corresponding set of normalized eigenfunctions. Arrieta, Hale and Han [8] showed that #~--+0

as

c~O

and that #~ is negative and bounded away from zero, which generalized a result of Hale and Vegas [27]. The methods used there as well as refinements of Arrieta [2], [3] yield the following theorem.
Theorem 14. I f
- no u

u rift,

where Re is defined by (17), then the following conclusions hold for any m." e---+0 and the corresponding eigenfunetions can be chosen so that lira []r

e---+0

- r

-- 0.

R e m a r k 15. We remark there can be many different channels and many different open sets connected by these channels. The results will be the same except there are more eigenvalue problems for the limit as e --+ 0.
4.3. Thin domains
Hale and Raugel [25] have considered some properties of the dynamics of reaction diffusion equations on thin domains and, as a byproduct of the investigation, also have given results on the convergence of eigenvalues and eigenfunctions of the Laplacian with mixed boundary conditions. We describe a special case of their results for a particular case of a thin domain over a line segment for Neumann boundary conditions. For a more complete and more general discussion, see Raugel [55].
Let
R~ = { ( x , y ) CI~ 2 : 0 < x <

1, 0 < y < G ( x , ~ ) } ,

(18)

where the function G c C 1 ([0, 1] x [0, co]) and satisfies
OG
G(x, 0) - 0, Go(x) - -0T(x, 0) > 0, x e [0, 1].

(19)

Let {Am, m > 1 } be the ordered set of eigenvalues of - A with homogeneous Neumann boundary conditions and let {~m, m > 1} be a corresponding set of normalized eigenfunctions. Hale and Raugel [25] show that the appropriate limit problem as ~ -+ 0 is the eigenvalue problem

Eigenvalues and Perturbed Domains
1

-~(Go(x)u~)~

115

- )~u in (0, 1),

a0(x)

u,-0

(20)

at x - 0 , 1 .

(21)

If {A~ m > 1} is the ordered set of eigenvalues of (20) and { ~ o m > 1} is a corresponding set of normalized eigenfunctions, they prove that the following statement is true.
Theorem 16. For any integer m, there arepositive constants eo(m), C ( m ) such that, for every integer n < m, 0 < c 1} the corresponding

in (0, 1) at x -

(22)

0,1.

The following result is due to Arrieta [2], [3].
Theorem 18. Let (A~, yzc (pe ~ )
n), n,

be the eigenpairs defined above. If

r/, - sup{elg'~(x)l 9 x e [0, 11},

then, for any integer m, there are positive constants eo (m), C (m) such that, for n _ 1} the ordered set of eigenvalues and {G~, n _> 1} the corresponding normalized eigenfunctions of this eigenvalue problem, then we have the following result.
Theorem 19. The following statements are true: limp~-um, m_>l,

e--+O

lim[~m - De(r e ---+O

~m)~m] -- 0 strongly in L2((O, 1)

weakly in HI((O, 1),

'

where

1

D~(f,g) -

f0

9~f9.

An example for which the above result applies is the function

9~(x) = 1 + p s i n ( x e - ~ ) , where p C (0, 1), c~ > 0. In this case, a - 1 and b ~ (1 - p2)-1/2 > 1 and the eigenvalues of the two dimensional problem are close to the eigenvalues of the operator ( - 1 / b ) O 2 with Dirichlet boundary conditions.

118

J. K. Hale

4.4. General variations
For Neumann boundary conditions, Lobo-Hidalgo and Sanchez-Palencia [40] proved that, if f~o C 9t~ and mn(Q~\~0)-+0 as e - + 0 , where mn is the Lebesgue measure in ~n, then every point of the spectrum o f - A % ~ is approximated by points of the spectrum o f - A ~N, whereas the contrary statement is false r in general; that is, there may be situations for which there are accumulation points of the spectrum of-A~vr which are not in the spectrum o f - A : ~ ~.
Arrieta [4] has considered perturbed domains in this general setting and has given conditions for which one has convergence of eigenvalues and eigenfunctions. The conditions are stated in such a way as to lead to proofs of the results in this section as well as many more. To be specific, let Q0 C 9~ and mn(Qe\Qo)-+0 as

e-~O,

let
R~ = ~ \ ~o,
For functions V~ E L ~ ( ~ ) with

F~ = OVto n OR~.

IIEIIL~(~) < c for some C > 0 independent of e, consider the Schr6dinger operators
- --AN

R~

+ E

R~

A D ( F ~ ) N -- --AD(F~)N + Ve

where the superscript denotes the domain on which the operator is applied, the subscript
N denotes homogeneous Neumann boundary conditions on the domain and the subscript
D(F~)N denotes homogeneous Dirichlet conditions on F~ and Neumann conditions on the remainder of the domain. We let H r, (R~) denote the space of H ~ functions which respect i the Dirichlet boundary conditions on F~.
Without loss of generality, we may suppose that

E_>0.
The objective is to show that eigenvalues and eigenfunctions of A ~ behave as the eigenvalues and eigenfunctions of A~ ~ and AnDir~)n . To achieve this, the following hypothesis is assumed:
(H) If u~ C H1 (~')e) with for some positive constant C1 independent of e, then there exists ~ C H ~ (R~) such that lira Ilu~ - ~IIL2(R~) - - 0 e--+O Eigenvalues and Perturbed Domains

119

Let
{A,~,ar

>_ 1},

{Am(f~o,e),m _> 1},

{Vm(Re),m >_ 1},

be the ordered set of eigenvalues counting multiplicity of the operators

and let

{ ~ m , ~ , m >_ 1},

{r

{~m(Re),rrt >_ 1},

_> 1},

be a corresponding set of orthonormal eigenvectors. Let
{ A ~ , m _> 1} = { A m ( a 0 , e ) , m _> 1} U {rm(Re),m >_ 1} be ordered (counting multiplicity), and define
~; = r in do, = 0 in Re if Am = a i ( a o , e ) ,
~ = 0 in f~o, ~j(Re) in Re, if A~ = Tj(Re).
Obviously, we have

~'n C H I (~-~0) l..JH ~ (-~e).
We say that cr~ > 0 divides the spectrum if there are positive constants ~, M, N such that, for e C (0, e0), we have
[ere-5, de + 5] n { A ~ , m >_ 1} = •,

ae 1, there exists an integer L k such that
Card{A~"

, ~ < k} < Lk,

120

J. K. Hale

is satisfied, then condition (H) is equivalent to the statements (i) and (ii) of Theorem 20.
Arrieta [4] shows that these results include all of the examples of the previous subsections.

References

1. Abraham, R., and Robbin, J., Transversal Mappings and Flows, Benjamin, 1967.
2. Arrieta, J.M., Spectral Properties of Schr6dinger Operators under Perturbation of the
Domain, Ph.D. Dissertation, Georgia Tech., 1991.
3. Arrieta, J.M., Neumann eigenvalue problems on exterior prturbations of the domain, J.
Diff. Eqns. 118 (1995), 54-103.
4. Arrieta, J.M., Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case,
Trans. Amer. Math. Soc. 347 (1995), 3503-3531.
5. Arrieta, J.M., Elliptic equations, principal eigenvalues and dependence on the domain,
Comm. PDE 21 (1996), 971-991.
6. Arrieta, J.M., Domain dependence of elliptic operators in divergence form, Resenhas
IME-USP 3 (1997), 107-123.
7. Arrieta, J.M., Spectral behavior and upper semicontinuity of attractors, in Differential
Equations Vol.1 (Fiedler, B., Gr6ger, K., Sprekels, J., Eds.), pages 11-196, World
Scientific, 2000.
8. Arrieta, J.M., Hale, J.K., and Han, W., Eigenvalue problems for nonsmoothly perturbed domains, J. Diff. Eqns. 91 ( 1991 ), 24-52.
9. Babu~ka, I., and Vyborny, R., Continuous dependence of eigenvalues on the domains,
Czech. Math. J. 15 (1965), 169-178.
10. Beale, J.T., Scattering frequencies of resonators, Pure Appl. Math. 26 (1973), 549-563.
11. Bensoussan, A., Lions, J.L., and Papanicolaou, G., Asymptotic Methods in Periodic
Structures, North-Holland, 1978.
12. Berestycki, H., Nirenberg, L., and Varadhan, S.R.S., The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm.
Pure. Appl. Math. 47 (1994), 47-92.
13. Brown, R., Hislop, ED., and Martinez, A., Eigenvalues and resonances for domains with tubes: Neumann boundary conditions, J. Diff. Eqns. 115 (1995), 458-476.
14. Chavel, I., and Feldman, E.A., Spectra of domains in compact manifolds, J. Funct.
Anal. 30 (1978), 196-222.
15. Courant, R., and Hilbert, D., Methods of Mathematical Physics, Vol. 1, WileyInterscience, New York, 1953. Translation from the German edition, 1937.
16. Ciuperca, I.S., Spectral properties of Schr6dinger operators on domains with varying order of thinness, J. Dyn. Diff. Eqns. 10 (1998), 73-108.
17. Dancer, E.N., The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Diff. Eqns. 74 (1988), 120-156.
18. Dancer, E.N., The effect of domain shape on the number of positive solutions of certain nonlinear equations, II, J. Diff. Eqns. 87 (1990), 316-339.

Eigenvalues and Perturbed Domains

121

19. Dancer, E.N., Domain variation for certain sets of solutions with applications, Top.
Meth. Nonlinear Anal Preprint.
20. Dancer, E.N., and Daners, D., Domain perturbation of elliptic equations subject to
Robin boundary conditions, J. Diff. Eqns. 74 (1997), 86-132.
21. Daners, D., Domain perturbation for linear and nonlinear parabolic equations, J. D/ft.
Eqns. 129 (1996), 358-402.
22. de Giorgi, E., and Spagnolo, S., Sulla convergenzia degli integrali dell'energia per operatori ellitici del 2 deg ordine, Boll. U. Mat. Ital. 8 (1973), 391-411.
23. Garabedian, E, and Schiffer, M., Convexity of domain functionals, J. Analyse Math. 2
(1952), 281-368.
24. Hadamard, J., M6moire sur le problbme d'analyse relatif/t l'6quilibre des plaques
61astiques encastr6es, in 6~uvres de J. Hadamard Vol. 2, 1968.
25. Hale, J.K., and Raugel, G., Convergence in gradient-like systems and applications,
ZAMP 43 (1992), 63-124.
26. Hale, J.K., and Raugel, G., A reaction-diffusion equation on a thin L-shaped domain,
Proe. Roy. Soe. Edinburgh 125A (1995), 283-327.
27. Hale, J.K., and Vegas, J.M., A nonlinear parabolic equation with varying domain,
Arch. Rat. Mech. Anal. 2 (1984), 99-123.
28. Hempel, R., Seco, L.A., and Simon, B., The essential spectrum of Neumann
Laplacians on some bounded singular domains, J. Funct. Anal. 102 (1991), 448-483.
29. Henry, D., Perturbation of the Boundary for Boundary Value Problems for Partial
Differential Equations, Sere. Brasileiro Anal. ATS 22, 1985.
30. Henry, D., Generic properties of equilibrium solutions by perturbation of the boundary, in Dynamics of Infinite Dimensional Systems (Chow, S.N., Hale, J.K., Eds.), pages
129-139, NATO ASI Series F 37, Springer-Verlag, 1987.
31. Henry, D., Perturbation of the Boundary in Partial Differential Equations, Lecture
Notes (1996). To appear in Cambridge University Press.
32. Hislop, ED., and Martinez, A., Scattering resonances of a Helmholtz resonator,
Indiana U. Math. J. 40 (1991), 767-788.
33. Jimbo, S., Singular perturbation of domains and the semilinear elliptic equation, J. Fae.
Sci. Univ. Tokyo 35 (1988), 27-76.
34. Jimbo, S., Singular perturbation of domains and the semilinear equation, II, J. Diff.
Eqns. 75 (1998), 264-289.
35. Jimbo, S., The singularly perturbed domain and the characterization for the eigenfunctions with Neumann boundary conditions, J. Diff. Eqns. 77 (1989), 322-350.
36. Jimbo, S., Perturbation formula of eigenvalues in a singularly perturbed domain, J.
Math. Soc. Japan 45 (1993), 339-356.
37. Jimbo, S., and Morita, Y., Remarks on the behavior of certain eigenvalues on a singularly perturbed domain with several thin channels, Comm. PDE 17 (1992), 523552.
38. Joseph, D., Parameter and domain dependence, Arch. Rat. Mech. Anal. 24 (1967), 325351.
39. Kesavan, S., Homogenization of elliptic eigenvalue problems, I, Appl. Math. Optim. 5
(1979), 153-167.

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J. K. Hale

40. Lobo-Hidalgo, M., and Sanchez-Palencia, E., Sur certaines propri6t6s spectrales des perturbations du domaine dans les probl6mes aux limites, Comm. PDE 4 (1979), 10851098.
41. L6pez-G6mez, J., The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Diff. Eqns. 127 (1996), 263-294.
42. Micheletti, A.M., Perturbazione dello spettro dell'operatore di Laplace in relazione ad una variazione del campo, Ann. Seuola Norm. Sup. Pisa XXVI Fasc. I (1972) 151-169.
43. Micheletti, A.M., Metrica per famiglie de domini limitati e propriet/l generiche degli autovalori, Ann. Scuola Norm. Sup. Pisa XXVI Fasc. III (1973), 633-694.
44. Micheletti, A.M., Perturbazione dello spettro di un operatoire ellittico tipo variazionale, in relazione ad una variazione del campo, Annali Mat. Pura Appl. XCVII,
Fasc. IV (1973), 261-281.
45. Micheletti, A.M., Perturbazione dello spettro di un operatore ellittico di tipo variazionale, in relazione ad una variazione del campo, II, Recerche Mat. 25 (1976),
187-200.
46. Ozawa, S., Singular variations of domains and eigenvalues of the Laplacian, Duke
Math. J. 48 (1981), 769-778.
47. Ozawa, S., Singular variation of domain and spectra of the Laplacian with small Robin conditional boundary, I, Osaka J. Math. 29 (1992), 837-850.
48. Ozawa, S., and Roppongi, S., Singular variation of domains and continuity property of eigenfunction for some semi-linear elliptic equations, II, Kodai Math. J. 18 (1995),
315-327.
49. Pereira, A.L., Auto valores do Laplaciano im regi6 es simO tricas, Ph.D. Thesis, Univ.
S~.o Paulo, Brasil, 1989.
50. Pereira, A.L., Appendix to Henry, D., [31 ], 1996.
51. Pereira, A.L., Eigenvalues of the Laplacian on symmetric regions, NoDEA 2 (1995),
63-109.
52. Peetre, J., On Hadamard's variational formula, J. Diff. Eqns. 36 (1980), 335-346.
53. Prizzi, M., and Rybakowski, K.P., The effect of domain squezzing upon the dynamics of reaction-diffusion equations, J. Diff. Eqns. 173 (2001 ), 271-320.
54. Rauch, J., and Taylor, M., Potential and scattering theory on wildly perturbed domains,
J. Funct. Anal. 18 (1975),27-59.
55. Raugel, G., Dynamics of partial differential equations on thin domains, in Dynamical
Systems (Johnson, R., Ed.), pages 208-315, Lectures Notes in Mathematics 1609,
Springer, 1995.
56. Rayleigh, J.W.S., Theory of Sound, Dover, 1945.
57. Saut, J.C., and Temam, R., Generic properties of nonlinear boundary value problems,
Comm. PDE 4 (1979), 293-319.
58. Serrin, J., A symmetry problem of potential theory, Arch. Rat. Mech. Anal. 43 (1971),
304-318.
59. Stummel, F., Perturbation of domains in elliptic boundary value problems, in
Applications of Methods of Functional Analysis to Problems in Mechanics (Germain,
P., and Nayroles, B., Eds.), pages 110-135, Lecture Notes Mathematics 503, 1976.
60. Uhlenbeck, K., Eigenfunctions of Laplace operators, Amer. J. Math. 98, (1972), 10731076.

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123

61. Vegas, J.M, A functional-analytic framework for the study of elliptic operators on variable domains, Proc. Roy. Soc. Edinburgh I16A (1990), 367-380.
62. Ward, M.J., Henshaw, W.D., and Keller, J.B., Summing logarithmic expansions for singularly perturbed eigenvalue problems, SIAMJ. Appl. Math. 53 (1993), 799-828.
63. Ward, M.J., and Keller, J.B., Nonlinear eigenvalue problems under strong localized perturbations with applications to chemical reactors, Stud. Appl. Math. 85 (1991), 128.
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SIAMJ. Appl. Math. 53 (1993), 770-798.

Ten Mathematical Essays on Approximation in Analysis and Topology
J. Ferrera, J. L6pez-G6mez, F. R. Ruiz del Portal, Editors
(~) 2005 Elsevier B.V. All rights reserved

125

Monotone Approximations and Rapid
Convergence
V. L a k s h m i k a n t h a m

Florida Institute of Technology, Department of Mathematical Sciences,
Melbourne, FL 32901 USA

Abstract

Starting from the classical method of successive approximations, in a general set up, this paper describes in detail monotone iterative technique and the method of generalized quasilinearization. The paper demonstrates how monotone approximation techniques cover a broad range of nonlinearproblems in a variety of situations. The paper also traces the research excursions of the author for the benefit of the readers.
Key words: monotone approximations, rapid convergence, nonlinear differential equations I. I n t r o d u c t i o n

One of the objectives in approximation theory is the investigation of the following problem. If E is a Banach space, a linear subspace S is prescribed, whose elements s C S are utilized for approximating u C E. The interesting problem, known as best approximation, is to find, for a fixed u E E, an element s C S, for which Ilu - sII is as small as possible.
To be precise, if the infimum is attained by one or more elements of S, in the relation d ( u , s) -

inf Iiu - sIJ

sCS

then these elements of S, are called best approximations of u in S. The classical approximation theory of functions of one variable has dealt with such a problem. Multivariate approximation theory concerns with the approximation of functions of several variables and therefore is more complicated, which is clear by experience in solving partial differential equations. In this contribution, we shall investigate monotone flows which approximate

126

V. Lakshmikantham

the solutions of various nonlinear dynamic equations starting from the classical successive approximations of ordinary differential equations.

2. Successive approximations
Consider the Cauchy problem

(1)

x' -- f (t, x), x(to ) -- Xo, to e R+,

where f C C[Ro, Rn], and

Ro - [(t, x) "to < t < to + a and Ilx - xotl ~ b].
We shall begin with the following existence and uniqueness result, that is perhaps not well known, under more general assumption than Lipschitz. It shows, at the same time, the convergence of successive approximations to the solution of (1) as well as the generation of monotone flows of the comparison function when it is also monotone. Moreover, this result demonstrates the power of the comparison principle and also has an interesting history which we shall indicate later.

Theorem 1. Assume that
(i) g C C[[to, to + a] x [0, 2b], R+],
O < g(t,u) 1, we note that c~ satisfies for each v E H~ (9t), v > 0, a.e. in

/~ [~ aij(X)(O~n)xiVxj dx_
+C(X)(~nV]
where
We now use the ellipticity condition and the fact that c(x) >_ N > 0 with v - a n to get
/[0'

(o~n)x [2 + N i o~n ,2]dx 0 a.e. in f~ by taking the limit as n --+ ec, we see that

B[p, v] - f ]f (x, p) + g(x, r)]vdx and f
B[r, v] - ./o [f(x, r) + g(x, p)]vdx.

Finally, we claim that p and r are the weak coupled minimal and maximal solutions of (15) that is, i f u is any weak solution of 15 such that C~o(X) < u(x) 0 a.e. in f~,

= L [ f ( x , c~k_l) - f~(x, c~k-1)c~k-1 + g(x, c~k-1) -- g~(x,/~k-1)c~k-1]vdx, where ~o(x) - ~(x) - f u ( x , ~ k - ~ )

-

9~(x, 9k-l).

We now use the ellipticity condition and (B3) with v - c~k to get

f [O ] OZk,x ]2 +N , o~k [ 2 ] d / < f [f(x,O~k-i)-- f~(x,O~k-1)OLk-i

+ g(x, ~k-~) -

9u(x, 3k-1)~k-~]~kdx.

We then get, since by (B3) the integrand on the right-hand side belongs to L2(f~), in view of the fact that ak-1, ilk-1 E H~(f~), and the estimate sup I] c~k I]H~(a)< ~ k

A similar argument implies that sup I/~k []H~(f~) < OO. k Hence there exist subsequences {c~k~}, {r } which converge weakly in H I (a) to p,r E Hl(f~), respectively. To verify that p,r are weak solutions of (15), we fix

V. Lakshmikantham

146

v e H~ (f~), v _> 0 a.e. and find that (Yk+l,/3k+1 satisfy (31) and (32) with F and G defined by (27) and (28). Taking limits as k --+ ec, we obtain

B[p, v] - ~ [f (x, p) + g(x, p)]vdx, and f
B[r, v] - ] [f (x, r) + 9(x, r)]vdx, showing that p, r are weak solutions of (15).
To prove that p - r - u is the unique solution of (15), it is enough to prove that r _< p a.e. in 9t, since we know that p < r a.e. in f~. Taking a - r , / 3 - p, and applying Theorem 7, we find that r _< p a.e. in f~ proving the claim.
(c) Quadratic convergence of {ak }, {/3k }. To prove the quadratic convergence of sequences {ak }, {/3k } to the unique solution u respectively, we set

Pk-1 -- U -- C~k+l,

qk+l -- ~k+l -- U,

SO that Pk+l(X) -- 0 on ~ and qk+l (x) -- 0 on Ft. We then have for v E H~(Ct), v > 0 a.e. in fl, using the fact that fu is nondecreasing in u and 9,, is nonincreasing in u,
B[Pk+l,

V] --

.~o[f (x, u ) + 9(x, u) - f(x, ak)

fu(x, O~k)(O~k+l

-- g ( x , OLk) -- g u ( X , l~k)(OLk+l -- O~k)]vdx
Ib

_ 1 and q >_ 1, then
9I(A,0)-0

and

Du91(A, 0 ) - 0

for each

A CR,

where Du stands for Fr6chet u-differentiation, and, hence,
~(A, 0) - 0

and

Du~(A, 0) - s

In order to emphasize that it is already known, the state (A, u) - (A, 0) will be called the trivial solution of Equation (1).

154

J. L6pez-G6mez

Under Assumption L, s is a Fredhom operator of index zero for each A C ~ i.e.,
R[12(A)] is a closed subspace of U and dim N[s

= codimR[s

< oc,

since it is a compact perturbation of the identity. Subsequently, for any T C s
N[T]
and R[T] will stand for the null space (or kernel) and the range (or image) of T, respectively. Thanks to the open mapping theorem, 12(A) is an isomorphism if dim N[12(A)] = 0 .
Let 2 denote the spectrum o f the family 12(A) --its elements are called eigenvalues-2:={AEIR

: dim N[12(A)] _> 1},

(4)

and if5 the set of non-trivial solutions of Equation (1),
(~ "-- [~'-1(0) n (I~ X (U \ {0}))] U { ()~,0) " ,,~ C ~ } .

(5)

The following concept plays a crucial role in nonlinear analysis.

Definition 1. Given Ao C I~ (A0,0) is said to be a bifurcation point of Equation (1) from the curve of trivial solutions (A, 0) if there exists a sequence
()~n, ~tn) E ~'-1(0) n (I[~ x (U \ {0})) ,

/t R 1,

such that lim (An, un) - (Ao, 0). n---+oo The following result collects some well known general properties of 2 and @ (cf. [32,
Section 6.1 ]).

Proposition 2. For each It E I~ \ ~, there exists an open neighborhood of (#, O) in 1Rx U, say B, such that B n G -- 25. Thus, ~ is a closed subset of IR and (Ao, O) cannot be a bifurcation point of Equation ( 1)from (A, u) - (Ao, 0)/fAo ~ ~. Moreover, G is a closed subset o f IR x U.
As a result, if (Ao, 0) is a bifurcation point from (A, 0), then
N := dim N[s

_> 1.

(6)

However, very simple algebraic examples show that (6) might not be sufficient for bifurcation. Indeed, for any Ao C IN the system
(A - AO)Ul + u~ - 0,
(A - ~ o ) u 2

- u i3 -

O,

fits into our abstract setting by choosing
U = ~2,

u = (Ul, u2),

(7)

Spectral Theory and Nonlinear Analysis

155

and s f i t ( A , u ) - ( - u 3 )3 U l
.

Although s
= 0, and, hence, (6) holds true with N - 2, multiplying the first equation of (7) by uz, the second one by Ul, and subtracting the resulting identities, gives
+

-

o

/Zl --

2t 2 - -

0.

and, therefore,
Consequently, (Ao, 0) is not a bifurcation point from (A, 0). However this example shows that (6) is not sufficient for bifurcation, if we change the nonlinearity, then (Ao, 0) might be a bifurcation point for the new equation from (A, 0). Indeed, if instead of (7), we consider the following system
( A - - A O ) ~ 1 - - ~t 3 - -

0,

(~

0,

-

~o)u2

-

u~

-

(8)

then, for each A > Ao,

provides us with a nontrivial solution of (8), and therefore (Ao, 0) is a bifurcation point from (A, 0). These examples actually show that Ao is not a nonlinear eigenvalue of the family ~(~) = ( ~ - ~o)~R~.
Definition 3. Ao is said to be a nonlinear eigenvalue of s if (Ao, 0) is a bifurcation point of(l) from (A, 0) for any fit(A, u) satisfying Assumption N.

In other words, Ao is a nonlinear eigenvalue of s if the fact that bifurcation occurs is exclusively based on the linear part of (1). It should be noted that even in the case when Ao is a nonlinear eigenvalue of s the local structure of the set of solutions of (1) around
(Ao, 0) will depend on the nature of the nonlinearity fit(A, u). Actually, the problem of the algebraic classification of all possible solution varieties of (1) around (Ao, 0), according to the nature of the nonlinearity fit(A, u), is one of the central problems in singularity theory and real algebraic geometry (e.g., [21 ]), but that analysis is outside the scope of this review.

3. A brief introduction to the topological degree

The topological degree is a generalized counter of the number of zeros that a compact perturbation of the identity has within an open and bounded set. The faster way to introduce it is by means of the uniqueness theorem of H. Amann and S. A. Weiss [ 1].

156

J. L6pez-G6mez

Theorem 4. Let 0 be the set of bounded open subsets of U, and, for each f~ E O, denote by I f (~) the set of compact perturbations of the identity on ~ with the topology of the uniform convergence. Consider the set of pairs

"P'- {(f,Q) EK(~) x 0 9 f-l(o) FlOf'/--O}.
Then, there exists a unique map
Deg 9 P --+ Z satisfying the following properties."
Dl D e g ( I , f l ) - l / f O E f ~ .
D2 I f ~ C (Q, ~1 and ~2 are two disjoint open subsets off~ and f E K(f~) with f - l ( 0 ) n [(~ \ ( a l U f~2)]- O, then, Deg (f, 9t) - Deg (f,

~1

) -3 Deg (f, 9t2).
L

D3 I f f~ E O, h" [0, 1] --+/~'(fl) is continuous, and
(h(t), f~) E ~

for each

t E [0, 1],

then
Deg (h(t), f~) - Deg (h(0), ~)

Jbr each t E [0, 1].

The integer Deg ( f , f~ ) is called the topological degree o f f in f~.
Property D I is usually refereed to as the normalization property; it establishes that the identity map has one zero in any bounded open set containing the origin. Property D2 entails three basic properties that any counter of zeros must satisfy. Indeed, by choosing
~--

~1 -- ~ 2 -- 0 ,

gives
Deg (f, 0) - 0.

(9)

Secondly, by choosing ft = f'tl U ft2, shows that
Beg (f, ftl U f~2) = Beg (f, 9tl) + Beg (f, f~2),

(10)

so establishing the additivity of the counter. Thirdly, by choosing f~2 = 0, provides us with the identity
Beg (f, 9t) = Beg (f, f~1),
(11)
so establishing the excision property of the counter. As an easy consequence, from these properties it is apparent that f - 1 (0) N (~ = ~

',,

Beg (f, 9t) = 0.

(12)

157

Spectral Theory and Nonlinear Analysis
In other words, f must have a zero in f~ if
Deg (f, f~) -~ 0.

Property D3 establishes the homotopy invariance of the counter, i.e., the fact that the number of zeros remains unchanged if the map is continuously deformed in such a way that it does not loose nor win zeros through the boundary. In other words, it establishes the continuity of the counter in the quotient space defined by the relation being homotopic; as it is an integer, it must be constant.
In the special case when U = C, 7 is a Jordan curve, f~.y is the component enclosed by
7 and f is an holomorphic function in O.y such that
/ - 1 ( 0 ) fq Tray')' = 0 , then, Deg (f, f~), equals the total number of zeros of f in f~/counted according to their orders. Therefore, the topological degree is indeed a generalized counter of the number of zeros that f possesses within f~ (cf. [33, Chapter 11 ] for an elementary introduction at undergraduate level).
The first construction of the topological degree in R N was carried over by L. E. J.
Brouwer [5] to prove his celebrated fixed point theorem. Brouwer's degree was later extended by J. Leray and J. Schauder [29] to cover the infinite dimensional setting dealt with here. Naturally, it yields to the simplest proof of Schauder's fixed point theorem [51 ] (cf.
A. N. Tychonoff [55]).
Now, we will describe the standard procedure to calculate the topological degree in applications. Subsequently, we will consider the subset of 79 consisting of all pairs (f, [2) with f C C 1 (~), such that f - 1 (0) M f~ is finite, possibly empty, and

Dr(u)

is an isomorphism for each

u E f - 1 (0) N f~.

Any of those pairs will be called regular. The set of regular pairs will be denoted by 7~.
The next result can be obtained from the infinite dimensional version of S. Smale [54] of
Sard's theorem [49].
Theorem 5. For any ( f , fl ) E 79 there exists a continuous map h : [0, 1]--+ K(gt)

such that h(0) = f ,

(h(1), f~) e "/r

(h(t), ft) e 7)

V t e [0, 1].

In other words, any compact perturbation of the identity is homotopic, within T to a
),
regular function.
Consequently, thanks to the homotopy invariance of the degree,
Deg (f, f~) = Deg (h(1), f~).

J. L6pez-G6mez

158

Moreover, it is independent of the smoothing homotopy. Thus, calculating the topological degree is reduced to calculate the degree of a regular pair (f, Ft) C 7~. In calculating the degree of a regular pair the following scheme should be followed. Suppose
(f, f~) E T~ and

f-l(0)

n Q -- {~1, ..., ~ N } .

(13)

Note that, thanks to (12),
Deg(f, fl)-0

if

f-l(0) nQ_e.

Due to Property D2, (13) implies
N

(f,B,(uj))

Deg ( f , O ) - Z D e g

(14)

j=l

for any sufficiently small e > 0 satisfying

B,(uj) C f~,

l < j < n,

and

[~,(ui) n [~,(uj) -- 0

if

i :/: j .

Actually, (14) is still valid for any (f, Ft) C 7~ having a finite number of zeros in Ft.
Throughout the rest of this paper Bl~(U) stands for the open ball of radius R > 0 centered at u. Note that, for each 1 _< j _< N,

Deg(f,B~(uj))

is independent of

e

as soon as

Be(uj) C f~ and

f-l(0)n

Be(~zj)

-

{~zj}.

More generally, if(f, f~) E 7) and u0 C f~ is an isolated zero o f f , then Deg (f,B,(uo)) is well defined and independent of e > 0 as soon as f - l ( 0 ) n/~'e (u0) -- {Uo} ; this value being called the index of f at uo, and denoted by Ind (f, Uo). Using this concept,
(14) can be rewritten in the form
N

Deg (f, Ft) - Z

Ind (S, uj).

(15)

j=l

As the index is a local concept and, thanks to the inverse function theorem, one can establish a local homotopy between f and its linearization at uj, D f(uj) (by neglecting f(u) - D f ( u j ) ( u - uj)), because D f ( u j ) is an isomorphism for each 1 < j < N, it turns out that (15) gives rise to
N

Deg (f, Ft) - Z

Ind (Df(uj), 0).

(16)

j=l

Finally, to complete the calculation of the degree the following result should be used. It is usually refereed to as the Leray-Schauderformula.

Spectral Theory and Nonlinear Analysis

159

Theorem 6. Suppose It~ is a linear compact operator and T := I - K is a linear isomorphism. Then,

Ind (T, O) = ( - 1)n(Z)

(17)

where n ( T ) is the sum o f t he algebraic multiplicities o f all negative eigenvalues o f T . In particular, if U = I~N, then (17) becomes into

Ind (T, 0) = sign det T in any basis ofI~ N .

Thanks to Theorem 6, (16) implies
N

Deg (f, f~) - Z ( - 1 )

n(Df(w)) .

(18)

j=l

Conversely, defining the degree of a pair (f, f~) C 7~ through (18) and extending it to the class 7) by means of Theorem 5, one is naturally driven to the standard analytic construction of the topological degree. In applications, to calculate the topological degree of a pair
(f, ft) C 7), one should follow the next steps:
9 Constructing the smoothing homotopy.
9 Once deformed f into a regular operator in ft, say F, finding out the zeros of F.
9 Calculating n ( D F ( u o ) ) for each u0 C F - l ( 0 ) V/ft.
More general developments of the Leray-Schauder degree for wider classes of fixed point equations were introduced by F. E. Browder, R. D. Nussbaum and W. V. Petryshyn in the late sixties (see N. G. Lloyd [30] and K. Deimling [9]) and by J. Mawhin [42]. More recently, P. M. Fitzpatrick and J. Pejsachowitz [ 16] developed a degree theory for quasilinear
Fredholm mappings, which has been substantially tided up in a series of very recent papers by R Benevieri and M. Furi, [2], [3], [4]. These theories remain aside the scope of this introductory review.

4. Topological characterization of nonlinear eigenvalues

In this section, the general assumptions and notations introduced in Section 1 are maintained. The following result characterizes the nonlinear eigenvalues of 12(A) by means of the topological degree.
Theorem 7. Suppose Ao E IR is an isolatedpoint o f ~. Then, the following assertions are true." 1 . - I f Ind (12(A), 0) changes as A crosses Ao, then Ao is a nonlinear eigenvalue o f 12(A).
2.- If s

is o f class C I and Ao is a nonlinear eigenvalue o f 12(A), then

Ind (t2(A), 0)) changes as A crosses Ao .

J. L@ez-G6mez

160

Consequently, if s E C 1, then A0 E N is a nonlinear eigenvalue of s the parity of the integer number

if and only if

changes as A crosses Ao. We have denoted by ma[T; #] the algebraic multiplicity of # as an eigenvalue of T, i.e.,
OO

m~[T; # ] - dim U N[(T - #I) j] - dim N [ ( T j=O

#I) ~'(T'u)]

where u(T, #) is the algebraic ascent of p as an eigenvalue of T. Since under our assumptions the complex, non real, eigenvalues appear by pairs, A0 is a nonlinear eigenvalue of s if and only if d(A) - Ua Ca( s

Z ma[~(A); pA] Z ma[J~(A); p)~]
)A[Re (z) ,))n[Izl >1]

changes as A crosses Ao (cf. (2)). Therefore, Ao is a nonlinear eigenvalue of s if and only if the parity of the dimension of the unstable manifold of zero as a fixed point of the compact family ~(A) changes as A crosses Ao. This feature has a huge relevance from the point of view of the dynamics of the discrete dynamical system defined by the fixed point equation (3).
Basically, Part 1 of Theorem 7 goes back to M. A. Krasnosel'skii [27] and J. Ize [22], and
Part 2 is attributable to J. Ize [23] and P. M. Fitzpatrick & J. Pejsachowicz [ 14], though the most pioneering classifications of nonlinear eigenvalues were given in terms of the parity of the generalized algebraic multiplicity of J. Esquinas and J. L6pez-G6mez [31], [11],
[ 12] and [ 10] (cf. [32] for further details), rather than in terms of the fixed point index.
Part 1 is based upon a very classical principle in bifurcation theory going back to H.
Poincar6, establishing that associated with any change of index there is a bifurcation phenomenon. It actually entails that a continuum of solutions emanate from the trivial state, as a result of the hyperbolic structure of the set of zeros of (1). It turns out that the change of index implies the set of zeros of any regular approximation of (1) around (A0,0) to be of hyperbolic type and, therefore, it must contain something else than the trivial state
(cf. the beautiful geometric proof of this nonlinear analysis principle given in [32, Chapter
3], which is based upon some ideas coming from S. N. Chow and J. K. Hale [6] and the references therein). Although they cannot reach the generality of the further topological characterizations found in J. Ize [23] and P. M. Fitzpatrick & J. Pejsachowicz [14], the pioneering proofs of Part 2 given by J. Esquinas and J. L6pez-G6mez enjoy the tremendous advantage, over the subsequent purely topological proofs, of being entirely constructive.
The proofs of [23] and [14] used rather sophisticated topological non-constructive tools
--obstruction theory techniques.

161

Spectral Theory and Nonlinear Analysis

5. Algebraic characterizations of nonlinear eigenvalues.
This section gives the construction scheme of a finite algorithm to calculate the change of Ind (s
0) as A crosses an isolated eigenvalue Ao C ~. This algorithm is based upon the construction of a generalized algebraic multiplicity of s at Ao which extends all classical concepts of generalized algebraic multiplicities available in the specialized literature. Besides its intrinsic interest, it connects two areas that have temporarily evolved through rather separated paths. Namely, spectral theory and bifurcation theory.
Subsequently, we denote by 'I~o(U) the space of Fredholm operators of index zero in U, not necessarily of the form (2). By definition, T E ~o (U) if R [ T ] is closed and dim N [ T ] - codim R [ T ] < ec .
The general assumption of this section is that

c

(u)),

c r

r E NU{oc},

r>_l,

and that is an isolated eigenvalue of the family s

AoE2

Given a family 9)t(A) of class C r, we will denote
_ 1 dJgY~(Ao)
9Jlj " - j! dAJ
'

O 0 and an integer u _> 1 such that for each A satisfying 0 < IA - Aol < ~ the operator s is an isomorphism and
C

II~ - 1 ( ~ ) II _ 1 for which this estimate holds true is called the order of A0.
Note that Ao = 0 is not an algebraic eigenvalue of the family s defined by (20). The following result allows us extending the concept of multiplicity introduced by Definition 9 to cover the case of general families of operators (cf. [32, Chapters 4, 5]). It characterizes the families that can be transversalized by means of a family of isomorphism.
Theorem 11. For each integer 1 _kn>_l.
Moreover, due to [45, pp. 907], the integer numbers k l , . . . , kn are independent of the canonical set of Jordan chains of s at A0. Thus, the following concept of multiplicity, attributable to P. J. Rabier [45], is consistent: The integers k l , . . . , kn are called thepartial multiplicities of s at Ao. The number n -

Y: j=l will be referred to as Rabier's multiplicity of s at Ao. Moreover, if ~ and ~ are two families of operators of class C r around Ao such that ~!(Ao) and ~'(Ao) arc isomorphisms, then Rabier's partial multiplicities of s at Ao equal those of the new family s := e(A)n(A);~(A),

A ~ Ao,

and, in particular,
RIO(A); Ao] = R[s
Ao].
Moreover, the following result, generalizing I. C. G6hberg et al. [19, Chapter S l] and P.
J. Rabier [46, Section 4] is satisfied (cf. J. L6pez-G6mez & C. Mora-Corral [34], [35]). It establishes the existence of a local Smith form in finite dimension; the precise concept of local Smith form being incorporated in the statement of the theorem.

Spectral Theory and Nonlinear Analysis

167

Theorem 14.

Suppose U - IRN and the length o f all Jordan chains o f s at Ao is uniformly bounded above by some natural number hi E l~t such that 1 _ .." >_ kn >_ I are Rabier's partial multiplicities of 12(A) at Ao.

The following very recent result coming from J. L6pez-G6mez & C. Mora-Corral [34],
[35] characterizes the existence of the local Smith form in terms of the concept of algebraic eigenvalue introduced by Definition 10, helping to reveal the importance that it might deserve in spectral theory.

Theorem 15.
9 s

Suppose U - ]t{N. Then, the following conditions are equivalent."

possesses a local Smith form at Ao.

9 det 12(A) has a zero offinite order at A - Ao.
9 Ao is an algebraic eigenvalue orE(A) o f order k 0 be sufficiently small so that

~+B~(o,o) c a ,

a*ea\

U

"-~,"+~

,

and pick p* - p*(3) satisfying the requirements of Proposition 19. Then, for each )~ satisfying (31), the topological degree Deg (~'(,k, .), f~), \ Bo. ) is well defined, and, by homotopy invariance, it is constant for )~ in between the first components of two consecutive points of/3. Moreover, it equals zero if
A E (-eo, m i n B - ~] U [maxB + ~,
Suppose

n ([~*,(20) X {0}) -- {~I,...,#M }-

.

J.. L6pez-G6mez

172

Then, using the homotopy invariance and the the additivity property of the degree, the following chain of identities is obtained
Deg (~" (,k*,-), f~)~. \ Bp. ) = Deg (~'(,k*, .), f~:~. ) - Deg (~'(A*, .), Bp. )
= Deg (~'(pl - ~, "), f~,l-~) - Ind (s

- ~), 0)

5
5
= Deg (~'(#1 + ~,'), ~Ul+~) - Ind (2(#1 - ~),0)
5
= Deg (a'(>l + ~, "), ~Ul+~ \ Bo*) + Ind (s

5
+ ~ ) , O) - Ind ( s

5

0).

Repeating this argument M times, it is apparent that
Deg (;~()~*,-), a),. \ Bo. ) - Z

Ind (s

j=l

+ ~), O) - Ind (s

M

: +2 Z

)

- ~), O)

(34)

P(/*J)'

j=l

since
5
Deg (~d(PM + -~, "), f~,M+~ \ Bo* ) - O .
Going backwards gives the second identity of (32). The proof of (33) is an easily consequence from (32). Suppose r n f~,.

=

{ ()~*, u~), ..., (,~*, uh) }.

Then, by the implicit function theorem, around any of these points ~ consists of a differentiable curve and, hence, the isolating open neighboring f~ can be constructed in such a way that h Deg (~()~*, .), ~ .

\ Bp. ) - Z

Ind (D~(,k*, uj), 0).

(35)

j=l

Therefore, since
Deg (~'(,~*, .), ftx. \ Bp. ) E 2N and Ind (Du~'()~*, uj), 0) E {-1, 1},

1 < j < h,

it is apparent that h > 2 and, thanks to (34) and (35), we find that
M

2[Z

P(ltj)] - [Deg (~(A*, .), ft,x. \ Bp.)]

j--1 h = IZ

Ind (Du~'(,~*, uj), 0)l

j=l

)~, such that the mappings pax,, hA,, and hA are U - near, for every )(' _> )(.
One defines the limit of an approximate inverse system X as an approximate mapping p- X --+ X, which has the following universal property"
Whenever h" Z --+ X is an approximate mapping, then there exists a unique mapping h" Z --+ X such that p h - h,
i.e., pa h - hA, for all A E A.
Uniqueness of approximate limits (up to natural homeomorphism) follows immediately.
If all Xa are Tychonoff spaces, the approximate limit exists. It suffices to take for X the subspace of the product
Ilxa,
a

which consists of all points z - (za), za C Xa, satisfying the condition lim pax, (za,) - za,

al>a

for every A C A.

For the mappings Pa" X -+ Xa, one takes the restrictions of the canonical projections

IaI x a

--+ Xa.

The next theorem shows that good properties of limits of inverse systems of compact spaces are also properties of limits of approximate systems of compact spaces (see [35] and [41 ]). In particular, we have the following theorem.
Theorem 31. Let C be any of the classes from Theorem 7. If in an approximate inverse system X all terms X a belong to C, then so does the limit X.

Approximate systems of compact polyhedra can be used in constructing complicated compact Hausdorff spaces. E.g., in [36] L. R. Rubin and I proved the following non-metric version of a well-known result of R. D. Edwards and J. J. Walsh.

190

S. Marde~iO

Theorem 32. Let X be a compact Hausdorff space X, whose cohomological dimension dimT~ X 1.

Then there exist a compact Hausdorff space Z, whose covering dimension dim Z < n and weight w(Z) A, there exists a mapping h" X~, --+ P such that the mappings hp~, and f are 12- near.
(R2)' There exists an open covering );' of P such that, for any A E A and mappings h, h'" Xx --+ P such that hpx, h'p~ are ~;'-near mappings, there exists a A' _> A, such that the mappings hpxx,,, h'p~x,, are 12- near, for any A" _> A'.
An important feature of approximate resolutions is that they share all the good properties of resolutions. In particular, the analogue of Theorem 29 holds also for approximate resolutions. This is so because approximate resolutions can be characterized by properties (B 1) and (B2) from Theorem 30 (see [44]). Similarly, the analogue of Theorem 26 holds also for

Approximating Topological Spaces by Polyhedra

191

approximate resolutions [44]. Theorems 33 and 34, which were valid only in the compact case, now have the following analogues.
Theorem 35. ([61]) Every topological space X of dimension dim X f, f~ such that
'
fpPJ'(~)g(l,) ~ f~P:f'(l,)g(~),

for # _< #'.

(4)

In the case of strong shape, the appropriate category is the coherent homotopy category
CH(pro-Top). Its morphisms are equivalence classes of coherent mappings f" X ~ Y.
The latter are given by increasing functions f" M --+ A, by mappings

f . . x (.i -+ Y. and also by n-homotopies fuopl...un, for n >_ 1, where f~0~l...Pn " Xf(pr~) X A n ~

Y~o

Approximating TopologicalSpaces by Polyhedra

193

is a mapping, An is the standard n-simplex and u o < U l < ... < # n

is any increasing sequence of indices in M. The 1-homotopies fuu, realize (3), the 2homotopies

Xf(tt2) • A2 ----kY#o connect the 1- homotopies q~o~l ft, l~2, f~o~2 and f~oul (Pf(ttl)f(tt2) fttottltt2 "

• 1), etc. In an alternative, but equivalent approach, one defines CH(pro-Top) (in this case usually denoted by
Ho(pro-Top)) as the localization of the category pro-Top at homotopy level equivalences,
i.e., at morphisms [f] of pro-Top, which have a representative f: X --+ Y, where f is the identity mapping and every fa : Xa --+ Y), is a homotopy equivalence.

In both shape categories a morphism F : X --+ Y is given by two polyhedral expansions p: X -+ X,

q: Y -+ Y

and by a morphism [f] : X -+ Y. In the case of ordinary shape, [f] is from the category proH(Top) and in the case of strong shape, it is from CH(pro-Top). The conditions imposed on expansions insure that F does not depend on the particular choice of the expansions p and q. More precisely, if p ' : X --+ X ~ and q': Y -+ Y ' is another choice of polyhedral expansions, then there exist unique isomorphisms
[i]:X-+X'

and

[j]:Y-+Y'

[i][p] = [p']

and

[j][q] = [q'].

such that
If F ' : X ' -+ Y' is given by p': X - - + X ' ,

q':Y--+Y'

and by a morphism
[ff] : X' -+ Y', then F' is identified with F whenever
[j][f] = [f'][i].

For a more detailed description of the above mentioned homotopy categories of inverse systems see [33]. Here we will describe precisely the two kinds of expansions.
We say that a mapping p- (p~): X - + X

is a homotopy expansion provided the following two properties of Morita are fulfilled.
(M1) If P is a polyhedron and f : X -+ P is a mapping, then there exists a A E A and there exists a mapping h: X~ -+ P such that hp), '-,~ f .

S. Marde2id

194

(M2) If A C A and h0, hi" X,x --+ P are mappings such that

hop~ ~ hips, then there exist a A' > A such that

hopxx, ~ hlpxx,.
The following diagrams illustrate properties (M1) and (M2).
P~,

X~, i h

.,
- - ~

X

X~x

f

P~

.,

X xx' X,X,

P

p

Resolutions and homotopy expansions are related by the following theorem (see [40]).
Theorem 39. Every resolution p" X --+ X is a homotopy expansion.
An immediate consequence of Theorems 28 and 39 is the following theorem, which plays an essential role in the construction of the category Sh(Top), because it insures the existence of desired polyhedral approximations of spaces.
Theorem 40. Every topological space X admits" a polyhedral homotopy expansion p" X - + X.
A mapping p - - (p~)" X --+ X is a strong homotopy expansion, for short a strong expansion, provided it has Morita's property (M1) as well as the following strong form of property (M2).

($2) If A C A, h0, hi" Xx --+ P are mappings, and
F'XxI-+P
is a homotopy which connects hop~ and hlpx, then there exist a A' _> A and a homotopy H'X~, xI--+P, which connects hopxa, and hlpa~,. Moreover, the homotopies
H(pa, x l ) ,

F'XxI--+P

are connected by a homotopy (X x I) x I --+ P, which is fixed on X x OI, i.e.,

H(px, x 1) ~ F (rel (X x OI)).

Approximating Topological Spaces by Polyhedra

195

The following diagram illustrates property ($2). pxxl X), x OI

.

X x OI
Xa,

x

OI

h

C
H

~'"

P

X~,xI

o~176176176176176
)~
___ rel ( X x OI

.,

XxI
F

Clearly, every strong expansion is a homotopy expansion. Therefore, the following theorem is a strengthening of Theorem 39.
Theorem 41. Every resolution p: X -+ X is a strong expansion.

An immediate consequence of Theorems 28 and 41 is the following theorem, which plays an essential role in the construction of the category SSh(Top), because it insures the existence of desired polyhedral approximations of spaces.
Theorem 42. Every topological space X admits a strong expansion p: X -+ X which consists o f polyhedra.

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S. Marde~ik

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26. Marde~i6, S., On covering dimension and inverse limits for compact spaces, Illinois J.
Math. 4 (1960), 278-291.
27. Marde~i6, S., Locally connected, ordered and chainable continua (Croatian with
English summary), RadJugoslav. Akad. Znan. Umjetn. 319 (1960), 147-166.
28. Marde~i6, S., e-mappings and inverse limits, Glasnik Mat.-Fiz. Astron. 18 (1963),
195-205.
29. Marde~i6, S., Approximate polyhedra, resolutions of maps and shape fibrations, Fund.
Math. 114 (1981), 53-78.
30. Marde~i6, S., On approximate inverse systems and resolutions, Fund. Math. 142
(1993), 241-255.
31. Marde~i6, S., Coherent and strong expansions of spaces coincide, Fund. Math. 158
(1998), 69-80.
32. Marde~i6, S., Strong Shape and Homology, Springer Monographs in Mathematics,
Springer, Berlin, 2000.

Approximating Topological Spaces by Polyhedra

197

33. Marde~i6, S., Extension dimension of inverse limits, Glasnik Mat. 35 (55) (2000), 339354.
34. Marde~i6, S., and Matijevi6, V., P-like spaces are limits of approximate P-resolutions,
Top. and its Appl. 45 (1992), 189-202.
35. Marde~i6, S., and Rubin, L.R., Approximate inverse systems of compacta and covering dimension, Pacific J. Math. 138 (1989), 129-144.
36. Marde~i6, S., and Rubin, L.R., Cell-like mappings and non-metrizable compacta of finite cohomological dimension, Trans. Amer. Math. Soc. 311 (1989), 53-79.
37. Marde~i6, S., and Segal, J., e- mappings onto polyhedra, Trans. Amer Math. Soc. 109
(1963), 146-164.
38. Marde~i6, S., and Segal, J., e-mappings and generalized manifolds, Michigan Math. J.
14 (1967), 171-182.
39. Marde~i6, S., and Segal, J., e-mappings and generalized manifolds II, Michigan Math.
J. 14 (1967), 423-426.
40. Marde~i6, S., and Segal, J., Shape Theory. The Inverse System Approach, NorthHolland, Amsterdam, 1982.
41. Marde~i6, S., and Segal, J., 7~-like continua and approximate inverse limits, Math. Jap.
33 (1988), 895-908.
42. Marde~i6, S., and Ugle~i6, N., Morphisms of inverse systems require meshes, Tsukuba
J. Math. 20 (1996), 357-363.
43. Marde~i6, S., and Ugle~i6,U., On iterated inverse limits, Top. and its Appl. To appear.
44. Marde~i6, S., and Watanabe, T., Approximate resolutions of spaces and mappings,
Glasnik Mat. 24 (1989), 587-637.
45. Matijevi6, V., Approximate polyhedral resolutions with irreducible bonding mappings,
Rend. del 'Istituto di Matem. Univ. Trieste 25 (1993), 337-344.
46. Matijevi6, V., Spaces having approximate resolutions consisting of finite-dimensional polyhedra, Publ. Math. Debrecen 46 (1995), 301-314.
47. Morita, K., On expansions of Tychonoff spaces into inverse systems of polyhedra, Sci.
Rep. Tokyo Kyoiku Daigaku, A 13 (1975), 66-74.
48. Morita, K., On shapes of topological spaces. Fund. Math. 86 (1975), 251-259.
49. Morita, K., Resolutions of spaces and proper inverse systems in shape theory, Fund.
Math. 124 (1984), 263-270.
50. Nagami, K., Finite-to-one closed mappings and dimension, II, Proc. Japan Acad. 35
(1959), 437-439.
51. Nemec, A.G., and Pasynkov, B.A., Two general approaches to factorization theorems in dimension theory (Russian), Dokl. Akad. Nauk SSSR 233 (1977), no. 5,788-791.
52. Pasynkov, B.A., On polyhedral spectra and dimension of bicompacta, in particular of bicompact groups (Russian), Dokl. Akad. Nauk SSSR 121 (1958), 45-48.
53. Pasynkov, B.A., Factorization of mappings onto metric spaces (Russian), Dokl. Akad.
Nauk SSSR 182 (1968), 268-271.
54. Pasynkov, B.A., Factorization theorems in dimension theory (Russian), Uspehi Mat.
Nauk 36 (1981), 147-175.
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56. Pasynkov, B.A., A factorization theorem for the cohomological dimensions of mappings (Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1991), 26-33.
57. Pontryagin, L., The theory of topological commutative groups, Ann. of Math. 35
(1934), 361-388.
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59. Skordev, G., and Smirnov, Ju.M., The factorization and approximation theorems for
Aleksandrov-Cech cohomology in the class of bicompacta (Russian), Dokl. Akad.
Nauk SSSR 220 (1975), 1031-1034.
60. Stone, A.H., Paracompactness and product spaces, Bull. Amer. Math. Soc. 54 (1948),
977-982.
61. Watanabe, T., Approximate resolutions and covering dimension, Top. and its Appl. 38
(1991), 147-154.
62. Zarelua, A.V., The method of the theory of rings of functions in the construction of bicompact extensions (Russian), in Contributions to Extension Theory of Topological
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Berlin, 1969.

Ten Mathematical Essays on Approximation in Analysis and Topology
J. Ferrera, J. L6pez-G6mez, F. R. Ruiz del Portal, Editors
@ 2005 Elsevier B.V. All rights reserved

199

Periodic Solutions in the Golden Sixties: the Birth of a Continuation T h e o r e m
J. Mawhin
Department of Mathematics, Universit6 Catholique de Louvain,
Chemin de Cyclotron 2, B- 1348 Louvain-la-Neuve, Belgium

Abstract

This paper describes the genesis of a continuation theorem introduced by the author in the late sixties, for proving the existence of periodic solutions for ordinary differential equations. After describing the state of the art in this domain both for weakly and strongly nonlinear differential equations, and in particular the work of Cesari, Hale and Cronin, one finds the description of the two stages in obtaining the theorem: first a combination of Cesari's method, a priori bounds and Brouwer degree, then a more direct approach based on Leray-Schauder's degree. A comparison is made with the evolution of Leray's approach to nonlinear functional equations in the thirties, as well as a connection with approximation theory.
Key words: periodic solutions, continuation theorems, alternative method, topological degree, integral equations

1. Introduction

In the early sixties, the study of T-periodic solutions of T-periodic ordinary differential equations or systems was still divided into two almost separated worlds: the case of weakly nonlinear differential equations x' (t) = A(t)x + c9(t, x, c)

(1)

of arbitrary dimension n, where c is a small parameter, and the case of strongly nonlinear differential equations x' (t) = f (t, x(t)),
(2)

J. Mawhin

200

in dimension smaller or equal to 2. For example, in the Introduction of [12], Jack Hale writes: "Our knowledge of nonlinear systems is still far from being complete. For the case where the system of differential equations has order 2 (that is, one degree of freedom), much more is known than for higher-order systems. The reason for this is that analyticaltopological methods may be applied very nicely for systems of order 2, whereas for higher dimensions, the techniques of topology are not sufficiently developed. For systems of order greater than 2, the differential equations are usually assumed to contain a given parameter, and some type of perturbation technique is used to discuss the behavior of solutions."
This remark is well documented by most fundamental treatises published at this time, like those of Lefschetz (1959) [ 18], Minorsky (1962) [25], Cesari (1963) [3], Sansone and
Conti (1964) [32], Hale (1969) [13], which all separate the study of systems with small parameters and systems of dimension 2.

2. Weakly nonlinear systems
In the case of systems with a small parameter (1), it was necessary to distinguish between the nonresonant case, in which the associated linear system

Z(t) =A(t)x(t)

(3)

only has the trivial T-periodic solution, and the resonant one, in which (3) has non-trivial
T-periodic solutions. Notice that the case where

A(t) - O is always resonant.
In the nonresonant situation, the system (1) always has at least one (small) T-periodic solution when IcI is sufficiently small (a perturbation of the trivial solution when c = 0).
This is easily shown by applying the Schauder fixed point theorem if 9 is continuous, the
Banach fixed point theorem if9 is locally Lipschitzian in x or the implicit function theorem if 9 is of class C 1 with respect to x, to the equivalent integral equation

x(t) -- e

a(t, s)g(s, x(s), c) ds,

where G(t, s) is the Green matrix associated to the periodic problem for (3).
In the resonant situation, the system (1) may or may not have, for small Ic[, T-periodic solutions, depending upon the nature of the nonlinear function g and upon the non-trivial
T-periodic solutions of (3). Consider for example the resonant case where A(t) = O, for which the nontrivial T-periodic solutions of (3) are the constant functions. If x(t; c) is a
T-periodic solution of (1) with c r 0 which tend to a constant a when c -+ 0 (which is

Periodic Solutions in the Golden Sixties: the Birth of a Continuation Theorem

201

always the case if 9 is smooth enough), it is easy to see, by integrating both members of the identity x' (t; e) = eg(t, x(t; e), c)
(4)
over [0, T], using periodicity, and letting c -+ 0, that a must verify the system of equations fo T

9(8, a, 0) ds

O.

(5)

Thus, under mild regularity conditions upon 9, the existence of a solution to (5) is necessary for the existence of a T-periodic solution to (1) when Ie ] is small.
To show that such a condition is sufficient when sufficiently strengthened, one uses an idea coming back to the work of A.M. Lyapunov and E. Schmidt on nonlinear integral equations, and referred as the Lyapunov-Schmidt or alternative method. We present it in the functional analytic version developed in the fifties by Cesari and Hale (see [3,12]).
Assume that the Jacobian matrix Oxg(t, x, e) exist and is continuous. One can write the system (4) with e ~: 0 in the equivalent form
- cH

[

g(., ~ + ~(.), c) - ~

g(s,-2 + Y(s), e) as

fo T g(~, ~ + ~(~), ~) ds

o,

]

,

(6)

(7)

where
- - ~lf0Y X(~) a~,

~(t) .- x(t) -

~,

and H is the linear mapping associating to any T-periodic function with mean value zero its (T-periodic) primitive with mean value zero. Using the implicit function theorem in the subspace CT of T-periodic functions having mean value zero, it is easy to see that, for each fixed -2 in a fixed ball, and IGI sufficiently small, the problem (6) has a unique T-periodic solution ~ (-; -2, c) C CT such that

~(.;~,0) =0.
Thus -2 + ~(t; -2, c) will be a T-periodic solution of (4), if -2 satisfies the second equation
(7) with ~" replaced by ~, namely r ( ~ , c) . -

/o

g(s, ~ +

~(~; ~, c), ~) ds

- 0.

(8)

Using now the implicit function theorem in ]~n to study equation (8), we easily see that the system (4) has, for sufficiently small ]c], a T-periodic solution, if the system
G(-2) "-

/o

(9)

9(s,-~, O) ds - O

has a solution ~ such that

det O~G(-~) - det

[/o

O~g(s,-2, O) ds

]

r O.

(10)

202

J. Mawhin

The problem has thus been reduced to the solution of the system (9) of n equations in n unknowns, the so-called bifurcation equation.

3. Cesari's m e t h o d for strongly nonlinear systems

In the early sixties, Cesari proposed an extension of his approach to the case of systems of type (2), when f satisfies a Lipschitz condition in a suitable ball, namely, for some
L > 0 a n d R > 0,
] f ( t , x ) - f ( t , y ) ] 0 is an integer, define the operator Pm by m c o s s w t + G sin s~t).

P m x ( t ) "- ao + E ( a ~ s--1 Then, f o r each x C CT, one has

[ [ H ( I - P~)xll _< v~co-'a(m)l]xll

(12)

where
OO

s=m+l

so that

1

< o(m) <

m + 1

1

The proof of this inequality is a good exercise in the theory of Fourier series.
If x - G~ + ~'m with
-gin - Pmx,

"s

-- ( I -

Pro)x,

denotes a T-periodic function, then, extending the idea of A. M. Lyapunov and E. Schmidt, the T-periodic solutions of system (2) are the solutions of the system of equations
"s

- H(I-

P m ) f ( ' , - Y m ( ' ) + "s

P m x ' - - P m f ( ' , x m ( ' ) + "s

(13)
(14)

Periodic Solutions in the Golden Sixties." the Birth of a Continuation Theorem

203

Notice that the system (13) is nothing but the integrated form of

( I - Pm)x' - ( I - P m ) f ( ' , ~ m ( ' ) + Ym(')).
Now, if

r < R,

K - s u p { I f ( t , x ) I ' t E I~, Ixl < R},

115mll 0 such that the following conditions hold.
1. For each e C ]0, 1], any possible r-periodic solution o f (4) is such that Ilxl[ < r.
2. Each possible zero a o f G is such that ]a I < r.

3. degB[G , B(r), O] r O, where l G(a) "- -~ f o r g(t, a, O) dt.
Then, for each e E [0, T], the system

x' (t) = cg(t, x(t), c) has at least one T-periodic solution x C B(r) C CT.

Such a result, connecting for the first time the conditions of the small parameter method in a resonant case with an existence result for strongly nonlinear systems of arbitrary dimension, was conceptually appealing. But it was necessary to test its practical applicability with some examples. In this time, the "paradigm" for nonlinear oscillations was the forced
LiOnard equation y" + f (y)y' + g(y) = e(t),

Periodic Solutions in the Golden Sixties: the Birth o f a Continuation Theorem

207

and its special case with f constant, the so-called Duffing equation. In [ 10], Gomory had obtained some results when f and 9 are polynomials, and Theorem 1 not only allowed new and simpler proofs of Gomory's results, but also several extensions. Another consequence was various improvements and corrections of results --quite difficult to read because of their sketchy style--obtained for dissipative Duffing equation and some special Li6nard equation in 1964 and 1965 by Faure [7,8], using Leray-Schauder theory. The same approach had been used in 1960 by Ezeilo [6], in 1965 by Sedziwy [34], in 1965-66 by
Villari [36-38] and in 1966 by Reissig [26] to study the T-periodic solutions of some third order nonlinear differential equations. With the exception of Villari, all those authors had deduced the a priori bound for the T-periodic solutions from the ultimate boundedness of all solutions. The application of Theorem 1 improved some of those existence conditions in several ways, showing that strongly nonlinear differential systems in dimension higher than two could be successfully treated.
My thesis advisor at the University of Li6ge was Paul Ledoux, a worldly renowned expert in the stability and vibrations of stars, whose (never realized) hope, in hiring me, had been the application of nonlinear methods to vibrating stars. Although not at all a specialist in applying topological methods to nonlinear ordinary differential equations, Ledoux was bold enough to propose me collecting the above results in a PhD Thesis. This was done with enthusiasm and the adviser received the complete text (about 250 pages) during the
Spring 1968. It was only in the Fall that he called me to discuss the matter. The thesis [20] was (successfully) defended in February 1969, and a shortened version [21 ] was published the same year.

6. Applying Leray-Schauder's degree
We have seen in the previous section that the Leray-Schauder method had been successfully applied by various authors to problems of T-periodic solutions of some ordinary differential equations. The very first paper in this direction seems to be the one of Stoppelli
[35] in 1952, dealing with the equation y" + ]y'ly' + q(t)y' + y - p ( t ) y 3 = e(t),

where p(t) > 0. His results are described in [32] but, despite of this, Stoppelli's paper remained unnoticed untill some unfounded criticism in 1963 by Derwidu6 [5], who misused
Leray-Schauder's method in confusing the existence of an a priori bound for all solutions with the requirement that each solution is bounded. He claimed as a consequence that each system (2) with a right-hand member globally Lipschitzian in x must have a T-periodic solution! Between 1964 and 1966, Reissig [27-29] showed in successive steps, using LeraySchauder theory, the following existence result for (1), published in book form in [30].

Theorem 2. Let
A : ~ -+ L(I~n, ~ ~ )

and

9 : [O,T] • I~~ • [0, 1]-+ ~

208

J. Mawhin

be T-periodic in t and continuous. Assume that the following conditions hold.
1. There exist r > 0 such that, for each c C [0, 1], any possible T-periodic solution of each system x' (t) = A ( t ) x ( t ) + c9(t, x(t), c), c C [0, 1], is such that [[x l] < r.
2. The system x'(t) = A ( t ) x ( t ) has only the trivial T-periodic solution.
Then, for each e E [0, 1], the system x' (t) = A ( t ) x ( t ) + c9(t, x(t), e) has at least one T-periodic solution x with IIxll <

r

This result can be seen as a strongly nonlinear version of the existence theorem of the small parameter method in the non-resonant case. To prove it, Reissig wrote the problem in the equivalent fixed point form x(t) -- e

/o

G(t, s)g(s, x(s), e) ds,

with right-hand side compact on bounded subsets of CT, where G(t, s) is the Green matrix of the T-periodic problem associated to (3), and applied the elementary version of LeraySchauder fixed point theorem mentioned at the end of Section 4. Notice that, in contrast to the case of Theorem l, the Leray-Schauder degree associated to situations covered by
Theorem 2 must be one.
By comparing Theorems 1 and 2, I was somewhat puzzled by the fact that Theorem 1 required for its proof the right-hand member to be locally Lipschitzian with respect to x, a condition absent in Theorem 2. I suspected that the assumption came from the method and not from "nature". Early in 1969, playing with the operators introduced by the Cesari-Hale method, I realized that the T-periodic solutions of system (4) for c ~ 0 where the solutions of the fixed point problem in CT x = Pox + J P o g ( . , x ( . ) , e ) + e H ( I -

Po)g(.,x('),e),

(18)

where J : En __+ it~n is any automorphism, and that the right-hand member of (18) defined a compact nonlinear operator M on bounded subsets of CT. Hence Leray-Schauder degree could be applied directly to I - M, leading to the following improved version of Theorem 1.
Theorem 3. Let

g : ]t{ • E n • [0, 1] --+ ~n,

( t , x , e ) --+ g(t, x,c),

be T-periodic in t, and continuous. Assume there exists an open bounded set f~ C CT such that the following conditions hold.
1. For each c C ]0, 1], any possible T-periodic solution of (4) is such that x r 09t.

Periodic Solutions in the Golden Sixties: the Birth o f a Continuation Theorem

209

2. Each possible zero a o f G is such that a ~. OQ N I1~ . n 3. deg B [G, 9t VI~n, 0] r 0, where

lfo

G(a) " - -~

g(t,a,O)dt.

Then, for each c E [0, T], the system

x' (t) = c9(t, x(t), c) has at least one T-periodic solution x C ~.

By noticing that the T-periodic solutions of problem (1) were the solutions of the fixed point problem x = Pox + JPo[A(.)x(.) + e9(., x(.)], c) + c H ( I -

Po)[A(.)x(.) + c9(., x(.), c)],

it was also immediate to deduce from Leray-Schauder degree theory the following extension of Theorem 2.
Theorem 4. Let

A:R-+L(]R~,R

n)

and

g : ] R x ~ n x[O, 1 ] - + ~ ~

be T-periodic in t and continuous. Assume that the following conditions hold.
1. There exist an open bounded neighbourhood o f zero Q C CT such that, f o r each e E [0, 1], any possible T-periodic solution o f each system x' (t) = A ( t ) x ( t ) + eg(t, x(t), e),

e E [0, 1]

is such that x ~ OQ.
2. The system x' (t) = A ( t ) x ( t ) has only the trivial T-periodic solution.
Then, for each e C [0, 1], the system x' (t) = A ( t ) x ( t ) + eg(t, x(t), e) has at least one T-periodic solution x E Q.

Another easy consequence was a result just proved by Gfissefeldt [ 11 ] for systems odd with respect to x.
The whole thesis could be rewritten by replacing the use of Cesari's method and Brouwer degree by a direct application of Leray-Schauder degree to the compact operator M introduced above. As a consequence, some unnecessary local Lipschitz conditions on the nonlinear terms could be dropped in the main theorems and in the applications. But the thesis was already in the hands of the committee, and I could only keep the improvement for myself until the defense was performed. The new approach was published in the Fall

210

J. Mawhin

1969 [22], and appeared in book form in [31 ], including a version for the general resonant case with

A(t) g~ O.
Results of the type of Theorems 1 to 3 are generally referred as continuation theorems, in the sense that some solutions existing for c = 0 are indeed "continued" until c = 1. Given for example a problem of type (2), one imbeds it into a family of problems of type (4), with g chosen in such a way that
9(t,x, 1 ) - f ( t , x ) .
Three years later, the above continuation theorems were extended to L-compact perturbations N of a Fredholm linear operator L of index zero in a normed space, within the frame of coincidence degree, an extension of Leray-Schauder degree to mappings of the form L-N
[23]. In the hands of many mathematicians, this theory, presented in book form in [9,24], has provided a large number of new existence and multiplicity theorems for various boundary value problems associated to ordinary, functional or partial differential equations.

7. Learning from history
Later, learning more about the history of mathematics, I have noticed two facts which are not unrelated to the above story, and which are connected with some aspect of approximation theory, like it has been the case several times in the previous sections.
First Fredholm's classical approach to study linear integral equations f x(t) - /

T

I (t, , ) x ( , )

- h(t),

(19)

Jo
0

consists in approximating the integral by Riemann sums, solving the finite-dimension problem and going to the limit. Its abstract generalization in a Banach space given by F. Riesz for the equation x- Kx = h
(20)
in a Banach space, with K linear and compact, corresponds to approximating of K by linear operators having finite-dimensional range.
In 1907, E. Schmidt [33] proposed the following alternative approach to study (19).
We sketch it, for simplicity, in the case of solutions belonging to L2(0, T). Let (Ck) be a complete orthonormal systems in L2(0, T) such that
OO

I((t, s) ,-~ E

ck(t)Fk(S).

k--1

Then, there exists a positive integer m such that

[I((t,s) - ~_~ ck(t)~k(s)l 2 dsdt < 1. k=l (21)

Periodic Solutions in the Golden Sixties." the Birth of a Continuation Theorem

211

Letting m -~m (t, ~) -

Z c~ (t)v~ k=l (~),

N

Km(t,s) - K(t,s) - Km(t,s),

xj - ~0 T x(s)cj (s) ds,

(j - 1, 2,...),

(22)

we notice that

fOT -Km(t,

s)x(s) ds - Em XkCk(t).

k--1
Thus we can write equation (19) in the form

fo

x(t) -

(23)

B2m(t, s)x(s) ds - h(t) + E XkCk(t), k=l and, by Cauchy-Schwarz inequality,

fo T

~0 T Km (t, s)x(s) as
~"

dt 0 is the length at an instant t. It is traditional to assume that c~ is periodic, say of period T > 0. This model leads to suggestive examples of resonance because the equilibrium 0 = 0 becomes unstable if c~(t) oscillates in an appropriate way. Although the system has only one degree of freedom 1, the study of the stability of
0-0
1 or one and a half if the dependence on time is counted

216

R. Ortega

is not elementary. This probably explains why it is customary to substitute the original equation by its linear approximation

+ c~(t)O = 0.

(2)

The main theme of these notes will be the validity of this procedure. It will turn out that the linearization principle leads to the right conclusions in most cases, but there are exceptions.
Sometimes the equation (2) can be unstable while the equilibrium 0 = 0 is stable for (1).
In contrast, we shall find that the third order approximation (of Duffing type)
9
1
o. + ~ ( t ) o - ~ ( t )

03 - o

(3)

is faithful. This means that the equilibrium 0 = 0 is stable for (1) if and only if the same holds for (3). It must be noticed that the positivity of c~ is crucial for this result. Indeed, if c~(t) can change sign, probably none of the approximations obtained by truncating the expansion of the sine function is faithful.
The idea of replacing a complicated equation by an approximation is central in Stability Theory. The first Lyapunov's method is the simplest instance. It can be applied to our equation to prove instability in the easiest cases but it does not help in the proofs of stability. This is so because the notion of asymptotic stability (considered in Lyapunov's first method) is strange to Hamiltonian mechanics. The study of the stability of the equilibrium requires sophisticated techniques (KAM theory) which use the information on nonlinear approximations. We refer to [2,6] for the perturbative case, where c~(t) = w 2 + c/3(t), and to [26,23] for the general case. On the other hand, the results on instability also use nonlinear approximations but are of a more elementary nature. Already in [ 15], Levi-Civita obtained instability criteria using the quadratic approximation. His results were presented for abstract mappings and applied to the study of a three body problem. The basic technique in [ 15] is a detailed analysis of the dynamics around the equilibrium and it could be adapted to the pendulum of variable length. Also, it would be possible to employ Lyapunov functions as in [32]. In these notes we shall show how to obtain instability criteria using a less standard approach. Topological degree will be employed to reduce instability proofs to the computation of certain indexes (localized versions of the degree). The rest of these notes is organized in six sections. The notion of stability and its connection with the dynamics of planar mappings is discussed in w The next section, w analyzes the linearized equation and the symplectic group Sp(I~ 2). In particular, the conjugacy classes in this group are found. The basic facts about degree theory are collected in w The degree is useful to define the index of the equilibrium of our differential equation, as shown in w Some links between stability and index can be found in w Finally, in w several characterizations of the stability of the equilibrium of the pendulum are presented. They are obtained in terms of the index, the third approximation or the conjugacy classes of Sp(ll~ 2 ). The notes are concluded with some discussions about equations with more degrees of freedom.

The stability of the equilibrium: a search for the right approximation

217

2. Perpetual stability and discrete dynamical systems
We shall work with the class of differential equations

-- f (t, O)

(4)

where f is defined around 0 = 0, say f : IR x ( - e , e) --+ I~ with e > 0. The function f satisfies f(t, 0) = 0 V t e l R and so 0 = 0 is an equilibrium of the equation. In addition, f is continuous, T-periodic in t and there is uniqueness for the initial value problem associated to (4).
Given a point

(0o, c~o) E (-e, c) x ~, the solution satisfying
0(0) = 0o

and

t)(0) = C~o

will be denoted by O(t; 0o, coo). In general one cannot say that this solution is defined in the whole real line but it is at least defined in a large interval for small values of 10ol and I~ol.
The equilibrium 0 = 0 is said to be stable if given any neighborhood of the origin in
IR , say H, there exists another neighborhood 12 such that if (0o, COo)belongs to 12, then the
2
solution O(t; 0o, wo) is defined in ( - c o , ec) and

(O(t;Oo,c~o),O(t;Oo,wo)) E H

Vt

E

R.

This is the notion of perpetual stability, often employed in Hamiltonian dynamics (see
Chapter 3 of [32]). The reader who is familiar with stability theory will notice that it means
Lyapunov stability for the future and the past. Two simple examples are the equations
0+0=0

and

6/-0=0.

The equilibrium is stable only for the first.
Let us now consider the difference equation
~n--k-1 = M ( ~ n )

(5)

where

M:DQI~2

-+I[{2

is a one-to-one and continuous mapping defined in an open set 7). It is also assumed that the origin lies in 7) and it is a fixed point of M. Given an initial condition ~o E 7), the solution is defined on some subset I of Z. The fixed point ~ = 0 is said to be stable if for each neighborhood H(0), there exists another neighborhood 12(0) such that if ~o E l; then {~n} is defined in Z and
(n e H Vn e g.

218

R. Ortega

To practice with this definition the reader can consider the linear mappings M defined by the matrices

sin)

n[o]-

,

- sin O

H+[O] -

(osh snh)

cos |

,

sinh |

Or

cosh |

In the first case ~ = 0 is stable while in the second it is unstable.
There is of course a complete analogy between the definitions of stability for the continuous and discrete situations. Now we are going to immerse the study of stability for differential equations in the theory of difference equations. This is a central idea in dynamical systems that goes back to Poincar6.
The mapping

P(Oo,~Vo) = (O(T; Oo,~Vo), O(T; Oo,~Vo)) is well defined in a neighborhood of
0o = ~o = 0 and, due to the uniqueness for the initial value problem, it is one-to-one and continuous.
Moreover, the iterates p n are obtained by evaluating the solutions at time t = nT. This property is crucial to prove that the equilibrium 0 = 0 is stable for (4) if and only if the fixed point 0o = ~v0 = 0 is stable for the mapping M = P. The mapping P is usually called the Poincar6 map associated to the equation (4) and it has an important property: it preserves area and orientation. For smooth equations this is equivalent to the identity det P' (Oo, cVo) = 1 and it is a consequence of Liouville's theorem in Hamiltonian mechanics. The general case can be treated with the techniques in [31 ], Chapter IX.
To finish this section we notice that the notion of stability is invariant under changes of variables. For example, if ~ is a local homeomorphism fixing the origin, the change
= ~(~) transforms ~nA-1 -- M(~.rt)

into n,+l -M*

with
M* - ~-1 o M o ~, and the stability of ( - 0 and r / - 0 are equivalent.

3. The linear equation and the symplectic group
The linear equation oo 0 + a(t)O - O,

(6)

The stability o f the equilibrium: a search for the right approximation

219

where a(t) is continuous and T-periodic, is called Hill's equation and there are many studies about it. The book by Magnus and Winkler [20] is a classical reference. After passing to a first order system

(o)

-

A(t) -

,

(0 1)

03

o

we find the matrix solution X (t) satisfying x(o) = i

(oo) (oo)

(I is the 2 x 2 identity matrix). The Poincar6 map associated to (6) is linear, namely
P

- L

,

W0

L - X (T).

~d0

We present two examples. For the harmonic oscillator (a = 1) and a fixed period T, L is the rotation R[T] defined in the previous section. For the repulsive case (a ~ - 1 ) , L is the matrix H+ [T].
Liouville's theorem implies that the matrix solution X (t) always satisfies det X (t) = 1.
This property motivates our interest in the symplectic group. The group of 2 x 2 matrices with nonzero determinant will be denoted by G1 (I~2 ). The subgroup of G1 (I~2 ) composed by the matrices satisfying det L = 1 is the symplectic group, denoted by Sp(I~ 2 ). Given a matrix L in Sp(I~ 2 ), the eigenvalues p l , #2 satisfy
PlP2

=

1

and one can distinguish three cases:

9 elliptic: #1 - ~-,

[ P l l - 1, #1 ~: 4-1

9 hyperbolic: . x , . 2 e ]1[, 0 < 1#1[ < 1 < [.21
9 parabolic: #1 = #2 = 4-1
The conjugacy classes in the group Sp(I~ 2 ) can be described according to this classification. For an elliptic matrix L there exists Q c Sp(]~ 2 ) such that Q - 1 L Q is a rotation

Q - 1 L Q = R[O],

O r (0, 7r)U (~-, 27r).

A hyperbolic matrix is conjugate to a matrix in one of the two families
H+ [O] -

o sinhO)

,

sinh O

+ cosh O

O r (0, oo).

220

R. Ortega

Finally, a parabolic matrix will be conjugate to one of the six matrices
1 -4-1)

z,-I, P+, P_,-P+,-P_

where P+ 0

1

All these facts can be proven from the theory of Jordan canonical forms. In fact that theory can be seen as the classification of the conjugacy classes in G1 (I~2 ). There is a more subtle point which does not follow from Jordan canonical form. From the point of view of the group Gl(I~2), the rotations RIO] and R[Zzr - | are conjugate. This is not true in the symplectic group, for if Q r Gl (~2) satisfies
Q-1R[O]Q - R[2~- - o] then det Q < 0.
In view of this property we can say that the angle O is a symplectic invariant. Similar situations appear in the parabolic case for the matrices P+ and P_ (or - P + and - P _ ) .
More details and geometric insights about this group can be found in the paper by Broer and Levi [5]. The reader can deduce from the previous discussions that the origin 0 = 0 is stable for (6) if and only if the monodromy matrix X (T) is elliptic or parabolic with
+I.

Hill's equation is invariant under translation and rescaling of time. This means that the change t-A(s+r), x-x(s), with A > 0 and r r IR, transforms the Hill's equation in another equation of the same type, namely d20 ds 2

+ ct* (s)O - 0

(7)

with
~*(s) - A ~ ( A ( s + 7-)).
The new period is
T
A
We have made reference to this class of changes because they have a remarkable property, they are sufficient to arrive at the canonical form of monodromy matrices. More precisely,
T

'k"

_._

~ .

Proposition 1. Given c~(t), continuous a n d T-periodic, there exists r E ]~ and A > 0 such that the monodromy matrix associated to (7) is one o f the matrices:

9 RIO], O C (0, ~-) U (Tr, 2~-) (elliptic case)
9 H i [ 0 ] , 0 ~= 0 (hyperbolic case)
9 I, - I, P+, P _ , - P+, - P_ (parabolic case).

The stability of the equilibrium: a search for the right approximation

221

The proof of this result can be seen in [25], Proposition 8, for the elliptic case and in
[23], Lemma 2.1, for the parabolic case. Recently Yan and Zhang have found in [33] new applications of this result in the elliptic case. For the reader interested in details a proof in the hyperbolic case is presented.

Proof Assume that the eigenvalues are 1/1 and #2. We find Floquet solutions associated to these eigenvalues. These are non-trivial solutions satisfying
~ ( t + T ) -- # l ~ ( t ) ,

~ ( t + T ) -- # 2 ~ ( t ) .

The product
II = p ~ is T-periodic and so there exists T C R with

h(~) = o.
The linear independence of g) and ~ implies that ~(T) and ~(T) do not vanish. We select
F(7-) = ~(T) = 1 and define u- 1

~(~+~),

~-

1

~(~-~).

Then

U(T)=I,

V(T)=O,

/~(T)=0,

and +(T) # 0 .

Here one uses the definition of'r. From now on we shall assume

+(~) > 0.
If this derivative is negative we exchange the roles of ~ and ~. The function
U 2 -- V2 = ]'I

is T-periodic and so

U(T + T) z - V(T + T) z = 1.
From

I)(~ + T) = 0, we find that

~(~ + T)~(~ + T) - ~(~ + T)~(~ + T) = 0.
The Wronskian formula implies that
~(~ + T)u(v + T) - ~(~ + T)v(~ + T) = +(~).
From these equations one obtains
~(~+T)=~(~)v(~+T),

~(~+T)=~(~)u(v+T).

After the change

s=t+T,

222

R. Ortega

the monodromy matrix takes the form Q - 1 M Q with

0 )

\

u ( r + T)
M

V(T + T)

+

T)

+

Q -- (+(7")1/20

]

__

T)

/~(T)-I/2

We notice that M is of the type H i [ O ] with u ( r + T) - + cosh O,

v ( r + T) - sinh O.

The matrix Q is eliminated with a change of scale.

[--I

4. Degree theory and index of zeros

Let us fix 9t, bounded and open subset ofI~ a , d > 1. The degree is defined for continuous mappings from ~ into I~a which do not vanish on the boundary. More precisely, given
F E C(-~,IRa),

F(~) 7(= 0 V~ E Oft,

(8)

we can assign to it an integer which will be denoted by deg(F, f~). Among many other properties of degree we mention:
9 Existence. If deg(F, f~) :/= 0 then F(~) - 0 has at least one solution in [}.
9 Invariance by homotopy. If

f'~x

[0,1]-+I~ a,

f-f(~,A),

is continuous and f ( . , A) satisfies (8) for each A E [0, 1], then deg()C( ., A), f~)

is independent of A.

9 Excision. If A" is a compact subset of f~ and F(~) 7(: 0 for ~ E/x', then

deg(F, f~) - deg(F, f~ \ Ix').
In the properties of existence and excision it was assumed that F satisfied (8). There are many books about degree theory and we refer to [ 18] or [30] for more details.
Given an open set H c ]Ra and F C C(H, IRa), let us assume that ,~, C H is an isolated root of F(~) - 0. This means that
F(~,) - 0 and, for some 6 > O,
F(~)-r

if0<

I~-~*l _ 2. We are lead to the following consequence, Corollary 3. Assume that 0 - 0 is an equilibrium o f (9) which is isolated (period nT, n >_ 2 ) a n d stable. Then 7nT(O) -- 1.
This result is of practical value. It allows to obtain instability criteria via degree theory.
In fact, if one of the indexes is different from 1, we can say that 0 - 0 is unstable. As an example consider the equation
(11)

+ 0 + c(t)O 2 -- 0

and assume that c(t) has period
271

3
The linearization (0 + 0 - 0) has monodromy matrices (for periods T, 2T and 3T),
&

X ( T ) - R[2.-~],

X(2T)-

R[~],

X(3T)-

R[27c]- I.

This implies that, for periods T and 2T, it is possible to compute the index by linearization.
Namely, 0 - 0 is isolated (period 2T) and
")/T(0) - - sign{det ( I - X ( T ) ) } - 1,
~/2T(0) -- sign{det ( I - X ( 2 T ) ) } - 1.
To compute the third index we must employ the discussions about the degenerate case in
Section 5. Since
X(3T) - I and r (t) -- cos t,

r (t) -- sin t,

for period 3T the function H becomes
H (0o, Wo) - -~ fo 2~ c(t) (0o cos t + Wo sin t) 3 dt.
1

In complex notation,
- 0o + iwo,

equals
H(~, ~) -

lfo2

~ - 0o - icvo,

c(t)(~e - u +

The function c(t) has period -~ and this implies that
~o 27r c(t)eitdt

O.

) dt.

230

R. Ortega

Some computations lead to

-

1 (7~3+ 7~3)

-

where
27r

7 -The derivative

fo

(12)

1 (Hoo + iH~,o)

-

3

c(t)e 3itdt.

--2

is ~ and so, if-y :/: 0, the only critical point of H is the origin. It follows that 0 - 0 is isolated (period 3T) and
"~3T(0) -- i n d [ V H , 0] -- - 2 .
The conclusion is that 0 - 0 is unstable as soon as the quantity defined by (12) does not vanish. This example shows that the linearization procedure is not valid for a general equation of the type (9). In this example 0 - 0 was stable for the linearization and unstable for the original equation. The reader who is familiar with hamiltonian dynamics will have recognized the phenomenon of resonance at the roots of the unity. In this case it was the third root
~--C

7r.......&~
2

3

and we refer to [32,19] for more details.

7. The pendulum of variable length
Consider again the equation

+ a(t)sin 0 = 0

(13)

where a(t) is continuous, T-periodic and positive. We shall compute the second index
"Y2T(0). Let us start with the linearization principle. If pl and ]22 are the Floquet multipliers of the linearized equation (period T), the eigenvalues of
X (2T) = X (T) ~ are ]22 and ]222. In the elliptic case,
Pl = #2-,

/1,1 ~ +1,

and
727'(0) - sign{(1 - ]22)(1 - ]222)} - signll - ]2212 - 1.
In the hyperbolic case,
[Pl[ < 1 < []22[

and
72T(0) -- sign{(1 -- ]2~)(1 -- ]222)} -- --1.

The stability of the equilibrium: a search for the right approximation

231

In the parabolic case
Pl

= P2 =

+1,

we notice that X ( 2 T ) must be conjugate in Sp(]R 2 ) to one of the matrices I, P+, P_.
Going back to the methods of computation in the degenerate case and considering the third order approximation

1
O"+ a(t)O - ~. a(t)03 _ 0

(14)

we obtain an expansion of the Poincar6 map like in Section 5, with
1
24 fo 2T a(t)(01 (t)Oo + 02(t)wo)adt.

H(Oo,wo) -

Since H has a strict maximum at the origin ~ -- 0, we can use Euler's theorem for homogeneous functions to deduce that
~ - V H ( ~ ) = 4H(~) < 0

V~ # 0.

This inequality implies that ~ = 0 is the only critical point of H and so we can discuss the case x(2T)

= i.

More precisely, 0 -- 0 is isolated (period 2T) with
72T(0) = ind[VH, 0] = 1.
Here we have used a typical property of the index for gradient operators (see [12] or [1 ]).
Let us now assume that X (2T) is conjugate to P+ or P_. We apply the proposition in
Section 3 and find a change of independent variable t = ~ ( s + T)

such that the equation (13) becomes a* d2Ods--Z a*(s)O +

3!(s)03 + . . . .

O,

a*(s) - ,~2a(,k(s + 7))

and the monodromy matrix X * ( 2 T * ) of the linearization is precisely P+ or P _ . This transformation in time does not alter the index of 0 = 0. For if P * is the Poincar6 map of the new equation, then

L-1p*L = P with L(Oo, wo) = (0o,)~wo).
The commutativity theorem for degree shows that the indexes of I - P and I - P * coincide.
Incidentally we notice that the stability of 0 = 0 is also preserved. We apply once again the discussions of Section 5 and conclude that
"Y2T(0)- u s i g n { - ~1I f0 T* a * ( s ) O ; ( s ) 4 d s } . with u = 1 i f X ( 2 T ) ~ P+ and u = - 1 i f X ( 2 T ) ,-~ P_.

-u

232

R. Ortega

At this point the reader may think that the computation of other indexes ")/kT(O) could lead to more instability criteria. However this is not the case, as can be seen after computing all indexes. The next step is to discuss the stability of 0 = 0. This can be done but the techniques which are required go beyond the scope of these notes. The details can be seen in [26,23], the second paper in collaboration with Nflfiez. Summing up the previous discussions and the results in these papers one obtains
Theorem 4. The following statements are equivalent:
(i) 0 -- 0 is stable f o r (13)
(ii) 0 - 0 is isolated (period 2T) and 72T(0) -- 1
(iii) 0 = 0 is stable f o r the Duffing equation (14)
(iv) the monodromy matrix X (2T) is conjugate in Sp(It~2 ) to RIO], f o r some 0 E R or to P_.
We notice that the assertion (iv) is the answer to the question posed in the introduction of these notes9 The linearization procedure is valid for the pendulum of variable length excepting when
X ( 2 T ) ,,~ P_.
In this situation 0 = 0 is stable for the original equation (13) but unstable for the linearization. The analysis leading to Theorem 4 is not exclusive of the pendulum and can be applied to other equations. The crucial property is that the coefficient of the cubic term has a sign. Other results about stability using the third approximation can be found in
[28,16,17,22,24,29,34,13,14].
A natural question about Theorem 4 is its possible extension to more degrees of freedom.
To fix the ideas consider the system

01
O+AO+a(t)S(O)

-0,

O-

"
ON

where A is the N x N tridiagonal matrix, coming from the discretization of the Laplacian,
(-21

0 ...

0

1-21...0
0

0 ~
0

1 -2 9

0

0

A =~

,
0

0 ...-2

\ 0 and 0
0

0

...

1
1 -2j

e >0,

The stability of the equilibrium: a search for the right approximation

233

sin 01 s(o) -

9

sin 0N
It is not clear if the approach in these notes can be extended. In principle one cannot expect a result like Theorem 2 because we are in more dimensions and the index of a stable equilibrium can be any number. However, our system is analytic and Hamiltonian and we are in a rather special situation. Is there a version of Theorem 2 applicable to this example?
In any case it is probable that one can obtain instability criteria for the third approximation using Lyapunov functions. The stability is more delicate. Probably the notion of perpetual stability is too demanding as to obtain reasonable results. There is the notion of formal stability, associated to the Birkhoff normal form [3], which seems easier to study. This formal stability implies (via KAM theory) the notion of stability introduced by Moser in his conference in ICM Berlin 99 [21 ]. We finish these notes by recalling Moser's definition of stability in measure: instead of requiring that all orbits of a certain neighborhood are bounded for all times, one asks that most orbits (in the sense of measure) are bounded.

References

1. Amann, H., A note on degree theory for gradient mappings, Proc. Amer. Math. Soc. 85
(1982), 591-595.
2. Arnold, V.I., and Avez, A., Ergodic Problems of Classical Mechanics, Addison Wesley,
1989.
3. Birkhoff, G.D., Dynamical Systems, Amer. Math. Soc., 1927.
4. Bonati, C., and Villadelprat, J., The index of stable critical points, Topology Appl. 126
(2002), 263-271.
5. Broer, H.W., and Levi, M., Geometrical aspects of stability theory for Hill's equation,
Arch. Rat. Mech. Anal. 131 (1995), 225-240.
6. Broer, H.W., and Vegter, G., Bifurcational aspects of parametric resonance, Dynamics
Reported (N.S.) 1 (1992), 1-53.
7. Brown, M., A new proof of Brouwer's lemma on translation arcs, Houston Math. J. 10
(1984), 35-41.
8. Dancer, E.N., and Ortega, R., The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Diff. Eqns. 6 (1994), 631-637.
9. Erle, D., Stable equilibria and vector field index, Topol. Appl. 49 (1993), 231-235.
10. Krasnosel'skii, M.A., Translations Along Trajectories of Differential Equations, Trans.
Math. Monographs 19, Amer. Math. Soc., 1968.
11. Krasnosel'skii, M.A., Perov, A.I., Povolotskiy, A.I., and Zabreiko, EP., Plane Vector
Fields, Academic Press, 1966.
12. Krasnosel'skii, M.A., and Zabreiko, P.P., Geometrical Methods of Nonlinear Analysis,
Springer-Verlag, 1984.
13. Lei, J., and Zhang, M., Twist property of periodic motion of an atom near a charged wire, Letters in Math. Phys. 60 (2002), 9-17.

234

R. Ortega

14. Lei, J., Li, X., Yan, R, and Zhang, M., Twist character of the least amplitude periodic solution of the forced pendulum, SIAMJ. Math. Anal. 35 (2003), 844-867.
15. Levi-Civita, T., Sopra alcuni criteri di instabilita, Annali di Matematica V (1901), 221307.
16. Liu, B., The stability of the equilibrium of a conservative system, J. Math. Anal. App.
202 (1996), 133-149.
17. Liu, B., The stability of the equilibrium of reversible systems, Trans. Am. Math. Soc.
351 (1999), 515-531.
18. Lloyd, N.G., Degree Theory, Cambridge Univ. Press, 1978.
19. Meyer, K., Counter-examples in dynamical systems via normal form theory, SIAMRev.
28 (1986), 41-51.

20. Magnus, W., and Winkler, S., Hill's Equation, Dover, 1979.
21. Moser, J., Dynamical systems-past and present, Doe. Math. J., DMV Extra Volume
ICM I (1998), 381-402.
22. Nfifiez, D., The method of lower and upper solutions and the stability of periodic oscillations, Nonl. Anal. 51 (2002), 1207-1222.
23. N~fiez, D., and Ortega, R., Parabolic fixed points and stability criteria for nonlinear
Hill's equation, Z. Angew. Math. Phys. 51 (2000), 890-911.
24. Nfifiez, D., and Torres, E, Periodic solutions of twist type of an earth satelite equation,
Discr Con. Dyn. Syst. 7 (2001), 303-306.
25. Ortega, R., The twist coefficient of periodic solutions of a time-dependent Newton's equation, J. Dynam. Diff. Eqns. 4 (1992), 651-665.
26. Ortega, R., The stability of the equilibrium of a nonlinear Hill's equation, SlAM J.
Math. Anal 25 (1994), 1393-1401.
27. Ortega, R., Some applications of the topological degree to stability theory, in
Topological Methods in Differential Equations and Inclusions (Granas, A., and Frigon,
M., Eds.), pages 377-409, Kluwer Academic, 1995.
28. Ortega, R., Periodic solutions of a newtonian equation: stability by the third approximation, J. Diff. Eqns. 128 (1996), 491-518.
29. Torres, P., Twist solutions of a Hill's equation with singular term, Adv. Nonl. Studies 2
(2002), 279-287.
30. Rothe, E.H., Introduction to Various Aspects of Degree Theory in Banach Spaces,
Math. Surveys 23, Amer. Math. Soc., 1986.
31. Sell, G., Topological Dynamics and Ordinary Differential Equations, Van NostrandReinhold, 1971.
32. Siegel, C., and Moser, J., Celestial Mechanics, Springer-Verlag, 1971.
33. Yan, R, and Zhang, M., Higher order nonresonance for differential equations with singularities, Math. Meth. in App. Sci., 26 (2003), 1067-1074.
34. Zhang, M., The best bound on the rotations in the stability of periodic solutions of a newtonian equation, J. London Math. Soc. 67 (2003), 137-148.

Ten Mathematical Essays on Approximation in Analysis and Topology
J. Ferrera, J. L6pez-G6mez, F. R. Ruiz del Portal, Editors
@ 2005 Elsevier B.V. All rights reserved

235

The Bishop-Phelps Theorem
R. R. Phelps
Department of Mathematics, University of Washington,
Box 354-350, Seattle, WA 98195, USA

Abstract

What follows is a history of the Bishop-Phelps theorem on the density of support functionals, together with a sketch of its proof This is followed by descriptions of a number of extensions and applications, plus two basic open questions.
Key words." Banach spaces, convex sets, support functionals, support points,
Bishop-Phelps

I. Introduction
The theorem of the title is an elementary but fundamental result about convex sets and continuous linear functionals on real Banach spaces. I will test the reader's patience by starting with a personal narrative, giving some historical background before describing various extensions and applications, including two open questions. The ideas underlying the proof of the original result will be presented, but most of this article will be descriptive, with appropriate references.
To start, suppose that E is a Banach space and that

B={xcE:

Ilxll_ O. Then there exists 5 > O, a cone I f as above and a point x E S such that
IIx - zl[ <

and

S n (K + x) n B (x) = Tx},

This was used by Browder to obtain theorems about "normal solvability" of certain nonlinear operators connected with partial differential equations. One takes S -- f ( F ) where f is a certain nonlinear mapping from another Banach space F into E. In addition to [6,7], see [26] for an exposition and additional references.

7. Locally convex spaces
Tenney Peck [18] dashed any hopes for finding support points without the hypothesis that E be a Banach space by proving the following result.

Theorem 5. I f F_7,is the product space o f an infinite sequence o f non-reflexive Banach spaces, then E contains a bounded closed convex nonempty subset which has no support points. Such a product space is, of course, a locally convex Fr6chet space. For the special locally convex space consisting of the dual space E* of a Banach space E, provided with its weak* topology, it is possible to obtain some norm-density theorems; see e.g., [ 17] and [27].

8. Miscellany
As noted in the introductory section, a closed convex subset C of a Banach space can always be represented as the intersection of the closed half-spaces which support it. The question as to which sets S of support points can be removed from C and still have C represented by the half spaces supporting it at C \ S has been investigated in [28]. The question as to whether vector-valued lower semicontinuous convex functions need have subdifferentials was answered in the negative in [29]; this is a consequence of an example of two proper lower semicontinuous functions on g2 with no common point of subdifferentiability.

References
1. Bishop, E., and Phelps, R.R., A proof that every Banach space is subreflexive, Bull
Amer. Math. Soc. 67 (1961), 97-98.

The Bishop-Phelps Theorem

243

2. Bishop, E., and Phelps, R.R., The support functionals of a convex set, in Convexity,
Proc. Symp. Pure Math. VII, pages 27-35, Amer. Math. Soc., 1963.
3. Bourbaki, N., Espaces Vectoriels Topologiques, 2nd Ed., Hermann, Paris, 1966.
4. Bourgin, R.D., Geometric Aspects of Convex Sets with the Radon-Nikodym Property,
Lecture Notes in Maths. 993, Springer-Verlag, 1983.
5. Brondsted, A., and Rockafellar, R.T., On the subdifferentiability of convex functions,
Proc. Amer. Math. Soc. 16 (1965), 605-611.
6. Browder, F.E., Normal solvability and the Fredholm alternative for mappings into infinite dimensional manifolds, J. Funct. Anal. 8 (1971), 250-274.
7. Browder, F.E., Normal solvability and ~-accretive mappings of Banach spaces, Bull
Amer. Math. Soc. 78 (1972), 186-192.
8. Diestel, J., Geometry of Banach Spaces. Selected Topics, Lecture Notes in Maths. 485,
Springer-Verlag, 1975.
9. Diestel, J., and Uhl, J.J., Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.
10. Ekeland, I., Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979),
443-474.
11. Klee, V.L., Extremal structure of convex sets II, Math. Zeit. 69 (1958), 90-104.
12. Lindenstrauss, J., On operators which attain their norm, Israel J. Math. 3 (1963), 139148.
13. Lomonosov, V., A counterxample to the Bishop-Phelps theorem in complex spaces,
Israel. J. Math. 115 (2000), 25-28.
14. Lomonosov, V., On the Bishop-Phelps theorem in complex spaces, Quaest. Math. 23
(2000), 187-191.
15. Luna, G., Connectedness properties of support points of convex sets, Rocky Mount. J.
Math. 16 (1986), 147-151.
16. Luna, G., Local connectedness of support points, Rocky Mount. J. Math. 18 (1988),
179-184.
17. Luna, G., The dual of a theorem of Bishop and Phelps, Proc. Amer. Math. Soc. 47
(1975), 171-174.
18. Peck, N.T., Support points in locally convex spaces, Duke Math. J. 38 (1971), 271-278.
19. Phelps, R.R., Convex Sets and Nearest Points, Proc. Amer. Math. Soc. 8 (1957), 780797.
20. Phelps, R.R., Convex Sets and Nearest Points II, Proc. Amer. Math. Soc. 9 (1958),
867-873.
21. Phelps, R.R., Subreflexive normed linear spaces, Archiv der Math. 8 (1957), 444-450.
22. Phelps, R.R., Some subreflexive Banach spaces, Archiv der Math. 10 (1959), 162-169.
23. Phelps, R.R., A representation theorem for bounded convex sets, Proc. Amer. Math.
Soc. 11 (1960), 976-983.
24. Phelps, R.R., Some topological properties of support points, Israel J. Maths. 13 (1972),
327-336.
25. Phelps, R.R., Support cones and their generalizations, Proc. Syrup. Pure Math. 7
(1962), 393-401.
26. Phelps, R.R., Support cones in Banach spaces and their applications, Adv. in Math. 13
(1974), 1-19.
27. Phelps, R.R., Weak* Support Points of Convex Sets in E*, Israel J. Maths. 2 (1964),
177-182.

244

R. R. Phelps

28. Phelps, R.R., Removable sets of support points of convex sets in Banach spaces, Proc.
Amer Math. Soc. 99 (1987), 319-322.
29. Phelps, R.R., Counterexamples concerning support theorems for convex sets in Hilbert space, Canadian Math. Bull. 31 (1988), 121-128.
30. Phelps, R.R., The Bishop-Phelps theorem in complex spaces: An open problem, in
Function Spaces (Jarosz, K., Ed.), pages 337-340, Lect. Notes in Pure and Appl. Math.
136, Dekker, N. Y., 1992.
31. Phelps, R.R., Convex Functions, Monotone Operators and Differentiability (2nd
Edition), Lecture Notes in Maths. 1364, Springer-Verlag, 1993.
32. Rainwater, J., Yet more on the differentiability of convex functions, Proc. Amer. Math.
Soc. 103 (1988), 773-778.
33. Verona, M.E., More on the differentiability of convex functions, Proc. Amer. Math.
Soc. 103 (1988), 137-140.

Ten Mathematical Essays on Approximation in Analysis and Topology
J. Ferrera, J. L6pez-G6mez,F. R. Ruiz del Portal, Editors
(~) 2005 Elsevier B.V. All rights reserved

245

An essay on some problems of approximation theory
A. G. R a m m
Mathematics Department, Kansas State University,
Manhattan, KS 66506-2602, USA

Abstract
Several questions of approximation theory are discussed."

1) can one approximate stably in L e~ norm f ' given approximation f 6, II f ~ - f of an unknown smooth function f such that I1 f'(x) IIL~ _3, is a bounded
>
domain, by linear combinations o f the products ulu2, where Um C N ( L m ) , m =
1, 2, Lm is a formal linearpartial differential operator, and N (Lm) is the null-space of Lm in D,
N ( L m ) : - { w : Lmw - 0 in D} ?
3) can one approximate an arbitrary L2 (D) function by an entire function of exponential type whose Fourier transform has support in an arbitrary small open set? Is there an analytic formula for such an approximation?
Key words: stable differentiation, approximation theory, Property C, elliptic equations,
Runge-type theorems, scattering solutions

I. Introduction

In this essay I describe several problems of approximation theory which I have studied and which are of interest both because of their mathematical significance and because of their importance in applications.

A. G. Ramm

246

1.1. Approximation of the derivative of noisy data
The first question I have posed around 1966. The question is: suppose that f ( x ) is a smooth function, say f C C ~ (R), which is T-periodic (just to avoid a discussion of its behavior near the boundary of an interval), and which is not known; assume that its 3approximation f6 E L ~ (R) is known,
Ilfd - f l l ~ < ~, where I1" I1~ is the

L ~ (IR) norm. Assume also that

IIf'll~

< ml < OO.

Can one approximate stably in L ~ (It{) the derivative f ' , given the above data {6, fz, m~ }?
By a possibility of a stable approximation (estimation) I mean the existence of an operator L6, linear or nonlinear, such that

IIL~f~ - f ' l l ~ < ~(g) ~ 0

sup

as ~ --+ O,

(1)

fCC~(R)

IIf-f,~ll~ _ 1, by a linear combination ofthe products UlU2, where um E N ( L m ) ,

re=l,2,

Lm is a formal linear partial differential operator, and N(Lm) is the null-space of Lm in D,
N(Lm) := { w : L ~ w = 0 in D} ?
This question has led me to the notion of Property C for a pair of linear formal partial differential operators { L1, L2 }.
Let us introduce some notations. Let D C IRn , n _> 3, be a bounded domain,

Lmu(X) "-

Z ajm(x)DJu(x)' Ijl 1 is an integer, ajm (x) are some functions whose smoothness properties we do not specify at the moment,

cOJ u DJu =

Ox l

'

IJl = j l -+-"""-+- jn.

Define
Nm := ND(L,~):= { w : Lmw = 0 i n D } , where the equation is understood in the sense of distribution theory. Consider the set of products {wl w2 }, where Wm C Nm and we use all the products which are well-defined. If
Lm are elliptic operators and aim (x) C C "Y(I[{ ), then, by elliptic regularity, the functions n Wm are smooth and, therefore, the products Wl w2 are well defined.

Definition 1. A pair {L1,L2} has Property C if and only if the set total in L p (D) for some p _> 1.

{WlW2}VwmENmis

In other words, if f E L p ( D ) , then

{ /D f (x)wl W2dx - O,

VWrn E Nm } =;" f - O,

where Vwm C Nm means for all Wm for which the products Wl w2 are well defined.

(5)

A. G. Ramm

248

Definition 2. If the pair {L, L} has Property C then we say that the operator L has this property. From the point of view of approximation theory Property C means that any function

f E LP(D) can be approximated arbitrarily well in LP(D) norm by a linear combination of the set of products Wl w2 of the elements of the null-spaces Nm.
For example, i f L - V 2 then N(~72) is the set of harmonic functions, and the Laplacian has Property C if the set of products h lh2 of harmonic functions is total (complete) in

LP(D).
The notion of Property C has been introduced in [9]. It was developed and widely used in [9] - [20]. It proved to be a very powerful tool for a study of inverse problems [14]-[ 17],
[19]-[20], [28]-[29].
Using Property C the author has proved in 1987 the uniqueness theorem for 3D inverse scattering problem with fixed-energy data [ 11 ], [ 12], [ 16], uniqueness theorems for inverse problems of geophysics [ 11 ], [ 15], [ 17], and for many other inverse problems [ 17]. The above problems have been open for several decades.

1.3. Approximation by entire functions of exponential type
The third question that I will discuss, deals with approximation by entire functions of exponential type. This question is quite simple but the answer was not clear to engineers in the fifties. It helped to understand the problem of resolution ability of linear instruments
[21 ], [22], and later it turned to be useful in tomography [30]. This question in applications is known as spectral extrapolation.
To formulate it, let us assume that D C R~ is a known bounded domain,

f'-(~)

" - -

/D f(x)ei~Xdx "- 5 f ,

f c L2(D),

(6)

and assume that f(~c) is known for ~c C D, where D is a domain in R~. The question is:

can one recover f (x) from the knowledge of f (~) in D ?
Uniqueness of f(x) with the data {f(~c), ~c C D} is immediate: f(~c) is an entire function of exponential type and if f(,~) - 0 in D, then, by analytic continuation, f(~c) - 0, and therefore f(x) = 0. Is it possible to derive an analytic formula for recovery of f(x) from
{f(~), ~c C D}? It turns out that the answer is yes ([23] - [24]). Thus we give an analytic formula for inversion of the Fourier transform f(~c) of a compactly supported function f(x) from a compact set D.
.....

From the point of view of approximation theory this problem is closely related to the problem of approximation of a given function h(~C) by entire functions of exponential type

249

An essay on some problems o f approximation theory

whose Fourier transform has support inside a given convex region. This region is fixed but can be arbitrarily small.
In Sections 2,3 and 4 the above three questions of approximation theory are discussed in more detail, some of the results are formulated and some o f them are proved.

2. Stable approximation of the derivative from noisy data.
In this section we formulate an answer to Question 1.1. Denote
IIf(l+a) II "-- TYtl+a, where 0 < a _< 1, and
Ilf(l+a)ll-

IIf'l}~

+

sup
9 ,yER

If'(x)

f ' (y)l
]X - y[~
-

(7)

Theorem 3. There does not exist an operator T such that sup [ITf6-

f'[[c~ _< ~7(5) -+ 0

(8)

a s 5 --+ O,

fElC(5,mj)

if j - 0 or j T - L s,j where

1. There exists such an operator if j > 1. For example, one can take
!

Ls j f 6 . - f 6 ( x + h j ( 5 ) ) - f S ( x - h j ( 5 ) )
'
2hj (5)
'
and then

sup

hj(5) . - (

j--1

IILs,jfd- f'll~ ~< cjS--~,

1

5
) s m j (j - 1)
'

< j < 2,

(9)

(10)

f E1C(5,mj )

where

j

Cj "----

j-1

1_ mj .

(11)

(j - 1 ) - - 7

Proof
1. N o n e x i s t e n c e of T for j - 0 and j - 1.
Let

f~(x) -- O,

Extend

fl (x)

mx(x- 2h)

f~ (x) " - -

OO

h -{-

2

.-

~(a)

-

2ggY~,~.

NOW (14), with m - m2, yields

~/2 > v/25rn2- e(5).

(20)

Thus, we have obtained:
C o r o l l a r y 5. Among all linear and nonlinear operators T, the operator

Tf

1

L6f

0 ~

y(x + h(~)) -

f(x

- h(~))

2h(5)

h(a) - ~ / 2 a
'

gin2

yields the best approximation o f f ' , f C 1C(5, my), and
")'2 " - inf

[ Tfa - f'l[~ - e(5) "- V/25my.

sup

(21)

T f EK:(di,my)

Proof We have proved that ")'2 _> c(6). If T - La then

as follows from the Taylor's formula: if

then

IInsf5 - fill O,

where

OCM'-{O'OcC

n, 0 . 0 - 1 } .

Here

n
O " W "-- Z

Ojwj '

j=l

(note that there is no complex conjugation above wj), the variety M is noncompact, and
[17], [19]"
1_

IIRm(x,O)llL~,~ 0 does not depend on 0, c depends on D and on IIqIIL~(Bo), where q -- ~, q -- 0 for Ix I > a, and D C IRn is an arbitrary bounded domain. Also
C

IIRm(X,O)IIL=(D) _< ~ ,

0 e M,

IOl--+ oo,

m-

1,2.

(26)

It is easy to check that for any s C IRn , n _> 3, and any k > O, one can find (many) 01 and c 02 such that k(01 -+-02) -- .~,
1011-+ co, 0 1 , 0 2 C M, n _> 3.
(27)
Therefore, using (27) and (25), one gets" lim 1011-+oc

ff)l ~2 - e i('z 9

(28)

01"~-02-'-~, OI ,02E M

Since the set {ei~x}v~eR,~ is total in LP(D), it follows that the pair {L1,L2} of the
Schr6dinger operators under the above assumptions does have Property C.
3.2. Approximating by scattering solutions
Consider the following problem of approximation theory [20].
Let k - 1 (without loss of generality), a C S 2 (the unit sphere in/~3), and u "- u(x, ~) be the scattering solution that is, an element o f N ( ~ 72 + 1 - q ( x ) ) which solves the problem:
[272+1-q(x)]u-0

u-exp(io~.x)+A(oz' where ct C S 2 is given.

,

c~) eilxl

-i7i-+o

in IRa,
(1)

~,

Ixl-+o~,

(29) c~' . =

X

Ixl'

(30)

An essay on some problems of approximation theory

255

Let w e N(V 2 + 1- q(x)):= N(L) be arbitrary, w C H12oc, H t is the Sobolev space. The problem is:

Is itpossible to approximate w in L2(D) with an arbitrary accuracy by a linear combination of the scattering solutions u(x, ct) ?
In other words, given an arbitrary small number c > 0 and an arbitrary fixed, bounded, homeomorphic to a ball, Lipschitz domain D C IRn, can one find u~(a) C L2(S 2) such that /.t

I I w - / u(x, C~)U~(c~)dc~IIL=(D) 0 is an arbitrary small number?
The engineers discussed this question in a different form:
Can one transmit with an arbitrary accuracy a high-frequency signal 9(~) by using lowfrequency signals f (~)? The smallness of a means that the "spectrum" f (x) of the signal f (~) contains only "low spatial frequencies".
From the mathematical point of view the answer is nearly obvious: yes. The proof is very simple: if an approximation with an arbitrary accuracy were impossible, then

o- fsg( ) (f B ei~Xf(x)dx)

d~

Vf c L2(Ba).

a

This implies the relation

O - - / " 9(~)ei~'xd~

Vx C Ba.

Since D is a bounded domain, the integral above is an entire function of x E C n which vanishes in a ball B,. Therefore this function vanishes identically and consequently
-

0.

This contradiction proves that the approximation of an arbitrary 9(~) C L2(D) by the entire functions

f ( c~ )

/o

f (x)ei~.X dx

a

is possible with an arbitrary accuracy in L2(D).
Now let us turn to another question:
.v

.....

How does one derive an analytic formula for finding f (x) if f (~) is given in D?
In other words,

How does one invert analytically the Fourier tran,sform f (~) of a compactly supported function f (x), s u p p f C Ba, from a compact D?
We discuss this question below, but first let us discuss the notion of apodization, which was a hot topic at the end of the sixties. Apodization is a method to increase the resolution

An essay on some problems of approximation theory

259

ability of a linear optical system (instrument) by putting a suitable mask on the outer pupil of the instrument. Mathematically one deals with an approximation problem: by choosing a mask 9(x), which transforms the function f (x) on the outer pupil of the instrument into a function 9(x)f(x), one wishes to change the image f(~) on the image plane to an image
3j (~) which is close to the delta-function 6(~), and therefore increase the resolution ability of the instrument. That the resolution ability can be increased without a limit (only in the absence of noise!) follows from the above argument: one can choose 9(x) so that
..,..

g(~)f(x) will approximate arbitrarily accurately accurately. 5j(~), which, in turn, approximates 6(() arbitrarily

This conclusion contradicts to the usual intuitive idea according to which one cannot resolve details smaller than the wavelength.
In fact, if there is no noise, one can, in principle, increase resolution ability without a limit (superdirectivity in the antenna theory), but since the noise is always present, in practice there is a limit to the possible increase of the resolution ability.
Let us turn to the analytic formula for the approximation by entire functions and for the inversion of the Fourier transform of a compactly supported function from a compact D.
Multiply (43) by

(2~)-n~ (~)~-~ x and integrate over D to get

fj(x) - fB f(y)6j(x - y)dy - (27r)n if~ f(~)Sje-i~XdC~

(44)

a

where 6j(() is the Fourier transform of 6j(x).
Let us choose 6j (x) so that it will be a delta-type sequence (in the sense defined above).
In this case fj (x) approximates f (x) arbitrarily accurately: lim j--+c~ IIf - f i l l - O,

(45)

where the norm [1" II is L 2(B ~) norm if f c L 2 (B a ), and 6' (B a )-norm if f E C (B a ).
If

IlfllCl(Ba)

< m l , then

I l f - fjllc(Bo)