Nonlinear Analysis Theory and Applications: Proceedings of the Seventh International Summer School held at Berlin, GDR from August 27 to September 1, 1979 [Reprint 2021 ed.] 9783112541845, 9783112541838

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ABHANDLUNGEN D E R AKADEMIE D E R W I S S E N S C H A F T E N D E R D D R Abteilung Mathematik — Naturwissenschaften — Technik Jahrgang 1981 • Nr. 2N

Nonlinear Analysis Theory and Applications Proceedings of the Seventh International Summer School held at Berlin, GDR from August 27 to September 1 , 1979

Herausgegeben von

Professor Dr. sc. Reinhard Kluge

Akademie der Wissenschaften der DDR Institut für Mathematik

Mit 34 Abbildungen

und 2 Tabellen

AKADEMIE-VERLAG • B E R L I N • 1981

Herausgegeben im Auftrag des Präsidenten der Akademie der Wissenschaften der DDR von Vizepräsident Prof. Dr. Heinrich Scheel

Erschienen im Akademie-Verlag, DDR -1080 Berlin, Leipziger Straße 3—4 ©Akademie-Verlag Berlin 1981 Lizenznummer: 202 • 100/422/81 Gesamtherstellung: VEB Druckhaus „Maxim Gorki", 7400 Altenburg Bestellnummer: 762 924 4 (2001/81/2N) • LSV 1065 Printed in GDR DDR 8 4 , - M ISSN 0138-1059

Preface From August 27 to 31, 1979, the seventh international summer school "Nonlinear Analysis. Theory and Applications" was held in Berlin at the Central Institute of Mathematics and Mechanics of the Academy of Sciences of the GDR. It continued the summer schools devoted to nonlinear problems organized alternately by the Czechoslovakian Academy of Sciences and by the Central Institute in Berlin. The first school took place at Babylon (CSSR) in 1971, the second at Neuendorf/ Hiddensee (GDR) in 1972, the third at Starà Lesnà(CSSR) in 1974, the fourth at Berlin (GDR) in 1975, the fifth at Berlin (GDR) in 1977, the sixth at Horni Bradlo (CSSR) in 1978. The 105 participants of the seventh summer school came from Czechoslovakia, France, Italy, Poland, Romania, Switzerland, the U.S.S.R., the U.S.A and the German Democratic Republic. In the 31 invited lectures and 23 communications special attention was given to qualitative aspects (direct problems, optimal control, inverse problems), to approximation methods, to numerical realizations on the computer and to applications. It is hoped that the 52 papers contained in this volume provide an informative survey of the scientific programme of the summer school. Berlin, December 1979

1*

Prof. Dr. sc. Reinhard Kluge Chairman of the summer school

Contents Invited Lectures H. Amann, Saddle points and nonlinear differential equations H. Benker, On optimal control problems described by operator equations in Banach and Hilbert spaces M. Biroli, Regularity results for some elliptic variational inequalities with bounded measurable coefficients and applications L. Bittner, Nonlinear equality constraints in optimization problems with set restrictions G. Bruckner, On the speed of c0-sequences in connection with a lemma of Teeplitz H. Gajewski, On an initial-boundary value problem for the nonlinear Schrodinger equation with selfconsistent potential K . Groger and R. Hiinlich, On initial value problems in thermodynamics . . . . C. GroBmann, Common properties of nonlinear programming algorithms basing on sequential unconstrained minimizations B. Heinrich, The finite-difference method for a mildly nonlinear elliptic problem . . . I. Hlavacek, Finite element analysis of the unilateral contact between elastic bodies R. Kluge, An inverse problem for "coefficients" in linear equations. Uniqueness and iterative solution J. Kolomy, Nonlinear operators and solvability of operator equations E. KrauB, On convergence properties of maximal monotone operators A. Langenbach and L. Recke, On perturbation of branching problems and secondary branching in the theory of plates W. Miiller, On a program for the computation of matter transport in river networks J. Naumann, Existence and regularity theorems for weak solutions to the equations of motion of visco-plastic media J. Necas, A necessary and sufficient condition for the regularity of weak solutions to nonlinear elliptic systems of partial differential equations R. Nehse, Problems in connection with the Hahn-Banach Theorem I. Pawlow, Parabolic problems with free boundaries: Existence and properties of solutions; optimal control problems 0. Pironneau, Optimization methods for non-linear partial differential equations . . B. Rousselet, Design sensitivity analysis in structural mechanics

11 19 29 41 53 75 89 107 119 129 139 149 155 165 181 189 201 211 221 233 245

6

Contents

K.-H. Schlüßler, On optimal control problems in civil-engineering and technology B . Silbermann, Das asymptotische Verhalten von Toeplitzdeterminanten für einige Klassen von Erzeugerfunktionen I. Singer, Optimization and best approximation J . Sokolowski, Control in coefficients for P D E G. Stoyan, Towards a general-purpose difference scheme for the one-dimensional linear parabolic equation J . Szlaza, Two-component flow round a solid sphere F. Unger, To the mathematical treatment of a technical heat control problem . . L. v. Wolfersdorf, On some optimal control problems for elliptic systems in the plane

255 267 273 287 297 315 327 65

Communications J . Danes, On the radius of a set in a Hilbert space 337 Z. Denkowski, On the variational method without coerciveness 339 S. Dietze, Special control problems and monotone operators 343 S. Gähler, A generalization of an optimality theorem 347 J . Haslinger and J . Lovisek, Mixed variational formulation of unilateral problems 351 A. Hoffmann, Sufficient conditions for optimality of a control problem with discrete time 357 T. B. Kopeikina, A controllability problem of nonlinear dynamical systems . . . 361 K . Litewska and J . Muszynski, On the convergence of the finite elements method for some initial problem 367 R. Lorentz, On the normality of Lagrange multiplier rules for optimal control problems .371 M. Niezgödka, Some methods of solving optimal control problems for free boundary processes 375 G. Porath, Ein Approximations-Iterationsverfahren für nichtlineare Volterrasche Integralgleichungen zweiter Art 379 L. Recke, Secondary axisymmetric buckled states of thin elastic spherical shells . . 385 T. Riedrich, Perturbation and branching of the solutions of nonlinear eigenvalue problems . 389 H. Schmeling, Zur Existenz der optimalen Steuerung bei Volterraschen Integralgleichungen 395 R . Schumann, The finite difference method for quasilinear elliptic equations with rapidly increasing coefficients 399 H.-D. Sparing, A constructive proof of the implicit function theorem for analytic operators ' . . . . : • ' . ' . . . . • 403 W. Sperber, Some remarks to existence problems of optimal control for parabolic equations 407 W. Sprößig, Construction of local solutions for Maxwell equations 413

Contents R. Steudel, An abstract equation modelling diffusion in solids with bidisperse pore structure D. Wegner, Bifurcation of axiallv compressed circular cylindrical shells with free edges W. Weinelt, On difference approximations of a class of «-dimensional variational inequalities J . Wolska-Bochenek, On some Stefan like problem M. Wulst, On a system of nonlinear partial differential equations arising in the theory of heat and mass transfer in a nonlinear second-order fluid

7 417 421 427 433 437

Saddle points and nonlinear differential equations Herbert Amann, Zurich

1. In this note we describe some recent results concerning the solvability of nonlinear equations in Hilbert spaces, which have partly been obtained jointly with E . Zehnder (cf. [1, 2]). The abstract results have been motivated by problems in the theory of nonlinear partial differential equations. For this reason we shall illustrate our general results by means of the following simple examples. Let Q be a bounded domain in R " with smooth bondary cQ and let / € C 1 (R) be given. Then we are looking for solutions of the nonlinear elliptic boundary value problem (BVP) -Au = f{u) in Q, 1 w = 0

on

8Q.)

For simplicity we assume that / is asymptotically linear, that is, the limit f'(oo) : = lim/'(£) I£I-h>O

(1.1)

exists. In the second example we are looking for 27r-periodic solutions of the nonlinear wave equation utt ~ uxx = f(u) for (t, x) 6 R X (0, it), 1 u(t, 0) = u(t, ji) = 0

for

ielR.

j

Here we assume that / g C (IR), satisfies (1.1), and that 2

/'(I) = M

vfeR.

B y standard arguments it can be shown that (E) and (W) are very special cases of the following abstract problem: Find a solution u € D(A) of the equation Au =

F(u),

where A is a self-adjoint linear operator in a real Hilbert space H, and F:H H is a continuous potential operator, that is, there exists a functional $ € Cl(H, R ) satisfying 0(0) = 0 and 0' = F.

(A)

Of course, problem (A) contains much more general realizations than (E) and (W). In particular we should like to mention that our abstract results can be applied to the

H. Amann

12

problem of the existence of periodic solutions to nonlinear Hamiltonian systems of ordinary differential equations, in which case we obtain interesting new results (cf. [2]). Consider now problem (E). In this case it is well known that A has a compact resolvent and that the eigenvalues A* of A form an increasing sequence tending to infinity,

Aq

^

••

each At having finite multiplicity. It is a consequence of the work of many authors, e.g. Ambrosetti, Ambrosetti-Mancini, Ambrosetti-Rabinowitz, Berger, Rabinowitz, Dancer, Amann, Hess, Thews, and many others (cf. [2] for more precise bibliographical remarks) that the solvability of problem (E) depends on the way the nonlinearity / "interacts" with the spectrum of A. To describe the most significant case, let us assume that /(0) = 0,

(1.2)

so that (E) has the trivial solution. In this case, the best results known to the author, guaranteeing the existence of a nontrivial solution, are due to Thews [8] and Hess [6]. Thews proves the existence of a nontrivial solution of (E) if and

/'(0) < At ^ A, < /'(oo) < Al+1

(1.3)

/'(*) ^

(1.4)

1

V e 6 1R

whereas Hess requires (1.3) and ( m - Vif) f > o

v f

e

a

.

«

)

In addition there are in both cases existence results if the roles of /'(0) and /'(oo) are interchanged (roughly: if all inequality signs are reversed). In the case of the nonlinear wave equation (W), much less seems to be known. If we presuppose (1.2), then the best result concerning the existence of a nontrivial 2jr-periodic solution known to the author is due to Mancini [7] (cf. [1.2] for further bibliographical remarks). In the case of problem (W), the linear operator A has a pure point spectrum extending from —oo to + o o : — oo 0 and ô > 0 such that ll«'(*)ll ^ » INI -

à

VzeZ.

(2.1)

This inequality implies that there are no ciritical points of a outside of some large ball and that there is also no ciritical point " a t infinity", more precisely, that a satisfies the Palais-Smale condition. In order to find critical points of a we consider now the flow defined by the differential equation z - -- a'(z)

(2.2)

on Z. It can be shown that a' is globally Lipschitz continuous so that this flow is well defined. Let S denote the set of all bounded solutions of (2.2). If S =|= 0, then, taking an arbitrary bounded trajectory, it must have an co-limit point which due to the fact that we are dealing with gradient flows, has to be a critical point of a. Thus it remains to show that 8 =|= 0. Since (2.1) implies the boundedness of 8, we can, following C. C. Conley [4], associate with S a generalized Morse index h(S), which is the homotopy type of a pointed topological space. On the basis of hypothesis (F^), and using Conley's theory, this index can be computed and it turns out that h(S) equals the homotopy type of a pointed m^dimensional sphere, where m^ is the positive Morse index of [A — F'(oo)] | Z, that is, the dimension of a maximal subspace of Z on which A — F'(oo) is positive definite. Since this generalized Morse index is distinct from the generalized Morse index of the empty set, we obtain the following existence theorem Theorem 2. If (F^) is satisfied, then the equation Au = F(u) has at least one solution. Suppose now that F(0) = 0. Then the equation Au = F(u) has the trivial solution and Theorem 2 does not give any information. In this case we are interested in the existence of nonzero solutions. Since it follows easily that a'(0) = 0, we are interested in the existence of nonzero critical points of a. For our further considerations we have to impose an additional regularity hypothesis (R), which implies a 6 C2(Z, ]R). This regularity condition, which is somewhat technical and cumbersome to state explicitly, is satisfied in all of our applications. Since a'(0) = 0, it suffices by the above considerations to prove the existence of a bounded trajectory of (2.2), which does not contain 0 in its closure. Indeed, in this case any co-limit point of such a trajectory is then a nonzero critical point of a. Using again Conley's generalized Morse theory, a relatively complicated theorem about certain normal forms of flows, and a certain amount of algebraic topology, the existence of a nonzero critical point of a can be shown, provided maa