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.] 9783112484326, 9783112484319

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ABHANDLUNGEN DER AKADEMIE DER WISSENSCHAFTEN DER DDR Abteilung Mathematik — Naturwissenschaften — Technik Jahrgang 1981 • Nr. 2N

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

Herausgegeben von

Professor Dr. sc. Reinhard Kluge bearbeitet von

Dr. habil. Wolfdietrich Müller

Akademie der Wissenschaften der DDR Institut für Mathematik

Mit 34 Abbildungen

und 2 Tabellen

A K A D E M I E - V E R L A G • 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 (¡„-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. Muller, 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 a n d technology B . Silbermann, D a s a s y m p t o t i s c h e Verhalten von Toeplitzdeterminanten f ü r einige Klassen von E r z e u g e r f u n k t i o n e n I . Singer, Optimization a n d best a p p r o x i m a t i o n J . Sokolowski, Control in coefficients for P D E G. Stoyan, Towards a general-purpose difference scheme for t h e one-dimensional linear parabolic equation J . Szlaza, Two-component flow r o u n d a solid sphere F . Unger, To t h e m a t h e m a t i c a l t r e a t m e n t of a technical heat control problem . . L. v. Wolfersdorf, On some optimal control problems for elliptic systems in t h e plane

255 267 273 287 297 315 327 65

Communications J . Danes, On t h e radius of a set in a Hilbert space Z. Denkowski, On t h e variational m e t h o d without coerciveness S. Dietze, Special control problems a n d monotone operators S. Gähler, A generalization of a n o p t i m a l i t y theorem J . Haslinger and J . Lovisek, Mixed variational formulation of unilateral problems A. H o f f m a n n , Sufficient conditions for o p t i m a l i t y of a control problem with discrete time T. B. Kopeikina, A controllability problem of nonlinear dynamical systems . . . K . Litewska a n d J . Muszynski, On t h e convergence of t h e finite elements m e t h o d for some initial problem R . Lorentz, On t h e normality of Lagrange multiplier rules for o p t i m a l control problems M. Niezgödka, Some methods of solving o p t i m a l control problems for free b o u n d a r y processes G. P o r a t h , E i n Approximations-Iterationsverfahren f ü r nichtlineare Volterrasche Integralgleichungen zweiter Art L. Recke, Secondary axisymmetric buckled s t a t e s of t h i n elastic spherical shells . . T. Riedrich, P e r t u r b a t i o n a n d b r a n c h i n g of t h e solutions of nonlinear eigenvalue problems H . Schmeling, Z u r Existenz der o p t i m a l e n Steuerung bei Volterraschen Integralgleichungen R . S c h u m a n n , The finite difference m e t h o d for quasilinear elliptic equations with rapidly increasing coefficients , H . - D . Sparing, A constructive proof of t h e implicit f u n c t i o n t h e o r e m for analytic operators . . W . Sperber, Some remarks to existence problems of optimal control for parabolic equations W . Sprößig, Construction of local solutions for Maxwell equations

337 339 343 347 351 357 361 367 371 375 379 385 389 395 399 403 407 413

Contents R. Steudel, An abstract equation modelling diffusion in solids with bidisperse pore structure D. Wegner, Bifurcation of axially compressed circular cylindrical shells with free edges W. Weinelt, On difference approximations of a class of w-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. I n this note we describe some recent results concerning t h e solvability of nonlinear equations in Hilbert spaces, which have p a r t l y been obtained jointly w i t h E . Zehnder (cf. [1, 2]). T h e a b s t r a c t results have been m o t i v a t e d b y problems in t h e t h e o r y of nonlinear p a r t i a l differential equations. F o r this reason we shall illustrate our general results b y m e a n s of t h e following simple examples. Let Q be a b o u n d e d domain in IRK with smooth b o n d a r y oQ a n d let / £ C^IR) be given. T h e n we are looking for solutions of t h e nonlinear elliptic b o u n d a r y value problem (BVP) -A « = /(«)

in

Q,

1

u = 0

on

8ii. f

F o r simplicity we assume t h a t / is asymptotically linear, t h a t is, t h e limit /'(oo) : = l i m / ' ( £ ) . |f|-*X>

(1.1)

exists. I n t h e second example we are looking for 27r-periodic solutions of t h e nonlinear wave equation Utt — uxx = f(u)

for

(t, I)ER

X (0, n), |

u{t, 0) = u(t, n) = 0

for

t € R.

j

2

Here we assume t h a t / £ C (IR), satisfies (1.1), a n d t h a t /'(f) + 0

Vf

eR.

B y s t a n d a r d a r g u m e n t s it can be shown t h a t (E) and (W) are very special cases of the following a b s t r a c t problem : Find a solution u g D(A) of the equation Au =

F(u), (A)

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 0 € C11(H, 1R) satisfying 0(0) = 0 and 0' = F. Of course, problem (A) contains m u c h more general realizations t h a n (E) a n d (W). I n particular we should like to mention t h a t our a b s t r a c t results can be applied t o t h e

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 fa of A form an increasing sequence tending to infinity,

< fa < fa < . . each fa having finite multiplicity. It is a consequence of the work of many authors, e.g. Ambrosetti, Ambrosetti-Mancini, Ambrosetti-Rabinowitz, Berger, Rabinowitz, Dancer, Amaim, 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

m

/'( fa > /'(oo), /'(oo) f'{oo) and decreasing if /'(0)
0 is also a solution for K = 0 if the control set U has the form U = U n |M/||M||2 SS \\u(K)\\2}. Proof. We have j(Q(K), u(K)) + K \\u{K)\\l ^ j(Q, u) + K ||«||f this inequality follows j(Q(K), u(K)) ^ j(Q, u) + K(M% -

||«(Z)||f) ^ j(Q, u)

\/ u 6 U.

yutU.

From q.e.d.

Remarks. 1. The importance of Lemma 2 lies in the fact that from the solution of problems without control constraints we can obtain solutions for problems with constraints. 2. From Lemma 3 we obtain for instance that for U =

U, =

MM|2 ^

1}

(24)

and |\u(K) | |2 = 1 u(K) also solves the problem for K = 0. Now are treated continuity properties of the optimal control and the optimal functional value with respect to K. Lemma 4. For problem (12), (13) holds lim jKn{Q(E.), Kn-+Kt+0

u(Kn)) = )Ka{Q(K0), u(K0))

( f o r K0 ^ 0).

Proof. We have jKo{Q(K0), u(K0)) = j(Q(K0), u(K0)) + K0 \\u(K0)\\v ^ j(Q(K„), u(Kn)) + K„ \\u(Kn)\\l j(Q{K0), u{K0)) + K„ ||M(if0)||f. From this inequalities the assertion immediately follows, q.e.d. Remarks. 1. Lemma 4 gives a typical assertion of the regularization theory. I t can also be generalized to lim

iKn(Q*SK»)>

= ?k,{Q(KO). u(Ko))

,

where (QC{K), ue(K)) satisfies the following inequalities h[Q{K),

u(K)) ^ jK{Qe(K),

ue(K)) ^ jK(Q(K), u(K)) +

e.

2. For the special problem (5) we can additionally prove the convergence of the sequences u(Kn)^u(K0), Q(Kn) Q(K0) for Kn K0 + 0 (if B, = H{) (see [12]).

On optimal control problems in Banach and Hilbert spaces

23

If furthermore is given problem (7) with p = 2 and U = H2 then in Lemma 1 hold strong inequalities in b), c) and d) for LTr =(= 0 (for LTr = 0 immediately follows u(K) = 0) and u(K) is continuous on [e, oo) (s > 0 arbitrary but fixed). This was proved by Balakrishnan [5] and Vinter [4]. 3. Lemma 4 also holds if only for K > K0 exists a solution (Q(K), u(K)) of the problem, i.e. we have lim

kn(Q(Kn),

u(Kn))

= I n f i m u m jK>(Q, u).

For quadratic problems we now can apply lemmas 2, 3 and 4 in order to give analytic solutions too for problems with constraints. Lemma 5. Let K

0 be fixed and u(ft) be a solution

of the problem, (ft ig 0) 1

Jp(u) = ||Lu - r\\\ + (K + 0) \\u\\l A Minimum.

(25)

Then holds for a given M > 0: a) If

S u p ||«(/?)[|2 iS M then

0>O

u(o) = u° is

the solution

(with

minimal

norm)

of the

problem

J{u) = \\Lu — r\\\ + K \\u\\\ i Minimum. HiSM b) If Sup ||m(/9)||2 > M then u° = u(fi0) with /30 from,

(26) I[m(/30)[|2 =

M is the solution

(with

0 > O

minimal

norm) of problem

(26).

Proof. For K = 0 Lemma 5 was proved by Balakrishnan and Vinter [4, 5]. For K > 0 the assertions a) and b) immediately follow from lemmas 2, 3 and 4 (with remark 2). Remark. If the operator L is compact we can give the solution of problem (25) as an infinite series of the form (for K + P > 0) m

+ p + h) •2

= E (LTr,

¡=i

(27)

This is the well-known expansion of the solution of the equation ( L T L + ( K + /?) /)«,(/?) = LTr (necessary and sufficent optimality condition for (25)) on eigenelements of the operator LTL. Applying Lemma 5 we must now maximize the function oo ll«08)lli = E (LTr-

rffHK

+ (S + A;)2

¿=1 with respect to ¡3 and we obtain for the optimal control u° of (26) OO 00 = £ (LTr, v*) rtI(K + Aj) resp. = E (LTr, i—i

1 2

i=i

¿/(K + 0O + A,)-

For K = 0 L additionally must be compact. For a not compact operator L also exists an expansion of the optimal control u(ß) into an infinite series (see [16]).

24

H.

BENKER

In the assertion b) of Lemma 5 the optimal control is a so-called bang-bang optimal control, i.e., u° belongs to the boundary of the control set. In the following we shall give a definition for bang-bang controls and derive sufficient conditions for the bangbang-ness of an optimal control. Definition. A control u is called weakly bang-bang if Xu £ U implies that A iS 1 (or equivalent: Au $ TJ y a > 1). Now we generalize a result of Rogak/Kazarinoff [6, 7]. Lemma 6. Let u° 6 U (U star-shaped)

be an optimal control for the problem

j{Q, u) = ¡10 - i2|]f + K\\u\\\ ± Minimum

with

Q = Lu + ef(Q).

(28)

UiU

Furthermore we assume that the solution of the state equation from (28) has the form Q = + eCe(u) with Cc € (II2 -> //j) bounded. If then (||L||2 + K) M < H / 2

and

(Lu, R) j 4= 0

for some

ueV

Lu (29)

hold with M = Sup ||w||| and m = Inf (—Lu, 2B) = —(Lv, 2R) then u° is weakly banguiU

uiU

bang for sufficiently small e.

Lemma 7. Under the same assumption as in Lemma 6, for Kn > 2 \\LTR\\ N (30) with n = Inf \u\ and N = Sup ||w[| (Inf and Sup over all bang-bang controls) the optimal ontrol u° is not bang-bang for sufficiently small e. The p r o o f s of lemmas 6 and 7 are similar to the proofs for e = 0 given by the author in [14] and therefore they are omitted. Remarks. 1. Lemmas 6 and 7 also hold for more general state equations if their solutions have the form Q — Lu + eCe(u). 2. For the special control set U = U1 = {uj\\u]2 have the form resp.

1} the conditions (29) and (30)

(||i!|2 + K) < 1127*11,

(29')

K >Min (2^112,11^},

(30')

which easily follows from the proofs of the lemmas 6 and 7. 4. Duality theorems and lower bounds Applying a general duality theorem of the approximation theory by the author [12, 13] were given the following duality conditions Inf (||0 - J2||f + K\\u\\%)HP = Sup (hESup (l,Q + Kl2u)\, ).U)IV IWIP-SI \ (O.UIEK I (O.a)ev Inf (¡¡Lu UiU

K IM\pylp = Sup (hr lllllp'Sl '

Sup (L% + Kit) «\ uiU I

(31) (32)

On optimal control problems in Banach and Hilbert spaces

and if the B{ = Hi are Hilbert spaces and U = Min (||Lu INasa

25

U1

= Max ((lu r), IWIp-si

r\\{ + K Ml)^

\\LTl, + Kl21|2).

(33)

Obviously the dual problems have not a simpler structure than the primal problems. But they are an important tool in order to compute lower bounds. Estimating the dual functionals from (33), by the author [12] were derived the following bounds (for r 4= 0, K > 0): for

\\r\\y(\\r\\l + \\LTr\\HK"^lo IMIi ^ Min (||Lu HltSi

r\\l + KMl)^

=

K\\r\\\l\\LTr\\2 for IMIj ( i . e . w ° = 0 )

p = 1 and

p^2

\\LTr\y\\r\h

> K

for p = 1 and H ^ y i M I , ^ K

(34) (with 1 jp + 1 \q = 1). The disadvantages of these bounds lie in the fact that we need the inverse operator A"1 (L = A~XB) and that for K = 0 they can not be applied. Therefore we now go another way, applying a simple inequality of the convex analysis.

Lemma 8. Let j(Q, u) be a convex and continuous functional on the Hilbert space H = Hl x H2. Then holds for arbitrary {(/>, u°) with AQ° + Bu° + f = 0 ?(Q°, u") ^ j(Q, u) T

if dj{Q, u) n (R(A ) x R(BT)) ATN

(N, AQ + Bu + f)3

(35)

=t= 0 and the triple (Q, u, N) satisfies the following equations

= lQ, BTN

(36)

= lu

for any I = (lQ, lu) 0, b) R £ D(A), c) \\AT(AR + /)||! + \\BT{AR + /)||2 > 0 then follows ' J° = Minimum (\\Q ,0+Ba+/=0VilV

111

AR + / 6 D{AT)

n D{BT)

II J V I /II4 + K ||«|||) > — ^ > 0. T ^ - \\A {AR + f)\\\ + \\BT(AR + f)\\l/K

and

137) ' '

Proof. Here we have IQ = 2(Q — R), lu — 2 K u . Therefore we obtain from Lemma 8 j(Q°, «»)

^ \\ATN\\\l4i + \\BTN\\II4K =

—\\ATN\\\j4:

-

\\BTN\\y4:K

(N, AQ + Bu + f)3 -

(N, AR + f)3 =

: S(N).

(38)

26

H . BENKER

If we set N = —fi(AR + /) and maximize S(N) with respect to /? > 0 then we obtain the bound (37). q.e.d. Remarks. 1. The bound (37) gives for Q = Lu + / (i.e. A = I , B = —L and / = —/) the bound from (34) so that (37) is a generalization with the advantage that we do not need the inverse operator AFor unbounded operators A arises here the disadvantage t h a t the assumption b) is not fulfilled in many cases (for bounded operators b) is satisfied in most cases). 2. If for unbounded operators the assumptions b) and c) are not fulfilled we can t r y to find an element d € D(AT) n D(BT) cr H3 for that hold (ATd, + (d, f)3 4= 0 and \\ATd\h T + \\B d\\2 > 0. If we have such a n element d then by setting N = ¡3d and maximizing (38) with respect to /? we obtain the bound \\AH\\\ + \\BTd\\\IK Now we give a bound for the case K = 0. Theorem 3. If an element N 4= 0 exists with a) N € D(AT), b )Ni Ker (A T ), c) N