Variational methods in nonsmooth analysis and quasilinear equations [Elektronische Ressource] / vorgelegt von Hamid Douik

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Variational Methods in Nonsmooth Analysisand Quasilinear EquationsVon der Fakult at furÄ Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westf alischen Technischen Hochschule Aachen zur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaften genehmigte Dissertationvorgelegt vonDiplom-Mathematiker Hamid Douikaus Zagora, MarokkoBerichter: Universit atsprofessor Dr. Jochen ReinermannUniversit Dr. Dr. h.c. Hubertus Th. JongenTag der mundlicÄ hen Prufung:Ä 09. Juli 2003Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.ÄiiContentsIntroduction1 Preliminaries 11.1 Recalls of nonsmooth critical point theory . . . . . . . . . . . . . . . . . . 11.1.1 The concept of weak slope . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Comparison with other slopes . . . . . . . . . . . . . . . . . . . . . 31.1.3 Mountain Pass Theorem . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Quasilinear elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 7Z ZnX11.2.1 Properties of f(u):= a (x;u)DuD udx¡ G(x;u)dx 7ij i j2 Ω Ωi;j=11 1;fi1.2.2 Uniform L - and C -Estimates for critical points of f . . . . . . 101.2.3 Comparison Principles . . . . . . . . . . . . . . . . . . . . . . . . . 162 Conservation of local minimizers 192.1 Subcritical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Remarks on the critical and supercritical cases . . . . . . . . . . . . . .

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• Mathematics

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• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2003
• Nombre de lectures: 10
• Langue: English

Extrait

Variational Methods in Nonsmooth Analysis
and Quasilinear Equations
Von der Fakult at fur? Mathematik, Informatik und Naturwissenschaften der Rheinisch-
Westf alischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Mathematiker Hamid Douik
aus Zagora, Marokko
Berichter: Universit atsprofessor Dr. Jochen Reinermann
Universit Dr. Dr. h.c. Hubertus Th. Jongen
Tag der mundlic? hen Prufung:? 09. Juli 2003
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.?iiContents
Introduction
1 Preliminaries 1
1.1 Recalls of nonsmooth critical point theory . . . . . . . . . . . . . . . . . . 1
1.1.1 The concept of weak slope . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Comparison with other slopes . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Mountain Pass Theorem . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Quasilinear elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Z ZnX1
1.2.1 Properties of f(u):= a (x;u)DuD udx¡ G(x;u)dx 7ij i j
2 Ω Ωi;j=1
1 1;fi1.2.2 Uniform L - and C -Estimates for critical points of f . . . . . . 10
1.2.3 Comparison Principles . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Conservation of local minimizers 19
2.1 Subcritical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Remarks on the critical and supercritical cases . . . . . . . . . . . . . . . 29
3 Applications to quasilinear elliptic Problems 33
3.1 Sub- and Super-solutions: Existence of a first solution . . . . . . . . . . . 35
3.2 Existence of a second solution . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Palais-Smale condition . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Mountain-pass solution . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Conservation of critical groups at the origin 43
4.1 Critical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 A generalized Morse Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Regular deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
14.4 Conservation of critical groups in the C (Ω) topology . . . . . . . . . . . . 590
Bibliography 63
iii CONTENTSIntroduction
Inthiswork,westudytheexistenceandmultiplicityofsolutionsforthequasilinearelliptic
problem
8 n nX X1> n< ¡ D (a (x;u)Du)+ D a (x;u)DuD u=g(x; u) in Ω‰Rj ij i s ij i j2(P) i;j=1 i;j=1>:
u=0 on @Ω;
nwhere Ω is a bounded open subset ofR with sufficiently smooth boundary @Ω, the
functions a : Ω£R ! R, g : Ω£R ! R satisfy some raisonable hypotheses, andij
u:Ω!R is the ”unknown” function for which we want to establish the existence.
The following question arises: in which class of functions should we search for u and
whichpropertiesshoulduhave? Theanswertothisquestiondependsonthebehaviourof
the coefficients a ; g, such as differentiability and growth. For the problem (P) there areij
regularity results of the following general type: if a ; g and Ω satisfy certain conditions,ij
then u belongs to a certain functional class. Such results also affect the method of the
treatment of the problem (P).
We suppose that a ; g are defined, in such a way that the functionalij
Z ZnX1
(1) f(u):= a (x;u)DuD udx¡ G(x;u)dxij i j2 Ω Ωi;j=1
Z t
1where G(x;t) := g(x;s)ds, is well defined for all u in the Sobolev space H (Ω). The0
0
1 1space H (Ω) is the closure of C (Ω), the space of infinitely differentiable real valued0 0
functions having compact support in Ω, with respect to the norm
Z Z 1‡ ·
22 2kuk 1;2 := u + jDuj :W (Ω)
Ω Ω
1;2 2Moreover, W (Ω) is the set of functions u2L (Ω) for which there exist functions
2g 2L (Ω),jfij•1, such that:fi
Z Z
fi jfij 1u(x)D »(x)dx=(¡1) g (x)»(x)dx; »2C (Ω; 0•jfij•1:fi 0
Ω Ω
iiiiv Introduction
1Furthermore we assume that f has a directional derivative at u2H (Ω) with respect0
1 1 1 1to H (Ω)\L (Ω) , i.e., for all »2H (Ω)\L (Ω) we have0 0
f(u+t»)¡f(u)0f (u)(») :=lim
t!0 tZ n nX X1
= a (x; u)DuD »dx+ D a (x; u)DuD u»dxi;j i j s i;j i j2Ω i;j=1 i;j=1Z
¡ g(x; u)»dx2R:
Ω
1 1 1 0Consequently for u2H (Ω) and »2H (Ω)\L (Ω) with f (u)» =0 we have0 0
Z Zn nX X1
(2) a (x; u)DuD »+ D a (x; u)DuD u» = g(x; u)»:i;j i j s i;j i j2Ω Ωi;j=1 i;j=1
Note, if u would be differentiable on Ω, then we can integrate by parts in (2) and we ob-
1 ¯tainthatusolvestheproblem(P)intheclassicalsense(i.e. u2C (Ω)withu=0on@Ω).
1An element u2H (Ω) satisfying (2) is called aweak solution for problem (P), or a0
solution in distributional sense. Hence, one way to find solutions of (P) is to look for
1 0 1 1points u2H (Ω) with f (u):» =0 for all »2H (Ω)\L (Ω). Such elements u are called0 0
critical points of f, and the value f(u) is called a critical level of f.
In this way the search for solutions of (P) will be focussed on the search for critical
points of f. An obvious idea would be to use arguments of Weierstrass type in the
1setting of the infinite-dimensional space H (Ω). That is, the existence of a minimizer0
1of the functional f in a certain set K ‰ H (Ω) might be guaranteed with the help of0
a compactness argument, e.g., K weakly compact, f coercive ( i.e. f(u ) ! 1 ifn
ku k!1 ). and a corresponding continuity argument , weakly semicontinuity ( i.e.n
f(u)•liminff(u ) if (u ) converges weakly to u).n n
0For such a minimizer we have f (u)=0, provided f and K satisfy appropriate differ-
entiability hypotheses, and so one gets a critical point of f.
However, in many problems the functional f is not bounded from below, or f is not coer-
cive. Inaddition,thecriticalpointsobtainedbythemethoddescribedabove(Weierstrass’
approach ) are extremals of the functional f. However, there might be more solutions
for the associated problem. Therefore, we are also interested in critical points which are
different from extremals.
Asystematicmethodforfindingothercriticalpointsofagivendifferentiablefunctional
is a method based on continuous deformations . The latter lies at the heart of theory of
critical points: Morse theory, minimax theory .
1;1The basic idea is as follows. Let f : H !R be a C ¡mapping on a Hilbert space
H. Then, one might try to deform lower level sets of f from a higher level into a lowerIntroduction v
one, as long as no critical level is in between. The deformation · is obtained by taking
the solution of the following Cauchy problem
(
d 0·(t)= ¡f (·(t))dt
·(0)= u0
Oneeasilyshowsthatthefunctionf alongalocalsolution·(t)isdecreasing. However,
a solution need not to exist for all t. A central argument in the proof of the existence of
a deformation is the Palais-Smale condition( a compactness argument): “ every sequence
N 0(u ) 2H with (f(u )) bounded and f (u )!0 has a convergent subsequence” .h h2N h h
Let us turn back to our functional (1) : although f is in general continuous under
1natural conditions on the a ’s and g, it is not differentiable on H (Ω), even not locallyij 0
Lipschitz continuous (see chap. 1 ). Therefore, we cannot use the above considerations of
1 1criticalpointtheory. Ontheotherhand,f isdifferentiableinthesubspaceH (Ω)\L (Ω),0
endowedwiththenormk:k:=k:k 1;2 +k:k , butnow, itdoesnotsatisfythePalais-W (Ω) 1
Smale condition condition.
To overcome these difficulties D. Arcoya and L. Boccardo have introduced in 1996 [7]
1 1the following generalized compactness condition on the subspace H (Ω)\L (Ω):0
1 1 + Nevery sequence (u ) in H (Ω)\ L (Ω) satisfying for some (k ) 2 (R ) ; (" ) # 0 theh h h0
conditions
(f(u )) is bounded ;h
ku k•2k for all h2N;h h
£ ⁄kvk0 1 1jf (u ):vj•" +kvk 1;2 for all v2H (Ω)\L (Ω)h h W (Ω) 0kh
1has a convergent subsequence in H (Ω).0
1 1The idea to use a compactness argument for the functional f on H (Ω)\L (Ω), has0
already been used in 1982 in an earlier work of M. Struwe [55]
Another approach for the treatment of the problem (P), that we use in this work, is
basedonthevariationalmethodforcontinuousfunctions. Thishasbeendeveloped
by M. Degiovanni and M. Marzocchi [29] and J.-N. Corvellec, M. Degiovanni, and M.
Marzocchi [25]. A fundamental tool is the concept of a weak slope at a point u2 X of
a continuous function f :X !R, where X is a metric space. Its denoted byjdfj(u) and
defined as the supremum of ?’s in [0;1) such that there exists – > 0 and
· :B(u;–)£[0; –]!X continuous such that for all (v; t)2B(u;–)£[0; –]:
d(·(v; t);v))•t; f(·(v; t))•f(v)¡?t:
This notion is independently introduced by Degiovanni-Marzocchi [29] and G. Katriel
[44] in 1994. It is a generalization of the norm of the derivative in case of a smooth
1 0function. In fact, for f 2 C (X;R), we have jdfj(u) = kf (u)k where X is a normedvi Introduction
space (see chap. 1). With respect to the latter concept an element u 2 X is called
a critical point, if jdfj(u) = 0. The correspending Palais-Smale condition is stated
Nas follows: every sequence (u ) 2 X with (f(u )) bounded and jdfj(u ) ! 0 has ah h2N h h
convergent subsequence .
For our functional (1) we obtain ( see Proposition 1.2.1 )
0 1 1jdfj(u)‚supff (u)(») : »2H (Ω)\L (Ω);k