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2003 | Buch

Elements of Nonsmooth Analysis Kapitel lesen Erstes Kapitel lesen

Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems

verfasst von: D. Motreanu, V. Rǎdulescu

Verlag: Springer US

Buchreihe : Nonconvex Optimization and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

This book reflects a significant part of authors' research activity dur­ ing the last ten years. The present monograph is constructed on the results obtained by the authors through their direct cooperation or due to the authors separately or in cooperation with other mathematicians. All these results fit in a unitary scheme giving the structure of this work. The book is mainly addressed to researchers and scholars in Pure and Applied Mathematics, Mechanics, Physics and Engineering. We are greatly indebted to Viorica Venera Motreanu for the careful reading of the manuscript and helpful comments on important issues. We are also grateful to our Editors of Kluwer Academic Publishers for their professional assistance. Our deepest thanks go to our numerous scientific collaborators and friends, whose work was so important for us. D. Motreanu and V. Radulescu IX Introduction The present monograph is based on original results obtained by the authors in the last decade. This book provides a comprehensive expo­ sition of some modern topics in nonlinear analysis with applications to the study of several classes of boundary value problems. Our framework includes multivalued elliptic problems with discontinuities, variational inequalities, hemivariational inequalities and evolution problems. The treatment relies on variational methods, monotonicity principles, topo­ logical arguments and optimization techniques. Excepting Sections 1 and 3 in Chapter 1 and Sections 1 and 3 in Chapter 2, the material is new in comparison with any other book, representing research topics where the authors contributed. The outline of our work is the following.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Elements of Nonsmooth Analysis
Abstract
In this Chapter we recall important definitions and results from the theory of generalized gradient for locally Lipschitz functionals due to Clarke [8], different nonsmooth versions of Palais-Smale conditions and basic elements of nonsmooth calculus developed by Degiovanni [9], [10]. A major part in Section 2 is devoted to the relationship between the Palais-Smale condition and the coerciveness in the nonsmooth setting.
D. Motreanu, V. Rădulescu
Chapter 2. Critical Points for Nonsmooth Functionals
Abstract
The present Chapter deals with critical point theory for three different nonsmooth situations. First, we set forth the critical point theory for locally Lipschitz functionals, following the approach of Chang [4] and Clarke [5]. Then a critical point theory is described for nonsmooth functionals expressed as a sum of a locally Lipschitz function and a convex, proper and lower semicontinuous function, using the development in Motreanu and Panagiotopoulos [26]. Finally, the critical point theory for continuous functionals defined on a complete metric space as introduced by Degiovanni and Marzocchi [7] is presented.
D. Motreanu, V. Rădulescu
Chapter 3. Variational Methods
Abstract
This Chapter deals with general techniques for studying the existence and multiplicity of critical points of nondifferentiable functionals in the so-called limit case (see Remark 3.1). There are proved nonsmooth versions of several celebrated results like: Deformation Lemma, Mountain Pass Theorem, Saddle Point Theorem, Generalized Mountain Pass Theorem. First, we present a general deformation result for nonsmooth functionals which can be expressed as a sum of a locally Lipschitz function and a concave, proper, upper semicontinuous function. Then we give a general minimax principle for nonsmooth functionals which can be expressed as a sum of a locally Lipschitz function and a convex, proper, lower semicontinuous functional. Here we are concerned with the limit case (i.e. the equality c = a, see Remark 3.1), obtaining results which are complementary to the minimax principles in Section 2 of Chapter 2. These general results are applied in the second Section of this Chapter for proving existence, multiplicity and location of solutions to various boundary value and unilateral problems with discontinuous nonlinearities.
D. Motreanu, V. Rădulescu
Chapter 4. Multivalued Elliptic Problems in Variational Form
Abstract
In Partial Differential Equations, two important tools for proving existence of solutions are the Mountain Pass Theorem of Ambrosetti and Rabinowitz [1] (and its various generalizations) and the Ljusternik-Schnirelmann Theorem [16]. These results apply to the case when the solutions of the given problem are critical points of an appropriate functional of energy f, which is supposed to be real and C 1, or only differentiable, on a real Banach space X. One may ask what happens if f, which often is associated to the original equation in a canonical way, fails to be differentiable. In this case the gradient of f must be replaced by a generalized one, which is often that introduced by Clarke in the framework of locally Lipschitz functionals. In this setting, Chang [4] was the first who proved a version of the Mountain Pass Theorem, in the case when X is reflexive. For this aim, he used a “Lipschitz version” of the Deformation Lemma. The same result was used for the proof of the Ljusternik-Schnirelmann Theorem in the locally Lipschitz case. As observed by Brézis, the reflexivity assumption on X is not necessary.
D. Motreanu, V. Rădulescu
Chapter 5. Boundary Value Problems in Non-Variational Form
Abstract
This Chapter is devoted to an initial boundary value problem for a parabolic inclusion with a multivalued nonlinearity given by a generalized gradient in the sense of Clarke [13] of some locally Lipschitz function. The elliptic operator is a general quasilinear operator of Leray-Lions type. Of special interest is the case where the multivalued term is described by the usual subdifferential of a convex function. Our main result is the existence of extremal solutions limited by prescribed lower and upper solutions. The main tools used in the proofs are abstract results on nonlinear evolution equations, regularization, comparison, truncation, and special test function techniques as well as the calculus with generalized gradients.
D. Motreanu, V. Rădulescu
Chapter 6. Variational, Hemivariational and Variational-Hemivariational Inequalities: Existence Results
Abstract
The celebrated Hartman-Stampacchia theorem (see [6], Lemma 3.1, or [9], Theorem I.3.1) asserts that if V is a finite dimensional Banach space, K ⊂ V is non-empty, compact and convex, A : K → V* is continuous, then there exists u ∈ K such that, for every v ∈ K,
$$\langle Au,v - u\rangle \geqslant 0.$$
(6.1)
D. Motreanu, V. Rădulescu
Chapter 7. Eigenvalue Problems with Symmetries
Abstract
In this Chapter we consider several classes of inequality problems involving hemivariational inequalities with various kinds of symmetry and possibly with constraints. We establish multiplicity results, including cases of infinitely many solutions. The proofs use powerful tools of non-smooth critical point theory combined with arguments from Algebraic Topology. Results in this direction in the framework of elliptic equations have been initially established by Ambrosetti and Rabinowitz (see [1], [16]), while pioneering results in the study of multiple solutions for periodic problems can be found in Fournier and Willem [6], and Mawhin and Willem [9].
D. Motreanu, V. Rădulescu
Chapter 8. Non-Symmetric Perturbation of Symmetric Eigenvalue Problems
Abstract
In this Chapter we establish the influence of an arbitrary small perturbation for several classes of symmetric hemivariational eigenvalue inequalities with constraints. If the symmetric problem has infinitely many solutions we show that the number of solutions of the perturbed problem tends to infinity if the perturbation approaches zero with respect to an appropriate topology. This is a very natural phenomenon that occurs often in concrete situations. We illustrate it with the following elementary example: consider on the real axis the equation sin x = 1/2. This is a “symmetric” problem (due to the periodicity) with infinitely many solutions. Let us now consider an arbitrary non-symmetric “small” perturbation of the above equation. For instance, the equation sin x = 1/2 + εx 2 has finitely many solutions, for any ε ≠ 0. However, the number of solutions of the perturbed equation becomes greater and greater if the perturbation (that is, |ε|) is smaller and smaller. In contrast with this elementary example, our proofs rely on powerful tools such as topological methods in nonsmooth critical point theory. For different perturbation results and their applications we refer to [1], [15], [20] (see also [9] for a nonsmooth setting) in the case of elliptic equations, [8] for variational inequalities and [3], [5], [6], [14], [16], [17], [18] for various perturbations of hemivariational inequalities. This abstract developments are motivated by important appications in Mechanics (see [12], [13]).
D. Motreanu, V. Rădulescu
Chapter 9. Location of Solutions for General Nonsmooth Problems
Abstract
The aim of the present Chapter is to study from a qualitative point of view a general eigenvalue problem associated to a variational-hemivariational inequality with a constraint for the eigenvalue. The basic feature of our approach is that we are mainly concerned with the location of eigensolution (u, λ), where u and λ stand for the eigenfunction and the eigenvalue, respectively. This is done in Section 2, where the location of eigensolutions is achieved by means of the graph of the derivative of a C 1 function. Section 1 presents a general existence result for variationalhemivariational inequalities with assumptions of Ambrosetti and Rabinowitz type. Section 2 deals with the exposition of our abstract location results. In Section 3 we discuss the location of solutions to variationalhemivariational inequalities by applying the abstract results. The case of nonlinear Dirichlet boundary value problems is contained.
D. Motreanu, V. Rădulescu
Chapter 10. Nonsmooth Evolution Problems
Abstract
In this Chapter one discusses existence, uniqueness, Lipschitz continuous dependence on initial conditions and stability of solutions for different evolution initial value problems written in the form of variational inequalities or equalities. Section 1 concerns the study of the Cauchy problem for a first order dynamical variational inequality. Section 2 contains an existence result for the solutions of a Cauchy problem for a second order evolution variational equation. In Section 3 one presents stability, asymptotic stability and unstability results for first order evolution variational inequalities.
D. Motreanu, V. Rădulescu
Chapter 11. Inequality Problems in BV and Geometric Applications
Abstract
The theory of variational inequalities appeared in the middle 60’s in connection with the notion of subdifferential in the sense of Convex Analysis (see e.g. [4], [10], [16] for the main aspects of this theory). All the inequality problems treated to the beginning 80’s were related to convex energy functionals and therefore strictly connected to monotonicity: for instance, only monotone (possibly multivalued) boundary conditions and stress-strain laws could be studied. Nonconvex inequality problems first appeared in [18] in the setting of Global Analysis and were related to the subdifferential introduced in [7] (see A. Marino [17] for a survey of the developments in this direction).
D. Motreanu, V. Rădulescu
Backmatter
Metadaten
Titel
Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems
verfasst von
D. Motreanu
V. Rǎdulescu
Copyright-Jahr
2003
Verlag
Springer US
Electronic ISBN
978-1-4757-6921-0
Print ISBN
978-1-4419-5248-6
DOI
https://doi.org/10.1007/978-1-4757-6921-0