Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems | springerprofessional.de

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Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems

verfasst von: Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Verlag: Springer New York

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book focuses on nonlinear boundary value problems and the aspects of nonlinear analysis which are necessary to their study. The authors first give a comprehensive introduction to the many different classical methods from nonlinear analysis, variational principles, and Morse theory. They then provide a rigorous and detailed treatment of the relevant areas of nonlinear analysis with new applications to nonlinear boundary value problems for both ordinary and partial differential equations. Recent results on the existence and multiplicity of critical points for both smooth and nonsmooth functional, developments on the degree theory of monotone type operators, nonlinear maximum and comparison principles for p-Laplacian type operators, and new developments on nonlinear Neumann problems involving non-homogeneous differential operators appear for the first time in book form. The presentation is systematic, and an extensive bibliography and a remarks section at the end of each chapter highlight the text. This work will serve as an invaluable reference for researchers working in nonlinear analysis and partial differential equations as well as a useful tool for all those interested in the topics presented.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Sobolev Spaces

Abstract

This chapter provides a comprehensive survey of the mathematical background of Sobolev spaces that is needed in the rest of the book. In addition to the standard notions, results, and calculus rules, various other useful topics, such as Green’s identity, the Poincaré–Wirtinger inequality, and nodal domains, are also discussed. A careful distinction between various properties of Sobolev functions is made with respect to whether they are defined on a one-dimensional interval or a multidimensional domain. Bibliographical information and related comments can be found in the Remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 2. Nonlinear Operators

Abstract

This chapter focuses on important classes of nonlinear operators stating abstract results that offer powerful tools for establishing the existence of solutions to nonlinear equations. Specifically, they are useful in the study of nonlinear elliptic boundary value problems as demonstrated in the final three chapters of the present book. The first section of the chapter is devoted to compact operators and emphasizes the spectral properties, including the Fredholm alternative theorem. The second section treats nonlinear operators of monotone type, possibly set-valued, among which a prominent place is occupied by maximal monotone, pseudomonotone, generalized pseudomonotone, and (S ₊)-operators. The cases of duality maps and p-Laplacian are of high interest in the sequel. The third section contains essential results on Nemytskii operators highlighting their main continuity and differentiability properties. Comments on the material of this chapter and related literature are given in a remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 3. Nonsmooth Analysis

Abstract

This chapter offers a systematic presentation of nonsmooth analysis containing all that is necessary in this direction for the rest of the book. The first section of the chapter gathers significant results of convex analysis, especially related to the convex subdifferential such as its property of being a maximal monotone operator. The second section has as its main focus the subdifferentiability theory for locally Lipschitz functions. Further information and references are indicated in a remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 4. Degree Theory

Abstract

This chapter provides the fundamental elements of degree theory used later in the book for showing abstract results of critical point theory or bifurcation theory as well as for the study of the existence and multiplicity of solutions to nonlinear problems. The first section of the chapter introduces Brouwer’s degree and its important applications such as Brouwer’s fixed point theorem, Borsuk’s theorem, Borsuk–Ulam, and Lyusternik–Schnirelmann–Borsuk theorems. The second section sets forth the Leray–Schauder degree theory for compact perturbations of the identity. The third section amounts to a description of the degree for (S)₊maps using Galerkin approximations and construction of the degree theory for multifunctions of the form f + A with f an (S)₊-map and A a maximal monotone operator. Comments and historical notes are given in a remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 5. Variational Principles and Critical Point Theory

Abstract

This chapter addresses variational principles and critical point theory that will be applied later in the book for setting up variational methods in the case of nonlinear elliptic boundary value problems. The first section of the chapter illustrates the connection between the variational principles of Ekeland and Zhong and compactness-type conditions such as the Palais–Smale and Cerami conditions. The second section contains the deformation theorems that form the basis of the critical point and Morse theories. These results are proved in the setting of Banach spaces relying on the construction of a pseudogradient vector field and by using the Cerami condition. The third section focuses on important minimax theorems encompassing various linking situations: mountain pass, saddle point, generalized mountain pass, and local linking. The fourth section studies critical points for functionals with symmetries providing minimax values corresponding to index theories whose prototype is the Krasnosel’skiĭ genus. The fifth section is devoted to generalizations: critical point theory on Banach manifolds and nonsmooth critical point theories. Comments and related references are available in a remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 6. Morse Theory

Abstract

This chapter represents a self-contained presentation of basic results and techniques of Morse theory that are useful for studying the multiplicity of solutions of nonlinear elliptic boundary value problems with a variational structure. The first section of the chapter contains the needed preliminaries of algebraic topology. The second section focuses on the Morse lemma and the splitting and shifting theorems. The third section is devoted to the Morse relations, including the Poincaré–Hopf formula, which involve the critical groups and critical groups at infinity. The fourth section sets forth efficient results for the computation of critical groups that are powerful tools in the study of multiple solutions. Here an original approach is developed, and improvements of known results are shown. Notes on related literature and comments are provided in a remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 7. Bifurcation Theory

Abstract

This chapter examines the bifurcation points of parametric equations, that is, values of a parameter from which the set of solutions splits into several branches. The deep connection between bifurcation points and the spectrum of linear operators involved in problems is pointed out. The presentation consists of two parts regarding the used approach: degree theory and implicit function theorem. In the latter, the theory of Fredholm operators is utilized in conjunction with the Lyapunov–Schmidt reduction method. Applications to ordinary differential equations are given. The proofs of the results presented in the chapter are complete, and novel ideas are incorporated. The basic references are mentioned in a remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 8. Regularity Theorems and Maximum Principles

Abstract

This chapter provides a comprehensive presentation of regularity theorems and maximum principles that are essential for the subsequent study of nonlinear elliptic boundary value problems. In addition to the presentation of fundamental results, the chapter offers, to a large extent, a novel approach with clarification of tedious arguments and simplification of proofs. The first section of this chapter treats two major topics related to weak solutions of nonlinear elliptic problems: boundedness and regularity. The second section has as its objective to report on maximum and comparison principles. It comprises two parts: local results and strong maximum principles. Comments and related references are given in a remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 9. Spectrum of Differential Operators

Abstract

This chapter provides a self-contained account of the spectral properties of the following fundamental differential operators: Laplacian, p-Laplacian, and p-Laplacian plus an indefinite potential, with any 1 < p < +∞. The first section of the chapter examines the spectrum of the Laplacian separately under Dirichlet and Neumann boundary conditions, taking advantage of the essential feature that this refers to a linear operator. The second section addresses the spectrum of the p-Laplacian, again considering separately the Dirichlet and Neumann boundary conditions. Here the methods are completely different with respect to the Laplacian because the p-Laplacian is a nonlinear operator for p ≠ 2, making use of topological tools such as the Lyusternik–Schnirelmann principle. The third section extends this study to the more general class of nonlinear operators expressed as the sum of p-Laplacian and certain indefinite potential. Powerful related techniques are developed, for instance, the antimaximum principle, which is presented in a novel form. The fourth section addresses the Fučík spectrum, which incorporates the ordinary spectrum. The last section contains comments and information on relevant literature.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 10. Ordinary Differential Equations

Abstract

This chapter examines the existence and multiplicity of periodic solutions for nonlinear ordinary differential equations. The first section of the chapter investigates a nonlinear periodic problem involving the scalar p-Laplacian for 1 < p < + ∞ in the principal part and a smooth potential. The results cover cases of resonance at any eigenvalue of the principal part. They are obtained through variational methods and Morse theory. The second section presents results on the existence of multiple solutions for a second-order periodic system in the form of a differential inclusion. The multivalued term is expressed as a generalized gradient of a locally Lipschitz function. The approach is based on nonsmooth critical point theory. Comments and relevant references are given in a remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 11. Nonlinear Elliptic Equations with Dirichlet Boundary Conditions

Abstract

This chapter studies nonlinear Dirichlet boundary value problems through various methods such as degree theory, variational methods, lower and upper solutions, Morse theory, and nonlinear operators techniques. The combined application of these methods enables us to handle, under suitable hypotheses, a large variety of cases: sublinear, asymptotically linear, superlinear, coercive, noncoercive, parametric, resonant, and near resonant. In many situations we are able to provide multiple solutions with additional information about their properties, for instance, constant-sign (i.e., positive or negative) solutions and nodal (sign-changing) solutions. The first section of the chapter is devoted to the study of nonlinear elliptic problems through degree theory. The second section focuses on the variational approach, specifically for investigating coercive problems and (p − 1)-superlinear parametric problems. The third section makes use of Morse theory in studying (p − 1)-linear noncoercive equations and p-Laplace equations with concave terms. The fourth section deals with general elliptic inclusion problems treated via nonlinear, possibly multivalued, operators. The last section highlights related remarks and bibliographical comments.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Chapter 12. Nonlinear Elliptic Equations with Neumann Boundary Conditions

Abstract

This chapter aims to present relevant knowledge regarding recent progress on nonlinear elliptic equations with Neumann boundary conditions. In fact, all the results presented here bring novelties with respect to the available literature. We emphasize the specific functional setting and techniques involved in handling the Neumann problems, which are distinct in comparison with those for the Dirichlet problems. The first section of the chapter discusses the multiple solutions that arise at near resonance, from the left and from the right, in the Neumann problems depending on parameters. The second section focuses on nonlinear Neumann problems whose differential part is described by a general nonhomogeneous operator. The third section builds a common approach for both sublinear and superlinear cases of semilinear Neumann problems. Related comments and references are given in a remarks section.

Dumitru Motreanu, Viorica Venera Motreanu, Nikolaos Papageorgiou

Backmatter

Titel: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems
verfasst von: Dumitru Motreanu
Viorica Venera Motreanu
Nikolaos Papageorgiou
Copyright-Jahr: 2014
Verlag: Springer New York
Electronic ISBN: 978-1-4614-9323-5
Print ISBN: 978-1-4614-9322-8
DOI: https://doi.org/10.1007/978-1-4614-9323-5

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