Semilinear Elliptic Equations for Beginners

Existence Results via the Variational Approach

verfasst von: Marino Badiale, Enrico Serra

Verlag: Springer London

Buchreihe : Universitext

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

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Über dieses Buch

Semilinear elliptic equations are of fundamental importance for the study of geometry, physics, mechanics, engineering and life sciences. The variational approach to these equations has experienced spectacular success in recent years, reaching a high level of complexity and refinement, with a multitude of applications. Additionally, some of the simplest variational methods are evolving as classical tools in the field of nonlinear differential equations. This book is an introduction to variational methods and their applications to semilinear elliptic problems. Providing a comprehensive overview on the subject, this book will support both student and teacher engaged in a first course in nonlinear elliptic equations. The material is introduced gradually, and in some cases redundancy is added to stress the fundamental steps in theory-building. Topics include differential calculus for functionals, linear theory, and existence theorems by minimization techniques and min-max procedures. Requiring a basic knowledge of Analysis, Functional Analysis and the most common function spaces, such as Lebesgue and Sobolev spaces, this book will be of primary use to graduate students based in the field of nonlinear partial differential equations. It will also serve as valuable reading for final year undergraduates seeking to learn about basic working tools from variational methods and the management of certain types of nonlinear problems.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction and Basic Results

Abstract

The first chapter is an introduction. After a brief historical overview, we introduce the fundamental notions to be used later and the first simple existence results.

In particular we present a review of differential calculus for functionals, with many examples, and we introduce the fundamental notion of weak solution that allows one to interpret solutions of elliptic problems as critical points of functionals.

Convex functionals and their main properties are then described, providing the first examples of existence theorems.

Finally we recall the main results from the linear theory, particularly some spectral properties of elliptic operators that will be used throughout the book.

Marino Badiale, Enrico Serra

Chapter 2. Minimization Techniques: Compact Problems

Abstract

Throughout this chapter we show how techniques based on minimization arguments can be used to establish existence results for various types of problems.

Our aim is not to describe the most general results, but to give a series of examples, and to show how simple techniques can be refined to treat more complex cases.

We start from sublinear problems, for which the energy functionals are bounded from below. Direct minimization arguments, based on convexity and coercivity allow one to establish rather easily the existence of a solution.

Next we turn to superlinear problems. For these types of nonlinearities direct minimization does not work anymore: the corresponding energy functionals are generally unbounded from below. We then present some methods of constrained minimization, where one restricts the functional to a subset of functions on which it is bounded from below, and tries to establish the existence of a minimum point. Special care must then be employed to show that the constrained minimum is truly a critical point of the unconstrained functional.

Marino Badiale, Enrico Serra

Chapter 3. Minimization Techniques: Lack of Compactness

Abstract

In this chapter we present some examples of problems where compactness is not guaranteed a priori. The lack of compactness can take different forms, but in the simplest case, it is manifest through the fact that minimizing sequences are maybe bounded, but not (pre-)compact in the function spaces where the problem is set.

The reasons for this often come from geometrical or physical aspects, for instance when the problem is set on an unbounded domain.

Here we confine ourselves to some more or less simple examples of problems with lack of compactness and we try to show some ways to overcome the obstacle.

As usual, we study equations with different hypotheses on the nonlinearity. In particular when dealing with critical growth problems we consider homogeneous nonlinearities through minimization on spheres, while in the first sections we study nonhomogeneous nonlinearities, applying the method of minimization on the Nehari manifold.

Marino Badiale, Enrico Serra

Chapter 4. Introduction to Minimax Methods

Abstract

This chapter is an introduction to a broad class of methods that have been shown to be extremely useful in a variety of contexts.

We confine ourselves to the simplest cases, but we try to motivate the ideas involved in the construction of the main tools, so that the interested reader can turn to the study of more complex problems with a minimum of background.

In the preceding chapters we have mainly looked for critical points of a functional as minimum points, either on the whole space, or suitably restricting the functional to sets where minima could be shown to exist. In this chapter, on the contrary, we concentrate on the search of critical points that are not global minima, for example saddle points. The procedures to do this, called minimax methods, are quite elaborate, and we introduce the main steps gradually.

Marino Badiale, Enrico Serra

Backmatter

Titel: Semilinear Elliptic Equations for Beginners
verfasst von: Marino Badiale
Enrico Serra
Verlag: Springer London
Electronic ISBN: 978-0-85729-227-8
Print ISBN: 978-0-85729-226-1
DOI: https://doi.org/10.1007/978-0-85729-227-8