Nonlinear elliptic partial differential equations. An introduction (Q1744619)

Nonlinear elliptic partial differential equations are one of the most attractive and investigated topics in mathematical analysis. This book presents in an elegant and simple way the basic tools and concepts that anyone interested in this topic should know. These tools and concepts include fixed point methods, finite dimensional approximation methods, variational methods, maximum principles and regularity results. After an introductory chapter, which collects some fundamental notions and results from Lebesgue integration theory, distribution theory, Hölder and Sobolev spaces, weak topologies, variational analysis and spectral theory, in Chapter II the author deals with the fixed point theory and gives the proofs of some of the main results in this field starting with the classical Brower fixed point theorem, which ensures the existence of fixed points of continuous mappings from the unit ball of \(\mathbb{R}^n\) into itself. The proofs of the main generalizations of the Brower fixed point theorem, which are the Schauder theorem, the Leary-Schauder theorem and the Tychonoff theorem, are also given. Finally, the Schauder theorem is applied to solve the equation \(-\Delta u=f(u)\) in the Sobolev space \(H_0^1(\Omega)\), where \(f\in C(\mathbb{R})\cap L^\infty(\mathbb{R})\). Chapter III deals with the superposition operators on \(L^p\) spaces and Sobolev spaces and their continuity properties. In Chapter IV the author describes the Galerkin method for solving variational problems in a separable infinite dimensional normed space \(E\). Such a space is the closure of a countable union of an increasing sequence of spaces \(E_m\), with dim\((E_m)=m\). The Galerkin method to solve a variational problem in \(E\) consists in solving the same problems in \(E_m\) and then in showing that the approximated solution in \(E_m\) converges in \(E\), as \(m\rightarrow +\infty\), to a solution of the original problem. An application of the Galerkin method to the problem \(-\Delta u+\partial_1 u=f\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(\Omega\subset \mathbb{R}^N\) is a bounded domain and \(f\in H^{-1}(\Omega)\), is given. The first part of Chapter V is devoted to the maximum principle for classical and weak solutions of elliptic equations. Next, the author reviews some of the basic regularity results for weak solutions of elliptic problems, and then he describes the so called sub-supersolution method for solving nonlinear elliptic problems by combining the maximum principle and the regularity results. The basic notions of the calculus of variations are introduced in Chapter VI. In particular the role of the weak lower semicontinuity and the convexity in solving minimization problems is emphasized. Applications of the calculus of variations to quasilinear problems are given. In Chapter VII, the author gives the proofs of the Ekeland's variational principle and some of its main corollaries. Next, the author introduces the Palais-Smale condition, the deformation lemma and the concept of pseudo-gradient, emphasizing their roles in global minimization problems (jointly to the Ekeland's variational principle) as well as in the proofs of the minimax principle and the mountain pass theorem. In the last chapter, the author gives the definitions of monotone and pseudo-monotone operators and illustrates their correlations and continuity properties. Then, the author presents some classical existence and uniqueness results for variational inequalities involving monotone and pseudo-monotone operators. Finally, a special class of pseudo-monotone operators, the so called Leray-Lions operators, acting between the Sobolev spaces \(W_0^{1,p}(\Omega)\) and \(W^{-1,p}(\Omega)\), is introduced. This class of operators is used to exhibit an example of a pseudo-monotone operator which is not monotone. At the end of each chapter, a number of exercises and problems are proposed. The bibliography is not complete but this is justified by the very large number of books and articles on these topics that are worthy of mentioning.