Elements of nonlinear analysis.

*(English)*Zbl 0964.35002
Birkhäuser Advanced Texts. Basel: Birkhäuser. vi, 256 p. (2000).

The author presents some modern aspects of nonlinear analysis. The book consists of 13 chapters. In the first, an elementary theory of elasticity and a simple population model with diffusion are discussed as a motivation. The second completes the necessary standard material from analysis (distributions, integration on boundaries, elements of Sobolev spaces) so that the book is more or less selfcontained. Chapters 3-8 are devoted to elliptic problems, Chapters 9 and 10 deal with the calculus of variations and the last three chapters are concerned with parabolic problems.

The main exposition starts in Chapter 3 with classical linear problems (the Dirichlet problem, the Lax-Milgram theorem and applications). The next chapter is devoted to elements of elliptic variational inequalities (a generalization of the Lax-Milgram theorem, applications). Some general aspects and methods of nonlinear elliptic problems are explained in Chapter 5 (a compactness method, a monotonicity method, a generalization of variational inequalities, multivalued problems). A simple theory of regularity for nonlocal variational inequalities is presented in Chapter 6. Further, the questions of uniqueness and nonuniqueness of solutions to quasilinear and monotone problems is studied (Chapter 7). Chapter 8 is concerned with the finite element method for elliptic problems (an abstract setting, simple finite elements, interpolation error, convergence results, approximation of nonlinear problems). Some modern aspects of the calculus of variations are discussed in the subsequent two chapters. The classical approach to minimizers is explained but mainly the situations when the convexity assumption is dropped and no minimizer exists are under consideration. Linear and nonlinear parabolic problems are studied in Chapters 11 and 12 (the tools of functional analysis for parabolic problems are described, local and nonlocal problems are considered etc.) In the last chapter, the asymptotic behaviour of solutions to parabolic problems is discussed (the case of one and several stationary points, blow-up).

The main exposition starts in Chapter 3 with classical linear problems (the Dirichlet problem, the Lax-Milgram theorem and applications). The next chapter is devoted to elements of elliptic variational inequalities (a generalization of the Lax-Milgram theorem, applications). Some general aspects and methods of nonlinear elliptic problems are explained in Chapter 5 (a compactness method, a monotonicity method, a generalization of variational inequalities, multivalued problems). A simple theory of regularity for nonlocal variational inequalities is presented in Chapter 6. Further, the questions of uniqueness and nonuniqueness of solutions to quasilinear and monotone problems is studied (Chapter 7). Chapter 8 is concerned with the finite element method for elliptic problems (an abstract setting, simple finite elements, interpolation error, convergence results, approximation of nonlinear problems). Some modern aspects of the calculus of variations are discussed in the subsequent two chapters. The classical approach to minimizers is explained but mainly the situations when the convexity assumption is dropped and no minimizer exists are under consideration. Linear and nonlinear parabolic problems are studied in Chapters 11 and 12 (the tools of functional analysis for parabolic problems are described, local and nonlocal problems are considered etc.) In the last chapter, the asymptotic behaviour of solutions to parabolic problems is discussed (the case of one and several stationary points, blow-up).

Reviewer: M.Kučera (Praha)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

47Jxx | Equations and inequalities involving nonlinear operators |

35J85 | Unilateral problems; variational inequalities (elliptic type) (MSC2000) |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

49J53 | Set-valued and variational analysis |