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Symmetrization and applications. (English) Zbl 1110.35002

Series in Analysis 3. Hackensack, NJ: World Scientific (ISBN 981-256-733-X/hbk). xii, 148 p. (2006).
This monograph is intended to illustrate the major role played by symmetrization techniques in the qualitative analysis of the problems described by nonlinear ordinary or partial differential equations. The motivation provided by the author for such a study is related to the classical isoperimetric inequality \(L^2\geq 4\pi A\), where \(L\) stands for the perimeter and \(A\) for the enclosed area of a domain in the plane. The 3-dimensional equivalent of this celebrated inequality (whose roots go back to antiquity) is \(S^3\geq 36\pi V^2\), where \(S\) is the surface area of a body and \(V\) denotes its volume. Since equality holds for the disc, respectively for the sphere, it is natural to ask us about the role played by symmetry in such kind of problems.
The book is composed of five chapters. The first one introduces the main notions and properties used in the work: Schwarz symmetrization, co-area formula, relationship between the classical isoperimetric inequality and Sobolev’s inequality, etc. The Pólya-Szegö inequality is discussed in the second chapter of the book, in connection with the classical isoperimetric inequality. The author also proves elementary versions of the co-area formula which enable him to deduce the Pólya-Szegö inequality by means of simple partial differential equations techniques. The third chapter is mainly devoted to comparison theorems. The central result in this part of the book is Talenti’s theorem, which compares the solution of a second order elliptic boundary value problem with that of an appropriate symmetrized problem. Several applications are also provided by the author. The fourth chapter discusses the important role played by symmetrization arguments in the analysis of some eigenvalues problems. The main results of this chapter are the following: the Faber-Krahn inequality, the Szegö-Weinberger inequality, Chiti’s theorem, Rayleigh’s conjecture for clamped plates and the Payne-Pólya-Weinberger conjecture. This study is extended in the next chapter, with a particular attention for the proof of some Payne-Rayner type inequalities. There are also discussed various isoperimetric inequalities involving positive solutions of some nonlinear problems.
The book is a nice introduction in the fascinated theory of some problems issued from mathematical physics which are described by symmetry properties. The reviewer appreciates that this work is a good source for researchers in nonlinear analysis, as soon as for graduate students.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
58D19 Group actions and symmetry properties
58J70 Invariance and symmetry properties for PDEs on manifolds
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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