Yoshida, Norio Oscillation theory of partial differential equations. (English) Zbl 1154.35001 Hackensack, NJ: World Scientific (ISBN 978-981-283-543-7/hbk). xi, 326 p. (2008). This is a book containing large material on oscillation theory for partial differential equations. The author contributed many results presented in this book. There is a well-developed theory for oscillatory properties of Sturm-Liouville equations, but these properties for partial differential equations are much less known.The book consists of 8 chapters.Chapter 1, Oscillation of elliptic equations. In this chapter oscillatory properties for linear and nonlinear elliptic equations are studied. Basic tools are various comparison theorems which reduce the problem to an oscillatory problem for a one-dimensional equation. Earlier [I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators. Translated from the Russian. Jerusalem: Israel Program for Scientific Translations; (London): Oldbourne Press. (1965; Zbl 0143.36505)] described the connection between the oscillatory properties of the solutions to the Schrödinger equation \(Lu:=[-\Delta+q(x)]u=0\) with the properties of the negative spectrum of \(L\). Some proofs, given in this book, are not quite self-contained: for example, the proof of Corollary 1.1.1 refers to Leighton’s oscillatory criterion, and this criterion is not formulated. There are results on the oscillatory properties of some inhomogeneous elliptic equations of the second order, and some oscillatory results for some homogeneous elliptic equations of higher orders. There is no information about topology of the nodal lines for the solutions of the Dirichlet problem for second order elliptic equations in a bounded domain.Chapter 2 deals with the oscillatory properties of linear parabolic equations. It is mentioned that there are no oscillatory results known for nonlinear parabolic equations.Chapter 3 deals with the oscillatory properties of hyperbolic equations, both linear and nonlinear. The results of Gazenave and Haraux, Kreith, the author and others are given.Chapter 4 deals with the oscillatory properties of the extensible beam and Timoshenko beam equations. These equations have been studied by many authors (Timoshenko and coauthors, Fitzgibbon, Narazaki, the author, and others, referenced in the book). The oscillatory properties of these equations have been studied by Feireisl and Hermann, Timoshenko and coauthors, and the author. Sufficient conditions for oscillatory properties are formulated in terms of the existence of a solution to some inequalities for a function of one variable.Chapter 5 deals with the oscillatory properties of the functional elliptic equations, that is, elliptic equations with a delay in the argument, e.g., \(\Delta u(x-\sigma)+p(x)u(x-\tau)=f(x)\). This Chapter is short, just six pages long. It contains a result due to Tramow, a result due to the author, and three open problems.Chapter 6 deals with the oscillatory properties of the functional parabolic equations. The results of many authors are reported in this chapter (Kitamura and Kusano, the author, Bainov and Minchev, Li, Lakshmikantham, Bainov and Simeonov, and others).Chapter 7 deals with the oscillatory properties of the functional hyperbolic equations. The results of Deng,Ge, and Wang, the author, and others are presented.Chapter 8 deals with the Picone identity and its applications. The Picone identity, given in Theorem 8.1 is too long to be presented in this review. It is used in a proof of a Sturmian-type comparison theorem.An extensive bibliography consists of 327 entries.This book can be used by graduate students studying partial differential equations. The students may be unhappy occasionally, because the book is not quite self-contained. This, however, is compensated by the large amount of information presented in this book. Reviewer: Alexander G. Ramm (Manhattan) Cited in 24 Documents MSC: 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs Keywords:linear hyperbolic equations; nonlinear hyperbolic equations; oscillation theory Citations:Zbl 0143.36505 PDFBibTeX XMLCite \textit{N. Yoshida}, Oscillation theory of partial differential equations. Hackensack, NJ: World Scientific (2008; Zbl 1154.35001)