Monotone operators in Banach space and nonlinear partial differential equations. (English) Zbl 0870.35004

Showalter, R. E.

Mathematical Surveys and Monographs. 49. Providence, RI: American Mathematical Society. xi, 278 p. (1997).

This is a new book concerning the theory of monotone operators and nonlinear semigroup theory. A great quality of this book is the large number of examples which are discussed to illustrate the power of the abstract theory in solving initial boundary value problems for partial differential equations.
The book is organized in four chapters and an appendix. The first chapter gives an overview of the subject in the case of linear problems in one space dimension. All the notions, methods and theoretical results are strongly motivated by various classical boundary value problems. The second chapter, entitled “Nonlinear Stationary Problems”, is devoted to the following subjects: Banach spaces, \(L^p\)-spaces, Sobolev spaces, elliptic boundary value problems, variational inequalities, convex functions, elliptic equations in \(L^1\). Chapter III, entitled “Nonlinear Evolution Problems”, is mainly concerned with the abstract theory of evolution equations, including degenerate equations, and of evolution variational inequalities. Chapter IV, entitled “Accretive Operators and Nonlinear Cauchy Problems”, includes the general theory of existence for Cauchy problems, including degenerate evolution equations. Finally, in the Appendix, the author discusses three applications: heat conduction, flow in porous media, and continuous mechanics.
The book is meant for advanced graduate students in mathematics or engineering science and for researchers in partial differential equations and related fields.

Reviewer: Gheorghe Moroşanu (Iaşi)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
47-02	Research exposition (monographs, survey articles) pertaining to operator theory
47H05	Monotone operators and generalizations
47H20	Semigroups of nonlinear operators

Keywords:

nonlinear stationary problems; nonlinear evolution problems; accretive operators and nonlinear Cauchy problems; initial boundary value problems