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**Methods for analysis of nonlinear elliptic boundary value problems. Translated from the Russian by Dan D. Pascali. Translation edited by Simeon Ivanov.**
*(English)*
Zbl 0822.35001

Translations of Mathematical Monographs. 139. Providence, RI: American Mathematical Society (AMS). xi, 348 p. (1994).

The author describes the motivation of the book: “The theory of nonlinear elliptic equations of arbitrary order is nowadays one of the most actively developing branches of the theory of partial differential equations. The study of nonlinear second-order elliptic equations has a history of almost one century and the main trends of investigations, namely, the regularity of solutions and the solvability of boundary value problems, are assigned by Hilbert’s nineteenth and twentieth problems. The research of S. N. Bernstein, J. Leray, J. Schauder, C. B. Morrey, E. de Giorgi, J. Moser, O. A. Ladyzhenskaya, N. N. Ural’tseva, and other authors has contributed not only to the solution of Hilbert’s problems, but also to creating many methods playing fundamental roles both in the theory of differential equations and in related parts of mathematics.”

“Nevertheless, the rich experience accumulated in the study of second order equations proved inefficient in the study of nonlinear elliptic equations of arbitrary order. The fact is that, in many cases, the methods for obtaining a priori estimates (namely such estimates play a key role in the nonlinear theory) have turned out to be inapplicable to higher order equations. This led to the necessity of working out new techniques for nonlinear elliptic partial differential equations of arbitrary order. This monograph is dedicated to the presentation of these methods, contained to a considerable extent in journal papers.”

In ten chapters the author treats the following main topics:

– methods of monotone-like operators and their applications to nonlinear elliptic boundary value problems;

– topological methods (degree theory for generalized monotone mappings) for investigation of nonlinear problems;

– applications of the topological arguments to solvability and behavior of solutions of nonlinear elliptic boundary value problems;

– semilinear boundary value problems and solvability conditions in the resonance case;

– regularity of generalized solutions;

– behavior of solutions of the Dirichlet problem for quasilinear elliptic equations near nonsmooth boundary and in some perforated domains.

A great part of the results are due to the author and most of them are presented with detailed proofs. The references include about 160 items covering the developments up to 1982-85. This is a nice and useful book representing a systematic approach to nonlinear elliptic partial differential equations with several interesting examples and applications.

“Nevertheless, the rich experience accumulated in the study of second order equations proved inefficient in the study of nonlinear elliptic equations of arbitrary order. The fact is that, in many cases, the methods for obtaining a priori estimates (namely such estimates play a key role in the nonlinear theory) have turned out to be inapplicable to higher order equations. This led to the necessity of working out new techniques for nonlinear elliptic partial differential equations of arbitrary order. This monograph is dedicated to the presentation of these methods, contained to a considerable extent in journal papers.”

In ten chapters the author treats the following main topics:

– methods of monotone-like operators and their applications to nonlinear elliptic boundary value problems;

– topological methods (degree theory for generalized monotone mappings) for investigation of nonlinear problems;

– applications of the topological arguments to solvability and behavior of solutions of nonlinear elliptic boundary value problems;

– semilinear boundary value problems and solvability conditions in the resonance case;

– regularity of generalized solutions;

– behavior of solutions of the Dirichlet problem for quasilinear elliptic equations near nonsmooth boundary and in some perforated domains.

A great part of the results are due to the author and most of them are presented with detailed proofs. The references include about 160 items covering the developments up to 1982-85. This is a nice and useful book representing a systematic approach to nonlinear elliptic partial differential equations with several interesting examples and applications.

Reviewer: V.Mustonen (Oulu)

### MSC:

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

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35J40 | Boundary value problems for higher-order elliptic equations |

47H11 | Degree theory for nonlinear operators |

47H05 | Monotone operators and generalizations |