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Book review

Author(s) Krylov, N.V.
Title Nonlinear Elliptic and Parabolic Equations of the Second Order
Publisher D. Reidel Publishing Company
Year of publication 1987
Reviewed by C.I.Gheorghiu

The rough contents are as follows: Ch.1 Auxiliary Notions; Ch.2 Nonlinear Equations with Constant 'Coefficients' in the Whole Space; Ch.3 A Priori Estimates in L_p for Solutions of Nonlinear Elliptic and Parabolic Equations, Ch.4 A Priori Estimates in C^alpha for Solutions of Linear and Nonlinear Equations; Ch.5 A Priori Estimates in C^(2+alpha) for Solutions of Nonlinear Equations; Ch.6 Existence Theorems for Solutions of Nondegenerate Equations; Ch.7 Degenerate Nonlinear Equations in the Whole Space; and Ch.8 Degenerate Nonlinear Equations in a Domain. The book also contains the editor's preface, a preface of author, two appendices, the bibliography and a subject index.

The aim of the work is to build up a solvability theory covering the Bellman equation, the Monge-Ampère equation and the quasilinear equations under natural conditions on their coefficients. Of course, the author constructs a differential theory without probabilistic or geometrical arguments. An important feature of the most nonlinear operators involved is the convexity with respect to the second derivatives of the unknown function.

The book is essentially based on the important results of the author in the classical theory of linear and nonlinear partial differential equations. It becomes very usefull for those who are familiar with the results on the classical theory of linear second-order partial differential equations (the author considers these results as familiar).

The massive material managed, is well organized in precise and clear statements. The treatment is classic (no word about weak solution or weak formulation!) but the book remains a valuable issue.

Purchasing it is a good investiment, we think, for university libraries.

Eventually, a word about the fact that a lot of valuable results initially written in Russian are now brought to a larger circulation by such publications.