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Nonlinear elliptic and parabolic equations of second order. (Нелинейные ѐллиптические и параболические уравнения второго порядка.) (Russian) Zbl 0586.35002

Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoĭ Literatury. 376 p. R. 4.20 (1985).
The monograph is devoted to second order elliptic and parabolic equations which are arisen in many applications and in general theory of differential equations. The solvability of general nonlinear degenerate elliptic Bellman equations on whole space in function classes with bounded derivatives is largely investigated. Note that in earlier works the author used probabilistic methods for attacking the problem. In the present book only methods of differential equations are used. The author’s idea is to use estimates for maximum of solutions of parabolic equations with \(L_ p\) norms of right hand side. Related estimates are established by Alexandrov for elliptic equations with geometric considerations. An existence theory for equations including Bellman equations, Monge-Ampère equations and quasilinear equations under natural assumptions for coefficients is constructed.
The book is based on known facts of the theory of linear elliptic and parabolic equations described in the books of O. A. Ladyzhenskaya and N. N. Ural’tseva [Linear and quasilinear equations of elliptic type. 2nd revised ed. (Russian) Moskva: Nauka (1973; Zbl 0269.35029)] and O. A. Ladyzhenskaya, N. N. Ural’tseva and V. A. Solonnikov [Linear and quasilinear equations of parabolic type. (Russian) Moskva: Nauka (1968; Zbl 0164.12302)].
The book is clear written and will be a useful tool for specialists on nonlinear partial differential equations.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B45 A priori estimates in context of PDEs
35K10 Second-order parabolic equations
35J10 Schrödinger operator, Schrödinger equation
35J60 Nonlinear elliptic equations
35J96 Monge-Ampère equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations