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Minimax solutions of first-order partial differential equations. (English. Russian original) Zbl 0878.35024
Russ. Math. Surv. 51, No. 2, 283-313 (1996); translation from Usp. Mat. Nauk 51, No. 2, 105-138 (1996).
This 31 pages survey paper, submitted in 1993, may be considered as an abridged version of the author’s already published monograph [Generalized Solutions of First-Order PDEs, Birkhäuser (1994; Zbl 0820.35003)] with, practically, the same introduction and contents. The article, as well as the book, proposes the study of first-order PDEs of the form: \[ F(x,u(x),Du(x))=0,\quad x\in G\subseteq \mathbb{R}^{n},\qquad u(x)=u_{0}(x),\quad x\in G_{0} \] in terms of the so called “minimax solutions”, as an alternative to the well known “viscosity solutions” introduced in l983 by M. G. Crandall and P. L. Lions [Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)]. Stemming from Krassovskij’s and the author’s previous results on differential games, the concept of minimax solution is proved to generalize the classical (differentiable) solution obtained by Cauchy’s method of characteristics and to coincide with the viscosity solutions in certain cases. Expressed in terms of a certain “differential inclusion of characteristics”, one of the several equivalent definitions gives a constructive characteristics of the minimax, hence of viscosity solutions. Giving in concise form the main ideas and results from the theory of minimax solutions, the present paper may prove to be a useful complement to the book, which, however, contains more detailed proofs, examples, and comments.

35F20 Nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35F25 Initial value problems for nonlinear first-order PDEs
35F30 Boundary value problems for nonlinear first-order PDEs
91A23 Differential games (aspects of game theory)
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