Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisited. (English) Zbl 0644.35016

Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis

J. Math. Soc. Japan 39, 581-596 (1987).

This paper is the result of distilling the essential points in the proofs (up to 1986) of uniqueness of viscosity solutions of first order nonlinear partial differential equations. It pulls together the central ideas involved in the proofs since the breakthrough paper of M. G. Crandall and P. L. Lions [Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)]. The original result of Crandall and Lions naturally attracted many authors, including especially the authors of this paper, to try to push the ideas of viscosity solutions as far as possible. All of the major goals of p.d.e. theory emanate, using viscosity solutions, from uniqueness, viz., existence, regularity, and approximation. Since 1986 the ideas expounded in the present paper, of course with several new ones, have now permeated second order, fully nonlinear, partial differential equations.

Reviewer: E.Barron

MSC:

35F20	Nonlinear first-order PDEs
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

uniqueness; viscosity solutions; first order; existence; regularity; approximation