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User’s guide to viscosity solutions of second order partial differential equations. (English) Zbl 0755.35015
Summary: The notion of viscosity solution of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.

MSC:
35D05Existence of generalized solutions of PDE (MSC2000)
35J60Nonlinear elliptic equations
35K55Nonlinear parabolic equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B50Maximum principles (PDE)
35B25Singular perturbations (PDE)
35K65Parabolic equations of degenerate type
35F20General theory of first order nonlinear PDE
49L25Viscosity solutions (infinite-dimensional problems)
35J25Second order elliptic equations, boundary value problems
35K20Second order parabolic equations, initial boundary value problems
35K15Second order parabolic equations, initial value problems