User’s guide to viscosity solutions of second order partial differential equations

1992/01/01 by Michael G. Crandall, Hitoshi Ishii, Pierre-Louis Lions · 36 citations
Mathematics · Engineering · #Fractional Differential Equations Solutions #Stability and Controllability of Differential Equations #Nonlinear Differential Equations Analysis

paper · doi:10.1090/s0273-0979-1992-00266-5

openalex publication_date 1992/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

The notion of viscosity solutions 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.

Citations

• Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations
• Two approximations of solutions of Hamilton-Jacobi equations
• Perron’s method for Hamilton-Jacobi equations
• Neumann type boundary conditions for Hamilton-Jacobi equations
• On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's

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• Hitoshi Ishii [wikipedia]