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

Michael G. Crandall, Hitoshi Ishii, Pierre Louis Lions

Research output: Contribution to journalArticle

2502 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)1-67
Number of pages67
JournalBulletin of the American Mathematical Society
Volume27
Issue number1
DOIs
Publication statusPublished - 1992
Externally publishedYes

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Viscosity Solutions
Second order differential equation
Partial differential equations
Partial differential equation
Viscosity
Fully Nonlinear
Continuous Dependence
Comparison Theorem
Uniqueness Theorem
Nonlinear Partial Differential Equations
Existence Theorem
Scalar
Theorem
Range of data
Framework

Keywords

  • Comparison theorems
  • Dynamic programming
  • Elliptic equations
  • Fully nonlinear equations
  • Generalized solutions
  • Hamilton-Jacobi equations
  • Maximum principles
  • Nonlinear boundary value problems
  • Parabolic equations
  • Partial differential equations
  • Perron’s method
  • Viscosity solutions

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

User’s guide to viscosity solutions of second order partial differential equations. / Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre Louis.

In: Bulletin of the American Mathematical Society, Vol. 27, No. 1, 1992, p. 1-67.

Research output: Contribution to journalArticle

Crandall, Michael G. ; Ishii, Hitoshi ; Lions, Pierre Louis. / User’s guide to viscosity solutions of second order partial differential equations. In: Bulletin of the American Mathematical Society. 1992 ; Vol. 27, No. 1. pp. 1-67.
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