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

Michael G. Crandall, Hitoshi Ishii, Pierre Louis Lions

Research output: Contribution to journalArticlepeer-review

3245 Citations (Scopus)


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
Issue number1
Publication statusPublished - 1992
Externally publishedYes


  • 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


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