The maximum principle for semicontinuous functions

Michael G. Crandall, Hitoshi Ishii

Research output: Contribution to journalArticle

67 Citations (Scopus)


The result of calculus which states that at a maximum of a twice differentiable function the gradient vanishes and the matrix of second derivatives is nonpositive plays a significant role in the theory of elliptic and parabolic differential equations of second order, where it is used to establish many results for solutions of these equations. The theory of viscosity solutions of fully nonlinear degenerate elliptic and parabolic equations, which admits nondifferentiable functions as solutions of these equations, is now recognized to depend on a "maximum principle" for semicontinuous functions, which replaces the calculus result mentioned above. This work contains a more general statement of this form together with a simpler proof than were available heretofore.

Original languageEnglish
Pages (from-to)1001-1014
Number of pages14
JournalDifferential and Integral Equations
Issue number6
Publication statusPublished - 1990
Externally publishedYes


ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Crandall, M. G., & Ishii, H. (1990). The maximum principle for semicontinuous functions. Differential and Integral Equations, 3(6), 1001-1014.