The maximum principle for semicontinuous functions

Michael G. Crandall, Hitoshi Ishii

Research output: Contribution to journalArticle

67 Citations (Scopus)

Abstract

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
Volume3
Issue number6
Publication statusPublished - 1990
Externally publishedYes

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Maximum principle
Maximum Principle
Calculus
Degenerate Elliptic Equations
Elliptic Differential Equations
Parabolic Differential Equations
Degenerate Parabolic Equation
Fully Nonlinear
Viscosity Solutions
Second derivative
Differentiable
Vanish
Differential equations
Viscosity
Gradient
Derivatives

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.

The maximum principle for semicontinuous functions. / Crandall, Michael G.; Ishii, Hitoshi.

In: Differential and Integral Equations, Vol. 3, No. 6, 1990, p. 1001-1014.

Research output: Contribution to journalArticle

Crandall, MG & Ishii, H 1990, 'The maximum principle for semicontinuous functions', Differential and Integral Equations, vol. 3, no. 6, pp. 1001-1014.
Crandall, Michael G. ; Ishii, Hitoshi. / The maximum principle for semicontinuous functions. In: Differential and Integral Equations. 1990 ; Vol. 3, No. 6. pp. 1001-1014.
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