### 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 language | English |
---|---|

Pages (from-to) | 1001-1014 |

Number of pages | 14 |

Journal | Differential and Integral Equations |

Volume | 3 |

Issue number | 6 |

Publication status | Published - 1990 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Differential and Integral Equations*,

*3*(6), 1001-1014.

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

Research output: Contribution to journal › Article

*Differential and Integral Equations*, vol. 3, no. 6, pp. 1001-1014.

}

TY - JOUR

T1 - The maximum principle for semicontinuous functions

AU - Crandall, Michael G.

AU - Ishii, Hitoshi

PY - 1990

Y1 - 1990

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84972560830&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972560830&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84972560830

VL - 3

SP - 1001

EP - 1014

JO - Differential and Integral Equations

JF - Differential and Integral Equations

SN - 0893-4983

IS - 6

ER -