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Zbl 0723.35015
Crandall, Michael G.; Ishii, Hitoshi
The maximum principle for semicontinuous functions. (English)
[J] Differ. Integral Equ. 3, No.6, 1001-1014 (1990). ISSN 0893-4983

Summary: 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 admit 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.

MSC 2000:: *35B50 Maximum principles (PDE)
35J65 (Nonlinear) BVP for (non)linear elliptic equations
35K60 (Nonlinear) BVP for (non)linear parabolic equations

Keywords: viscosity solutions; fully nonlinear

Cited in: Zbl 0884.49012 Zbl 0759.53035

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