Crandall, Michael G.; Ishii, Hitoshi The maximum principle for semicontinuous functions. (English) Zbl 0723.35015 Differ. Integral Equ. 3, No. 6, 1001-1014 (1990). 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. Cited in 4 ReviewsCited in 42 Documents MSC: 35B50 Maximum principles in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations Keywords:viscosity solutions; fully nonlinear PDFBibTeX XMLCite \textit{M. G. Crandall} and \textit{H. Ishii}, Differ. Integral Equ. 3, No. 6, 1001--1014 (1990; Zbl 0723.35015)