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The maximum principle for semicontinuous functions. (English) Zbl 0723.35015
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:
35B50Maximum principles (PDE)
35J65Nonlinear boundary value problems for linear elliptic equations
35K60Nonlinear initial value problems for linear parabolic equations