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Strong maximum principle for semicontinuous viscosity solutions of nonlinear partial differential equations. (English) Zbl 0907.35008
Summary: We derive a strong maximum principle for upper semicontinuous viscosity subsolutions of fully nonlinear elliptic differential equations whose dependence on the spatial variables may be discontinuous. Our results improve previous related ones for linear and nonlinear equations because we weaken structural assumptions on the nonlinearities. Counterexamples show that our results are optimal. Moreover they are complemented by comparison and uniqueness results, in which a viscosity subsolution is compared with a piecewise classical supersolution. It is curious to note that existence of a piecewise classical solution to a fully nonlinear problem implies its uniqueness in the larger class of continuous viscosity solutions.

MSC:
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35J60Nonlinear elliptic equations
35B50Maximum principles (PDE)
35R05PDEs with discontinuous coefficients or data