Strong maximum principle for semicontinuous viscosity solutions of nonlinear partial differential equations

Abstract

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 [18] and nonlinear [22] 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.