On the strong maximum principle for fully nonlinear degenerate elliptic equations

Abstract.

We prove a Strong Maximum Principle for semicontinuous viscosity subsolutions or supersolutions of fully nonlinear degenerate elliptic PDE's, which complements the results of [17]. Our assumptions and conclusions are different from those in [17], in particular our maximum principle implies the nonexistence of a dead core. We test the assumptions on several examples involving the p-Laplacian and the minimal surface operator, and they turn out to be sharp in all cases where the existence of a dead core is known. We can also cover equations that are singular for p = 0$ and very degenerate operators such as the $\infty$-Laplacian and some first order Hamilton-Jacobi operators.