On the Strong Maximum Principle for Fully Nonlinear Degenerate Elliptic Equations

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 diierent 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 1-Laplacian and some rst order Hamilton-Jacobi operators.