Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle

Abstract

We investigate positivity sets of nonnegative supersolutions of the fully nonlinear elliptic equations F ( x , u , D u , D 2 u ) = 0 F(x,u,Du,D^2u)=0 in Ω \Omega , where Ω \Omega is an open subset of R N \mathbb {R}^N , and the validity of the strong maximum principle for F ( x , u , D u , D 2 u ) = f F(x,u,Du,D^2u)=f in Ω \Omega , with f ∈ C ( Ω ) f\in \mathrm {C}(\Omega ) being nonpositive. We obtain geometric characterizations of positivity sets { x ∈ Ω : u ( x ) > 0 } \{x\in \Omega \,:\, u(x)>0\} of nonnegative supersolutions u u and establish the strong maximum principle under some geometric assumption on the set { x ∈ Ω : f ( x ) = 0 } \{x\in \Omega \,:\, f(x)=0\} .