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

We investigate positivity sets of nonnegative supersolutions of the fully nonlinear elliptic equations F(x,u,Du,D2u)=0F(x,u,Du,D^2u)=0 in Ω\Omega, where Ω\Omega is an open subset of RN\mathbb {R}^N, and the validity of the strong maximum principle for F(x,u,Du,D2u)=fF(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 uu and establish the strong maximum principle under some geometric assumption on the set {x∈Ω:f(x)=0}\{x\in \Omega \,:\, f(x)=0\}.