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) = 0 in Ω, where Ω is an
open subset of RN, and the validity of the strong maximum principle for
F(x, u,Du,D2u) = f in Ω, with f ∈ C(Ω) being nonpositive. We obtain
geometric characterizations of positivity sets {x ∈ Ω : u(x) > 0} of nonnegative
supersolutions u and establish the strong maximum principle under some
geometric assumption on the set {x ∈ Ω : f(x) = 0}.