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

Automated summary

In this study, we investigate the positivity sets of nonnegative supersolutions of fully nonlinear elliptic equations and establish the strong maximum principle under certain geometric assumptions. Geometric characterizations of the positivity sets of nonnegative supersolutions were obtained, contributing to a better understanding of the behavior of these solutions in open subsets of R-N.