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

Birindelli, Isabeau; Galise, Giulio; Ishii, Hitoshi

doi:10.1090/tran/8226

Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle
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Trans. Amer. Math. Soc. 374 (2021), 539-564 Request permission

Abstract:

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

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