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^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 n...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2021-01, Vol.374 (1), p.539-564
Main Authors: Birindelli, Isabeau, Galise, Giulio, Ishii, Hitoshi
Format: Article
Language:English
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Summary: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\}.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/8226