Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle
Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle
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DOI：
10.1090/tran/8226
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发表时间：
2019-11
影响因子：
1.3
通讯作者：
I. Birindelli;G. Galise;H. Ishii
中科院分区：
文献类型：
--
作者：
I. Birindelli;G. Galise;H. Ishii
我们研究 Ω \Omega 中完全非线性椭圆方程 F ( x , u , Du , D 2 u ) = 0 F(x,u,Du,D^2u)=0 的非负超解的正集，其中 Ω \Omega 是 R N \mathbb {R}^N 的开子集，以及 F ( x , u , Du , D 2 ) 的强极大值原理的有效性u ) = f F(x,u,Du,D^2u)=f in Ω \Omega ，其中 f ∈ C ( Ω ) f\in \mathrm {C}(\Omega ) 为非正数。我们获得了非负超解 u u 的正性集合 { x ε Ω : u ( x ) > 0 } \{x\in \Omega \,:\, u(x)>0\} 的几何表征，并在集合 { x ε Ω : f ( x ) = 0 } \{x\in \Omega \,:\, f(x)=0\} 上的某些几何假设下建立了强极大值原理。
We investigate positivity sets of nonnegative supersolutions of the fully nonlinear elliptic equations F ( x , u , D u , D 2 u ) = 0 F(x,u,Du,D^2u)=0 in Ω \Omega , where Ω \Omega is an open subset of R N \mathbb {R}^N , and the validity of the strong maximum principle for F ( x , u , D u , D 2 u ) = f F(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 u u and establish the strong maximum principle under some geometric assumption on the set { x ∈ Ω : f ( x ) = 0 } \{x\in \Omega \,:\, f(x)=0\} .