Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces
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| Format: | Preprint |
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2023
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| _version_ | 1866916612463919104 |
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| author | Björn, Anders Björn, Jana |
| author_facet | Björn, Anders Björn, Jana |
| contents | We study the condenser capacity $\mathrm{cap}_p(E,Ω)$ on \emph{unbounded} open sets $Ω$ in a proper connected metric space $X$ equipped with a locally doubling measure supporting a local $p$-Poincaré inequality, where $1<p<\infty$. Using a new definition of capacitary potentials, we show that $\mathrm{cap}_p$ is countably subadditive and that it is a Choquet capacity. We next obtain formulas for the capacity of superlevel sets for the capacitary potential. These are then used to show that $p$-harmonic Green functions exist in an unbounded domain $Ω$ if and only if either $X$ is $p$-hyperbolic or the Sobolev capacity $C_p(X\setminus Ω)>0$. As an application, we deduce new results for Perron solutions and boundary regularity for the Dirichlet boundary value problem for $p$-harmonic functions in unbounded open sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_05702 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces Björn, Anders Björn, Jana Analysis of PDEs Functional Analysis Primary: 31C45, Secondary: 30L99, 31C12, 31C15, 31E05, 35J08, 35J92, 46E36, 49Q20 We study the condenser capacity $\mathrm{cap}_p(E,Ω)$ on \emph{unbounded} open sets $Ω$ in a proper connected metric space $X$ equipped with a locally doubling measure supporting a local $p$-Poincaré inequality, where $1<p<\infty$. Using a new definition of capacitary potentials, we show that $\mathrm{cap}_p$ is countably subadditive and that it is a Choquet capacity. We next obtain formulas for the capacity of superlevel sets for the capacitary potential. These are then used to show that $p$-harmonic Green functions exist in an unbounded domain $Ω$ if and only if either $X$ is $p$-hyperbolic or the Sobolev capacity $C_p(X\setminus Ω)>0$. As an application, we deduce new results for Perron solutions and boundary regularity for the Dirichlet boundary value problem for $p$-harmonic functions in unbounded open sets. |
| title | Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces |
| topic | Analysis of PDEs Functional Analysis Primary: 31C45, Secondary: 30L99, 31C12, 31C15, 31E05, 35J08, 35J92, 46E36, 49Q20 |
| url | https://arxiv.org/abs/2310.05702 |