Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces
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| Format: | Preprint |
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2024
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| author | Capogna, Luca Gibara, Ryan Korte, Riikka Shanmugalingam, Nageswari |
| author_facet | Capogna, Luca Gibara, Ryan Korte, Riikka Shanmugalingam, Nageswari |
| contents | We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional $p$-Laplace non-homogeneous equation $(-Δ_p)^su =f$, with $0<s<1$, $1<p<\infty$, for data $f$ satisfying a weighted $L^{p'}$ condition in a doubling metric measure space $(Z,d_Z,ν)$ that is possibly unbounded. Our approach is inspired by the work of Caffarelli and Silvestre \cite{CS} (see also Mol{č}anov and Ostrovski{ĭ} \cite{MO}), and extends the techniques developed in
\cite{CKKSS}, where the bounded case is studied. Unlike in \cite{EbGKSS}, we do not assume that $Z$ supports a Poincaré inequality. The proof is based on the well-posedness of the Neumann problem on a Gromov hyperbolic space $(X,d_X, μ)$ that arises as an hyperbolic filling of $Z$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_18883 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces Capogna, Luca Gibara, Ryan Korte, Riikka Shanmugalingam, Nageswari Analysis of PDEs Metric Geometry 30L15, 31E05, 35R11, 35B65, 46E35 We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional $p$-Laplace non-homogeneous equation $(-Δ_p)^su =f$, with $0<s<1$, $1<p<\infty$, for data $f$ satisfying a weighted $L^{p'}$ condition in a doubling metric measure space $(Z,d_Z,ν)$ that is possibly unbounded. Our approach is inspired by the work of Caffarelli and Silvestre \cite{CS} (see also Mol{č}anov and Ostrovski{ĭ} \cite{MO}), and extends the techniques developed in \cite{CKKSS}, where the bounded case is studied. Unlike in \cite{EbGKSS}, we do not assume that $Z$ supports a Poincaré inequality. The proof is based on the well-posedness of the Neumann problem on a Gromov hyperbolic space $(X,d_X, μ)$ that arises as an hyperbolic filling of $Z$. |
| title | Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces |
| topic | Analysis of PDEs Metric Geometry 30L15, 31E05, 35R11, 35B65, 46E35 |
| url | https://arxiv.org/abs/2410.18883 |