Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929580111036416 |
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| author | Mourgoglou, Mihalis Tolsa, Xavier |
| author_facet | Mourgoglou, Mihalis Tolsa, Xavier |
| contents | Let $Ω\subset \mathbb{R}^{n+1}$ be a bounded chord-arc domain, let $\mathcal L=-{\rm div} A\nabla$ be an elliptic operator in $Ω$ associated with a matrix $A$ having Dini mean oscillation coefficients, and let $1<p\leq 2$. In this paper we show that if the regularity problem for $\mathcal L$ is solvable in $L^q$ for some $q>p$ in $Ω$, $\partial Ω$ supports a weak $p$-Poincaré inequality, and $Ω$ has very big pieces of superdomains for which the Neumann problem for $\mathcal L$ is solvable uniformly in $L^q$, then the Neumann problem for $\mathcal L$ is solvable in $L^p$ in $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20385 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains Mourgoglou, Mihalis Tolsa, Xavier Analysis of PDEs 42B37, 35J25 Let $Ω\subset \mathbb{R}^{n+1}$ be a bounded chord-arc domain, let $\mathcal L=-{\rm div} A\nabla$ be an elliptic operator in $Ω$ associated with a matrix $A$ having Dini mean oscillation coefficients, and let $1<p\leq 2$. In this paper we show that if the regularity problem for $\mathcal L$ is solvable in $L^q$ for some $q>p$ in $Ω$, $\partial Ω$ supports a weak $p$-Poincaré inequality, and $Ω$ has very big pieces of superdomains for which the Neumann problem for $\mathcal L$ is solvable uniformly in $L^q$, then the Neumann problem for $\mathcal L$ is solvable in $L^p$ in $Ω$. |
| title | Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains |
| topic | Analysis of PDEs 42B37, 35J25 |
| url | https://arxiv.org/abs/2407.20385 |