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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.04406 |
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| _version_ | 1866910357186936832 |
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| author | Figalli, Alessio Khademloo, Somayeh Kim, Sunghan Shahgholian, Henrik |
| author_facet | Figalli, Alessio Khademloo, Somayeh Kim, Sunghan Shahgholian, Henrik |
| contents | Given $Ω\subset \mathbb{R}^n$ with $n\geq 2$, $D\subset Ω$ open, and $u:Ω\to \mathbb{R}^m$, we study elliptic systems of the type $$ {\rm div} \big( ( A + (B- A)χ_D)\nabla u\big) = 0 \quad \text{in $Ω\cap B_1$,} $$ for some uniformly elliptic tensors $A$ and $B$ with Hölder continuous entries. We show that, given appropriate boundary data, the Lipschitz regularity of $u$ inside $B_1 \cap D$ is transmitted to $B_{1/2}\cap Ω$ up to the boundary of $Ω$. This corresponds to the boundary counterpart of the interior regularity results in Figalli-Kim-Shahgholian, Nonlinear Anal. 2022. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity near the fixed boundary for transmission systems Figalli, Alessio Khademloo, Somayeh Kim, Sunghan Shahgholian, Henrik Analysis of PDEs Given $Ω\subset \mathbb{R}^n$ with $n\geq 2$, $D\subset Ω$ open, and $u:Ω\to \mathbb{R}^m$, we study elliptic systems of the type $$ {\rm div} \big( ( A + (B- A)χ_D)\nabla u\big) = 0 \quad \text{in $Ω\cap B_1$,} $$ for some uniformly elliptic tensors $A$ and $B$ with Hölder continuous entries. We show that, given appropriate boundary data, the Lipschitz regularity of $u$ inside $B_1 \cap D$ is transmitted to $B_{1/2}\cap Ω$ up to the boundary of $Ω$. This corresponds to the boundary counterpart of the interior regularity results in Figalli-Kim-Shahgholian, Nonlinear Anal. 2022. |
| title | Regularity near the fixed boundary for transmission systems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.04406 |