Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the $p$-Laplacian
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909663490998272 |
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| author | Bayrami, Masoud Fotouhi, Morteza Shahgholian, Henrik |
| author_facet | Bayrami, Masoud Fotouhi, Morteza Shahgholian, Henrik |
| contents | For a given constant $λ> 0$ and a bounded Lipschitz domain $D \subset \mathbb{R}^n$ ($n \geq 2$), we establish that almost-minimizers of the functional $$
J(\mathbf{v}; D) = \int_D \sum_{i=1}^{m} \left|\nabla v_i(x) \right|^p+ λχ_{\{\left|\mathbf{v} \right|>0\}} (x) \, dx, \qquad 1<p<\infty,
$$ where $\mathbf{v} = (v_1, \cdots, v_m)$, and $m \in \mathbb{N}$, exhibit optimal Lipschitz continuity in compact sets of $D$. Furthermore, assuming $p \geq 2$ and employing a distinctly different methodology, we tackle the issue of boundary Lipschitz regularity for $v$. This approach simultaneously yields alternative proof for the optimal local Lipschitz regularity for the interior case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_09073 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the $p$-Laplacian Bayrami, Masoud Fotouhi, Morteza Shahgholian, Henrik Analysis of PDEs 35R35, 35J60 For a given constant $λ> 0$ and a bounded Lipschitz domain $D \subset \mathbb{R}^n$ ($n \geq 2$), we establish that almost-minimizers of the functional $$ J(\mathbf{v}; D) = \int_D \sum_{i=1}^{m} \left|\nabla v_i(x) \right|^p+ λχ_{\{\left|\mathbf{v} \right|>0\}} (x) \, dx, \qquad 1<p<\infty, $$ where $\mathbf{v} = (v_1, \cdots, v_m)$, and $m \in \mathbb{N}$, exhibit optimal Lipschitz continuity in compact sets of $D$. Furthermore, assuming $p \geq 2$ and employing a distinctly different methodology, we tackle the issue of boundary Lipschitz regularity for $v$. This approach simultaneously yields alternative proof for the optimal local Lipschitz regularity for the interior case. |
| title | Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the $p$-Laplacian |
| topic | Analysis of PDEs 35R35, 35J60 |
| url | https://arxiv.org/abs/2311.09073 |