Lipschitz regularity for almost minimizers of a one-phase $p$-Bernoulli-type functional in Carnot Groups of step two
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| Format: | Preprint |
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2024
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| _version_ | 1866910507573706752 |
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| author | Ferrari, Fausto Merlino, Enzo Maria |
| author_facet | Ferrari, Fausto Merlino, Enzo Maria |
| contents | In this paper, in a Carnot group $\mathbb{G}$ of step $2$ and homogeneous dimension $Q$, we prove that almost minimizers of the (horizontal) one-phase $p$-Bernoulli-type functional
$$ J_p(u,Ω):=\int_Ω\Big( |\nabla_{\mathbb{G}} u(x)|^p+χ_{\{u>0\}}(x)\Big)\,dx$$ whenever $p>p^\#:=\frac{2Q}{Q+2}$, are locally Lipschitz continuous with respect Carnot-Carathéodory distance on $\mathbb{G}$. This implies an Hölder continuous regularity from an Euclidean point of view. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_00084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lipschitz regularity for almost minimizers of a one-phase $p$-Bernoulli-type functional in Carnot Groups of step two Ferrari, Fausto Merlino, Enzo Maria Analysis of PDEs 35R35, 35R03 In this paper, in a Carnot group $\mathbb{G}$ of step $2$ and homogeneous dimension $Q$, we prove that almost minimizers of the (horizontal) one-phase $p$-Bernoulli-type functional $$ J_p(u,Ω):=\int_Ω\Big( |\nabla_{\mathbb{G}} u(x)|^p+χ_{\{u>0\}}(x)\Big)\,dx$$ whenever $p>p^\#:=\frac{2Q}{Q+2}$, are locally Lipschitz continuous with respect Carnot-Carathéodory distance on $\mathbb{G}$. This implies an Hölder continuous regularity from an Euclidean point of view. |
| title | Lipschitz regularity for almost minimizers of a one-phase $p$-Bernoulli-type functional in Carnot Groups of step two |
| topic | Analysis of PDEs 35R35, 35R03 |
| url | https://arxiv.org/abs/2407.00084 |