Serrin's overdetermined theorem within Lipschitz domains
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915854810087424 |
|---|---|
| author | Dong, Hongjie Zhang, Yi Ru-Ya |
| author_facet | Dong, Hongjie Zhang, Yi Ru-Ya |
| contents | Let $Ω\subset\mathbb R^n$ be a Lipschitz domain. We prove that, $Ω$ satisfies the following Serrin-type overdetermined system
$$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\mathbb R^n\setminus Ω,\quad Δu=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*Ω} - \mathbf{1}_Ω\,dx,$$
in the weak sense if and only if $Ω$ is a ball. Here $\mathscr H^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure. Moreover, a generalization of our method in the anisotropic setting is discussed. Our approach offers an alternative proof to [15] in the case of Lipschitz domains, introducing a novel viewpoint to settle [18, Question 7.1]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05155 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Serrin's overdetermined theorem within Lipschitz domains Dong, Hongjie Zhang, Yi Ru-Ya Analysis of PDEs 35N25 Let $Ω\subset\mathbb R^n$ be a Lipschitz domain. We prove that, $Ω$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\mathbb R^n\setminus Ω,\quad Δu=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*Ω} - \mathbf{1}_Ω\,dx,$$ in the weak sense if and only if $Ω$ is a ball. Here $\mathscr H^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure. Moreover, a generalization of our method in the anisotropic setting is discussed. Our approach offers an alternative proof to [15] in the case of Lipschitz domains, introducing a novel viewpoint to settle [18, Question 7.1]. |
| title | Serrin's overdetermined theorem within Lipschitz domains |
| topic | Analysis of PDEs 35N25 |
| url | https://arxiv.org/abs/2509.05155 |