An anisotropic Serrin's problem in general domains
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866914375068024832 |
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| author | Figalli, Alessio Zhang, Yi Ru-Ya |
| author_facet | Figalli, Alessio Zhang, Yi Ru-Ya |
| contents | Serrin's symmetry theorem shows that the classical overdetermined torsion problem forces the domain to be a ball. Extending this rigidity statement to merely Lipschitz (and more generally rough) domains in the weak formulation has been a long-standing and challenging problem, recently resolved by the authors in [12].
In this paper we address the corresponding question in the anisotropic setting: Given a uniformly convex $C^{2,γ}$ anisotropy $H$, we study the overdetermined problem for the anisotropic Laplacian $Δ_H u={\rm div}\big(H(\nabla u)\,DH(\nabla u)\big)$ on a bounded indecomposable set of finite perimeter $Ω$. Assuming the Ahlfors--David regularity of $\partial^*Ω$ and a global $β$-number square-function bound (a weak uniform rectifiability hypothesis), we prove that a weak solution exists if and only if $Ω$ is a translate and dilation of the Wulff shape, in which case the solution is unique and explicit. In particular, the result applies to Lipschitz domains.
While our approach follows the rough-domain strategy of [12] at a high level, the key Laplacian-specific ingredients exploited there have no direct analog for $Δ_H$, necessitating the development of new ideas and techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06119 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An anisotropic Serrin's problem in general domains Figalli, Alessio Zhang, Yi Ru-Ya Analysis of PDEs 35N25, 35J62 Serrin's symmetry theorem shows that the classical overdetermined torsion problem forces the domain to be a ball. Extending this rigidity statement to merely Lipschitz (and more generally rough) domains in the weak formulation has been a long-standing and challenging problem, recently resolved by the authors in [12]. In this paper we address the corresponding question in the anisotropic setting: Given a uniformly convex $C^{2,γ}$ anisotropy $H$, we study the overdetermined problem for the anisotropic Laplacian $Δ_H u={\rm div}\big(H(\nabla u)\,DH(\nabla u)\big)$ on a bounded indecomposable set of finite perimeter $Ω$. Assuming the Ahlfors--David regularity of $\partial^*Ω$ and a global $β$-number square-function bound (a weak uniform rectifiability hypothesis), we prove that a weak solution exists if and only if $Ω$ is a translate and dilation of the Wulff shape, in which case the solution is unique and explicit. In particular, the result applies to Lipschitz domains. While our approach follows the rough-domain strategy of [12] at a high level, the key Laplacian-specific ingredients exploited there have no direct analog for $Δ_H$, necessitating the development of new ideas and techniques. |
| title | An anisotropic Serrin's problem in general domains |
| topic | Analysis of PDEs 35N25, 35J62 |
| url | https://arxiv.org/abs/2603.06119 |