Rigidity for two-phase overdetermined problems in 𝕊n
Abstract
In this paper, we study rigidity phenomena associated with a class of overdetermined two-phase problems on the n-dimensional sphere 𝕊n. Specifically, we analyze the solutions of coupled elliptic equations defined on a domain Ω⊂ 𝕊n and its complement, subject to Dirichlet boundary conditions on the interface ∂Ω and a compatibility condition on the gradient. We utilize two different methods. The method of moving planes works when ∂Ω is contained inside an open hemisphere. The other method works when Ω is simply connected in 𝕊2. Both lead to the same conclusion. Namely, the existence of a solution to such overdetermined problems necessarily implies that Ω is a geodesic ball. In the latter, we prove this rigidity property for three distinct scenarios: a baseline problem with piecewise constant source terms, which is the two-phase version of the torsion problem; a generalization including a class of positive and regular nonlinearities; and finally an eigenvalue problem involving the first Dirichlet eigenvalue of the respective phases.
Related
- Rigidity results for overdetermined problems of p-Laplace equation with spatially dependent data
- Wulff Rigidity for Overdetermined Anisotropic k-Hessian Equations
- Serrin’s overdetermined problems on epigraphs
- Stability of Serrin-Type Overdetermined Problems for the p-Laplacian and Related Operators
- Overdetermined problems for sum Hessian equations in the space forms
BibTeX
@misc{bisterzo2026rigidity,
title = {{Rigidity for two-phase overdetermined problems in $\mathbbS^n$}},
author = {Andrea Bisterzo and Shigeru Sakaguchi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15103},
eprint = {2609.15103},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15103},
}
RIS
TY - UNPB
TI - Rigidity for two-phase overdetermined problems in 𝕊^n
AU - Andrea Bisterzo
AU - Shigeru Sakaguchi
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15103
UR - https://arxiv.org/abs/2609.15103
AB - In this paper, we study rigidity phenomena associated with a class of overdetermined two-phase problems on the n-dimensional sphere 𝕊^n. Specifically, we analyze the solutions of coupled elliptic equations defined on a domain Ω⊂ 𝕊^n and its complement, subject to Dirichlet boundary conditions on the interface ∂Ω and a compatibility condition on the gradient. We utilize two different methods. The method of moving planes works when ∂Ω is contained inside an open hemisphere. The other method works when Ω is simply connected in 𝕊^2. Both lead to the same conclusion. Namely, the existence of a solution to such overdetermined problems necessarily implies that Ω is a geodesic ball. In the latter, we prove this rigidity property for three distinct scenarios: a baseline problem with piecewise constant source terms, which is the two-phase version of the torsion problem; a generalization including a class of positive and regular nonlinearities; and finally an eigenvalue problem involving the first Dirichlet eigenvalue of the respective phases.
ER -