$\mathbb{S}^n$ 中两相超定问题的刚性

中文摘要 AI 辅助

本文研究了$n$维球面$\mathbb{S}^n$上一类超定两相问题相关的刚性现象。具体而言，我们分析了定义在区域$\Omega \subset \mathbb{S}^n$及其补集上的耦合椭圆方程的解，这些解在界面$\partial\Omega$上满足Dirichlet边界条件，并在梯度上满足相容性条件。我们采用了两种不同的方法。当$\partial\Omega$包含在某个开半球内时，移动平面法适用。另一种方法适用于$\Omega$在$\mathbb{S}^2$中单连通的情形。两种方法均得出相同结论：即此类超定问题解的存在性必然意味着$\Omega$是一个测地球。在后一种方法中，我们在三种不同情形下证明了这一刚性性质：一个具有分段常数源项的基准问题，即扭转问题的两相版本；一个包含一类正且正则非线性项的推广；以及一个涉及各相第一Dirichlet特征值的特征值问题。

英文摘要

In this paper, we study rigidity phenomena associated with a class of overdetermined two-phase problems on the $n$-dimensional sphere $\mathbb{S}^n$. Specifically, we analyze the solutions of coupled elliptic equations defined on a domain $Ω\subset \mathbb{S}^n$ and its complement, subject to Dirichlet boundary conditions on the interface $\partialΩ$ and a compatibility condition on the gradient. We utilize two different methods. The method of moving planes works when $\partialΩ$ is contained inside an open hemisphere. The other method works when $Ω$ is simply connected in $\mathbb{S}^2$. 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.

发表机构

• Scuola Normale Superiore（比萨高等师范学院）
• Tohoku University（东北大学）

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