四维Bernoulli问题的稳定性猜想

中文摘要 AI 辅助

我们证明了在$\mathbb R^4$中，单相Bernoulli问题的每一个全局经典稳定解都是一维的。特别地，若其自由边界非空，则该自由边界由一个或两个平行超平面组成。这解决了四维Bernoulli问题的稳定性猜想。作为推论，我们获得了$\mathbb R^4$中经典稳定解的局部Hessian和自由边界曲率的普适估计，以及$\mathbb R^4$中具有有限Morse指标的全局经典解的轴对称性。

英文摘要

We prove that every global classical stable solution to the one-phase Bernoulli problem in $\mathbb R^4$ is one-dimensional. In particular, if nonempty, its free boundary consists of one or two parallel hyperplanes. This settles the stability conjecture for the Bernoulli problem in dimension four. As consequences, we obtain universal local Hessian and free-boundary curvature estimates for classical stable solutions in $\mathbb R^4$, as well as axial symmetry of global classical solutions with finite Morse index in $\mathbb R^4$.

发表机构

• EPFL（洛桑联邦理工学院）
• ETH Zurich（苏黎世联邦理工学院）

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