发表机构
Scuola Normale Superiore; Università di Pisa（高等师范学院; 比萨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了单相自由边界问题中孤立奇点处爆破的唯一性，通过建立球面Weiss能量的无穷维Łojasiewicz不等式和epiperimetric不等式，完善了切锥唯一性的理论图景。
AI 中文摘要
我们证明了单相自由边界问题在每个奇点处爆破的唯一性，前提是其中一个爆破具有孤立奇点。该结果适用于Lipschitz平稳解，并且不需要对锥体施加任何可积性假设。从这个意义上说，我们的结果完善了单相问题中孤立奇点处切锥唯一性的图景。受Simon在极小曲面领域工作的启发，我们证明了球面Weiss能量的无穷维Łojasiewicz不等式。这为非线性极小化解导出了一个epiperimetric不等式，进而导致了唯一性结果。
英文摘要
We prove uniqueness of the blow-up at every singular point of the one-phase free boundary problem for which one blow-up has an isolated singularity. The result applies to Lipschitz stationary solutions and does not require any integrability assumption on the cone. In this sense, our result completes the picture for uniqueness of tangent cones at isolated singularities in the one-phase problem. Inspired by Simon's work in the minimal surface setting, we prove an infinite dimensional Łojasiewicz inequality for the spherical Weiss' energy. This yields an epiperimetric inequality for non-minimizing solutions, which leads to the uniqueness result.
