抛物型伯努利问题的奇点形成与自收缩子
Singularity formation and self-shrinkers for the parabolic Bernoulli problem
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中文总结 AI 辅助
该研究针对抛物型伯努利自由边界问题，引入I型爆破速率假设，分类径向自收缩子并分析其谱性质，证明切流唯一性，为理解该问题的奇点形成提供了关键理论结果。
中文摘要 AI 辅助
我们研究抛物型伯努利自由边界问题经典解的奇点形成及相关自相似剖面。受Ricci流和平均曲率流启发，我们对奇异自由边界点的爆破速率引入I型假设，在此假设下抛物型重标度是紧的，且每个切流都是椭圆型剖面方程的非平凡自收缩解。尽管不存在极小化结构，该证明仍建立了正集的收敛性，并在极限中恢复了伯努利条件。在次临界 regime 下，每个切流都是双平面。我们随后对所有径向自收缩子进行分类：除已知的球和外部剖面外，存在唯一的环形剖面。我们获得了其内外半径的维度一致界和精确渐近，包括其中点的定量向外偏移。这些精细估计在每个维度上完整分类了环形解的线性谱符号，由此得出球在环境对称下是动态稳定的，而环形是动态不稳定的，具有角量子数0、1、2、3的真实不稳定模式。最后，我们从紧径向剖面的谱非退化性出发，证明当一个切流是球或环形解时，该切流是唯一的；2≤n≤4000维下的一个临界谱符号通过区间算术的严格计算机辅助论证得到验证。
英文摘要
We study singularity formation for classical solutions of the parabolic Bernoulli free boundary problem and the associated self-similar profiles. Inspired by Ricci and mean curvature flows, we introduce a Type I assumption on the blow-up rate at a singular free boundary point, under which the parabolic rescalings are compact and every tangent flow is a non-trivial self-shrinking solution of the elliptic profile equation. The proof establishes convergence of the positivity sets and recovers the Bernoulli condition in the limit, despite the absence of a minimizing structure. In the subcritical regime, every tangent flow is a double plane. We next classify all radial self-shrinkers. Besides the known ball and exterior profiles, there is a unique annular profile. We obtain dimension-uniform bounds and sharp asymptotics for its inner and outer radii, including a quantitative outward bias of its midpoint. These delicate estimates yield a complete classification of the signs of the linear spectrum for the annular solution in every dimension. As a consequence, the ball is dynamically stable modulo ambient symmetries, whereas the annulus is dynamically unstable, with genuine unstable modes of angular degrees $0,1,2,3$. We finally prove, from the spectral nondegeneracy of the compact radial profiles, that whenever one tangent flow is a ball or an annular solution, the tangent flow is unique. One borderline spectral sign in dimensions $2\leq n \leq 4000$ is verified by a rigorous computer-assisted argument using interval arithmetic.
