发表机构
Rutgers University; University of Duisburg-Essen（罗格斯大学; 杜伊斯堡-埃森大学）
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Herrero和Velazquez关于Stefan问题径向解自由边界渐近性的猜想，通过势约化到常微分方程方法实现完全渐近刚性分类。
AI 中文摘要
我们研究经典两相和单相Stefan问题径向解的奇点。在一项引人注目的结果中，M.A. Herrero和J.J.L. Velazquez证明了存在一个解，其自由边界由$g(t)$描述，在维度$n\geq 3$下满足$$g(t)=c_n \sqrt{t} |\log t|^{\frac{-1}{n - 2}} (1 + o(1)), t\to 0,$$并且他们猜想这些渐近性在一般情况下成立。在此我们证明该猜想对所有径向解成立，这意味着在该类中完全渐近刚性。为了获得两相Stefan问题的这些精确渐近性，对于该问题频率公式和单调性公式是未知的，我们应用了arXiv:2608.18913中为Neumann Bernoulli问题引入的势约化到常微分方程方法。该方法的思想是自由表面的渐近形状由精确的常微分方程控制。我们在本文中能够将其应用于具有拟线性主部且在我们所考虑的范围内不是线性算子扰动的不同类型方程，这证实了该方法的通用性。
英文摘要
We investigate singularities of radial solutions of the classical two-phase and one-phase Stefan problem. In a remarkable result, M.A. Herrero and J.J.L. Velazquez show existence of a solution such that the free boundary described by $g(t)$ satisfies in dimension $n\geq 3$ $$g(t)=c_n \sqrt{t} |\log t|^{\frac{-1}{n - 2}} (1 + o(1)), t\to 0,$$ and they conjecture that these asymptotics hold in general. Here we prove the conjecture for all radial solutions, which means complete asymptotic rigidity in that class. In order to obtain these precise asymptotics for the two-phase Stefan problem, for which frequency formulas and monotonicity formulas are unknown, we apply the Potential Reduction to ODE method introduced in arXiv:2608.18913 for the Neumann Bernoulli problem. The philosophy of that method is that the asymptotic shape of the free surface is governed by a precise ordinary differential equation. The fact that we can apply it in this paper to an equation of different type with quasilinear principal part, which is in our regime not the perturbation of a linear operator, confirms the versatility of the method.
