C ∞ regularity in semilinear free boundary problems.

IF 1.4 2区数学 Q1 MATHEMATICS

Mathematische Annalen Pub Date : 2025-01-01 Epub Date: 2025-05-07 DOI:10.1007/s00208-025-03135-4

Daniel Restrepo, Xavier Ros-Oton

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引用次数: 0

Abstract

We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $Δ u = u^{γ - 1},$ with $γ \in (0, 1) .$ Our main results imply that, once free boundaries are $C^{1, α},$ then they are $C^{\infty} .$ In addition $u / d^{\frac{2}{2 - γ}}$ and $u^{\frac{2 - γ}{2}}$ are $C^{\infty}$ too. In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials $- Δ v = κ v / d^{2}$ in $Ω,$ where d is the distance to the boundary and $κ \leq \frac{1}{4} .$ Interestingly, we need to include even the critical constant $κ = \frac{1}{4},$ which corresponds to $γ = \frac{2}{3} .$

查看原文本刊更多论文

半线性自由边界问题的C∞正则性。

研究了Alt-Phillips问题Δ u = u γ - 1，且γ∈(0,1)的解和自由边界的高正则性。我们的主要结果表明，一旦自由边界是c1， α，那么它们就是C∞。另外u / d22 - γ和u 2 - γ 2也是C∞。为了实现这一点，我们需要为具有边界奇异Hardy势的线性方程的解建立精细的正则性估计- Δ v = κ v / d2在Ω中，其中d是到边界的距离，并且κ≤14。有趣的是，我们甚至需要包括临界常数κ = 14，它对应于γ = 23。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematische Annalen 数学-数学

CiteScore

2.90

自引率

7.10%

发文量

181

审稿时长

4-8 weeks

期刊介绍： Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.