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几何障碍型问题临界点的$C^{1,α}$ -正则性

On $C^{1,α}$-regularity for critical points of a geometric obstacle-type problem

Sujin Khomrutai, Armin Schikorra

发表信息：
J. Math. Pures Appl. 2020
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摘要：

本文研究了向量映射$u: \mathbb{B}^2 \subset \mathbb{R}^2 \to \mathbb{R}^N$ \[ \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u \in \mathbb{R}^N \backslash \mathbb{B}^N(0)$}. \]上几何障碍问题的临界点，主要结果是对任何$α< 1$，都有$C^{1,α}$ -正则性.从技术上讲，我们拆分了映射$u=λv$，其中$v: \mathbb{B}^2 \to \mathbb{S}^{N-1}$是矢量分量，$λ= |u|$是测量到原点的距离的标量分量。当$v$满足一个加权调和映射方程，权值为$λ^2$时，$λ$解决了\[ \int_{\mathbb{B}^2} |\nabla λ|^2+λ^2 |\nabla v|^2, \quad \mbox{subject to $λ\geq 1$}. \]的障碍问题，其中$|\nabla v|^2 \in L^1(\mathbb{B}^2)$。然后，我们在$λ$和$v$的规律性的增加之间打乒乓，以最终获得$C^{1,α}$结果。

英文摘要：

We consider critical points of the geometric obstacle problem on vectorial maps $u: \mathbb{B}^2 \subset \mathbb{R}^2 \to \mathbb{R}^N$ \[ \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u \in \mathbb{R}^N \backslash \mathbb{B}^N(0)$}. \] Our main result is $C^{1,α}$-regularity for any $α< 1$. Technically, we split the map $u=λv$, where $v: \mathbb{B}^2 \to \mathbb{S}^{N-1}$ is the vectorial component and $λ= |u|$ the scalar component measuring the distance to the origin. While $v$ satisfies a weighted harmonic map equation with weight $λ^2$, $λ$ solves the obstacle problem for \[ \int_{\mathbb{B}^2} |\nabla λ|^2+λ^2 |\nabla v|^2, \quad \mbox{subject to $λ\geq 1$}. \] where $|\nabla v|^2 \in L^1(\mathbb{B}^2)$. We then play ping-pong between the increases in the regularity of $λ$ and $v$ to obtain finally the $C^{1,α}$-result.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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