具有可变指数的双相函数最小值的部分正则性

Atsushi Tachikawa
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引用次数: 0

摘要

本文旨在研究具有可变指数的双相函数的最小化(\varvec{u:\Omega \subset \mathbb {R}^n \rightarrow \mathbb {R}^N}\ )的部分正则性:$$\begin{aligned}\varvec{int \left( \vert Du\vert _A^{p(x)} + a(x) { \vert }Du{ \vert }dx,} (end{aligned}$$其中 \(\varvec{{ \vert } \cdot { \vert }_A}\) 代表从正定充分连续张量场 \(\varvec{A:=\big (A_{\alpha \beta }^{ij} (x,u)\big )~~((x,u) \in \Omega \times \mathbb {R}^N)}\).我们证明,对于某个常数 \(\varvec{C^{1,\gamma }(\Omega _1;\mathbb {R}^N)}\) 和开放子集 \(\varvec{C^{1,\gamma }(\Omega _1\subset \Omega })),最小值 \(\varvec{u}\) 是在类 \(\varvec{C^{1,\gamma }(\Omega _1;\mathbb {R}^N)}\ 中的。我们还得到了一个关于 \(\varvec{\Omega \setminus \Omega _1}\) 的 Hausdorff 维度的估计值。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Partial regularity of minimizers for double phase functionals with variable exponents

The aim of this article is to study partial regularity of a minimizer \(\varvec{u:\Omega \subset \mathbb {R}^n \rightarrow \mathbb {R}^N}\) for a double phase functional with variable exponents:

$$\begin{aligned} \varvec{\int \left( \vert Du\vert _A^{p(x)} + a(x) { \vert } Du{ \vert } _A^{q(x)}\right) dx,} \end{aligned}$$

where \(\varvec{{ \vert } \cdot { \vert }_A}\) stands for the norm deduced from a positive definite sufficiently continuous tensor field \(\varvec{A:=\big (A_{\alpha \beta }^{ij} (x,u)\big )~~((x,u) \in \Omega \times \mathbb {R}^N)}\). We show that a minimizer \(\varvec{u}\) is in the class \(\varvec{C^{1,\gamma }(\Omega _1;\mathbb {R}^N)}\) for some constant \(\varvec{\gamma \in (0,1)}\) and open subset \(\varvec{\Omega _1 \subset \Omega }\). We obtain also an estimate for the Hausdorff dimension of \(\varvec{\Omega \setminus \Omega _1}\).

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