Regularity results up to the boundary for minimizers of p(x)-energy with \(p(x)>1\)

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• 作者：Atsushi Tachikawa ; Kunihiro Usuba
• 关键词：Mathematics Subject Classification49N60 ; 35J50 ; 58E20
• 刊名：manuscripta mathematica
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：152
• 期：1-2
• 页码：127-151
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1785
• 卷排序：152

文摘

We show partial regularity up to the boundary \(\partial \varOmega \) of a bounded open set \(\varOmega \subset \mathbb {R}^m\) for minimizers u for p(x)-growth functionals of the following type $$\begin{aligned} {\mathcal A}(u)=\int _\varOmega \left( A^{\alpha \beta }_{ij}(x,u) D_{\alpha }u^i(x) D_{\beta }u^j(x)\right) ^{p(x)/2}dx, \end{aligned}$$assuming that \(A^{\alpha \beta }_{ij}(x,u)\) are bounded uniformly continuous functions satisfying Legendre condition and that p(x) is a Hölder continuous function with \(p(x)>1\). When \(A^{\alpha \beta }_{ij}(x,u)\) are given as \(A^{\alpha \beta }_{ij}(x,u)=g^{\alpha \beta }(x)G_{ij}(x,u)\), we can also prove that minimizers have no singular points on the boundary.Mathematics Subject Classification49N6035J5058E20Dedicated to Professor Mariano Giaquinta on his 70th birthday