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manuscripta mathematica

, Volume 152, Issue 1–2, pp 127–151 | Cite as

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

  • Atsushi TachikawaEmail author
  • Kunihiro Usuba
Article

Abstract

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 Classification

49N60 35J50 58E20 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and TechnologyTokyo University of ScienceNodaJapan

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