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Everywhere 𝒞α-estimates for a class of nonlinear elliptic systems with critical growth

Miroslav Bulíček
• Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
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• Other articles by this author:
/ Jens Frehse
• Institute for Applied Mathematics, Department of Applied Analysis, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
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• Other articles by this author:
/ Mark Steinhauer
Published Online: 2012-11-23 | DOI: https://doi.org/10.1515/acv-2012-0002

Abstract.

We obtain everywhere 𝒞α-regularity for vector solutions to a class of nonlinear elliptic systems whose principal part is the Euler operator to a variational integral $\int F\left(u,\nabla u\right)\phantom{\rule{0.166667em}{0ex}}dx$ with quadratic growth in $\nabla u$ and which satisfies a generalized splitting condition that cover the case $F\left(u,\nabla u\right):=\sum _{i}{Q}_{i},$ where ${Q}_{i}:={\sum }_{\alpha \beta }{A}_{i}^{\alpha \beta }\left(u,\nabla u\right)\nabla {u}^{\alpha }·\nabla {u}^{\beta }$, or the case $F\left(u,\nabla u\right):=\prod _{i}{\left(1+{Q}_{i}\right)}^{{\theta }_{i}}.$ A crucial assumption is the one-sided condition ${F}_{u}\left(u,\eta \right)·u\ge -K$ and related generalizations. In the elliptic case we obtain existence of 𝒞α-solutions. If the leading operator is not necessarily elliptic but coercive, possible minima are everywhere Hölder continuous and the same holds also for Noether solutions, i.e., extremals which are also stationary with respect to inner variations. In particular if ${A}^{\alpha \beta }\left(u,\nabla u\right)={A}^{\alpha \beta }\left(u\right)$, our result generalizes a result of Giaquinta and Giusti. The technique of our proof (using weighted norms and inhomogeneous hole-filling method) does not rely on ${L}^{\infty }$-a priori estimates for the solution.

MSC: 35J60; 49N60

Revised: 2012-10-12

Accepted: 2012-10-24

Published Online: 2012-11-23

Published in Print: 2014-04-01

Award identifier / Grant number: GAČR 201/09/917

Funding Source: MŠMT

Award identifier / Grant number: LC06052

Citation Information: Advances in Calculus of Variations, Volume 7, Issue 2, Pages 139–204, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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