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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

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Volume 7, Issue 2


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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/ Jens Frehse
  • Institute for Applied Mathematics, Department of Applied Analysis, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
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/ Mark Steinhauer
Published Online: 2012-11-23 | DOI: https://doi.org/10.1515/acv-2012-0002


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 F(u,u)dx with quadratic growth in u and which satisfies a generalized splitting condition that cover the case F(u,u):=iQi, where Qi:=αβAiαβ(u,u)uα·uβ, or the case F(u,u):=i(1+Qi)θi. A crucial assumption is the one-sided condition Fu(u,η)·u-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 Aαβ(u,u)=Aαβ(u), 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-a priori estimates for the solution.

Keywords: Nonlinear elliptic systems; regularity; Noether equation; Hölder continuity

MSC: 35J60; 49N60

About the article

Received: 2012-04-23

Revised: 2012-10-12

Accepted: 2012-10-24

Published Online: 2012-11-23

Published in Print: 2014-04-01

Funding Source: Czech Science Foundation and the Collaborative Research Center (SFB) 611

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, DOI: https://doi.org/10.1515/acv-2012-0002.

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