Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

4 Issues per year


IMPACT FACTOR 2017: 1.676

CiteScore 2017: 1.30

SCImago Journal Rank (SJR) 2017: 2.045
Source Normalized Impact per Paper (SNIP) 2017: 1.138

Mathematical Citation Quotient (MCQ) 2017: 1.15

Online
ISSN
1864-8266
See all formats and pricing
More options …
Volume 7, Issue 2

Issues

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jens Frehse
  • Institute for Applied Mathematics, Department of Applied Analysis, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ 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 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.

Export Citation

© 2014 by Walter de Gruyter Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Lisa Beck, Miroslav Bulíček, and Jens Frehse
Journal of Differential Equations, 2015, Volume 259, Number 11, Page 6528

Comments (0)

Please log in or register to comment.
Log in