Annali di Matematica Pura ed Applicata (1923 -)

, Volume 194, Issue 4, pp 1025–1069 | Cite as

On Hölder continuity of solutions for a class of nonlinear elliptic systems with \(p\)-growth via weighted integral techniques

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Abstract

We consider weak solutions of nonlinear elliptic systems in a \(W^{1,p}\)-setting which arise as Euler–Lagrange equations of certain variational integrals with pollution term, and we also consider minimizers of a variational problem. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the independent and the dependent variables. We impose new structural conditions on the nonlinearities which yield \(\fancyscript{C}^{\alpha }\)-regularity and \(\fancyscript{C}^{\alpha }\)-estimates for the solutions. These structure conditions cover variational integrals like \(\int F(\nabla u)\,\mathrm{d}x \) with potential \(F(\nabla u):=\tilde{F} (Q_1(\nabla u),\ldots , Q_N(\nabla u))\) and positive definite quadratic forms \(Q_i\) in \(\nabla u\) defined as \(Q_i(\nabla u)=\sum \nolimits _{\alpha \beta } a_i^{\alpha \beta } \nabla u^\alpha \cdot \nabla u^\beta \). A simple example consists in \({\tilde{F}}(\xi _1,\xi _2):= |\xi _1|^{\frac{p}{2}} + |\xi _2|^{\frac{p}{2}}\) or \(\tilde{F}(\xi _1,\xi _2):= |\xi _1|^{\frac{p}{4}}|\xi _2|^{\frac{p}{4}}.\) Since the quadratic forms \(Q_i\) need not to be linearly dependent, our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. As a by-product, we also prove a kind of Liouville theorem. As a new analytical tool, we use a weighted integral technique with singular weights in an \(L^p\)-setting for the proof and establish a weighted hole-filling inequality in a setting where Green-function techniques are not available.

Keywords

Nonlinear elliptic systems Regularity Noether equation Hölder continuity Liouville theorem 

Mathematics Subject Classification (2010)

35J60 49N60 

Notes

Acknowledgments

The authors thank the Collaborative Research Center (SFB) 611 and the Hausdorff Center for Mathematics for their support. The support of the project MORE (ERC-CZ project no. LL1202 financed by the Ministry of Education, Youth and Sports, Czech Republic) is also acknowledged. M. Bulíček is thankful to the Karel Janeček Endowment for Science and Research for its support. M. Bulíček is a researcher in the University Centre for Mathematical Modelling, Applied Analysis and Computational Mathematics (Math MAC) and a member of the Nečas Center for Mathematical Modeling.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Miroslav Bulíček
    • 1
  • Jens Frehse
    • 2
  • Mark Steinhauer
    • 3
  1. 1.Faculty of Mathematics and Physics, Mathematical InstituteCharles University Sokolovská 83Praha 8Czech Republic
  2. 2.Department of applied analysis, Institute for Applied MathematicsUniversity of BonnBonnGermany
  3. 3.Mathematical InstituteUniversity of Koblenz-LandauKoblenzGermany

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