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\({\mathcal{C}^{\alpha}}\)-regularity for a class of non-diagonal elliptic systems with p-growth

  • Miroslav Bulíček
  • Jens Frehse
Article

Abstract

We consider weak solutions to nonlinear elliptic systems in a W 1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere \({\mathcal{C}^{\alpha}}\)-regularity and global \({\mathcal{C}^{\alpha}}\)-estimates for the solutions. These structure conditions cover variational integrals like \({\int F(\nabla u)\; dx}\) with potential \({F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}\) and positively definite quadratic forms in \({\nabla u}\) defined as \({Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}\). 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 Q i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L p -setting.

Mathematics Subject Classification (2000)

35J60 49N60 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Faculty of Mathematics and Physics, Mathematical InstituteCharles UniversityPraha 8Czech Republic
  2. 2.Department of Applied Analysis, Institute for Applied MathematicsUniversity of BonnBonnGermany

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