${\mathcal{C}^{\alpha}}$ -regularity for a class of non-diagonal elliptic syst
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- 作者：Miroslav Bulí?ek (1)
Jens Frehse (2) - 关键词：35J60 ; 49N60
- 刊名：Calculus of Variations and Partial Differential Equations
- 出版年：2012
- 出版时间：4 - March 2012
- 年：2012
- 卷：43
- 期：3
- 页码：441-462
- 全文大小：328KB
- 参考文献：1. Bensoussan, A., Frehse, J.: Regularity results for nonlinear elliptic systems and applications. In: Applied Mathematical Sciences, vol. 151. Springer, Berlin (2002)
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Jens Frehse (2)
1. Faculty of Mathematics and Physics, Mathematical Institute, Charles University, Sokolovská 83, 186 775, Praha 8, Czech Republic
2. Department of Applied Analysis, Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany - ISSN：1432-0835
文摘
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.