On H?lder continuity of solutions for a class of nonlinear elliptic systems with \(p\) -growth via weighted integral techniques
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- 作者：Miroslav Bulí?ek ; Jens Frehse…
- 关键词：Nonlinear elliptic systems ; Regularity ; Noether equation ; H?lder continuity ; Liouville theorem ; 35J60 ; 49N60
- 刊名：Annali di Matematica Pura ed Applicata
- 出版年：2015
- 出版时间：August 2015
- 年：2015
- 卷：194
- 期：4
- 页码：1025-1069
- 全文大小：846 KB
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Jens Frehse (2)
Mark Steinhauer (3)
1. Faculty of Mathematics and Physics, Mathematical Institute, Charles University Sokolovská 83, 186?75?, Praha 8, Czech Republic
2. Department of applied analysis, Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115?, Bonn, Germany
3. Mathematical Institute, University of Koblenz-Landau, Campus Koblenz Universitstr. 1, 56070?, Koblenz, Germany - 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics - 出版者：Springer Berlin / Heidelberg
- ISSN：1618-1891
文摘
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.