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Existence, uniqueness
and optimal regularity results for very weak solutions to
nonlinear elliptic systems
Miroslav Bulíček, Lars Diening and Sebastian Schwarzacher |
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Vol. 9 (2016), No. 5, 1115–1151
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Abstract |
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We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are sufficient and in many cases also necessary for building such a theory. We provide a unified approach that leads qualitatively to the same theory as the one available for linear elliptic problems with continuous coefficients, e.g., the Poisson equation. The result is based on several novel tools that are of independent interest: local and global estimates for (non)linear elliptic systems in weighted Lebesgue spaces with Muckenhoupt weights, a generalization of the celebrated div-curl lemma for identification of a weak limit in border line spaces and the introduction of a Lipschitz approximation that is stable in weighted Sobolev spaces. |
Keywords
nonlinear elliptic systems, weighted estimates, existence,
uniqueness, very weak solution, monotone operator,
div-curl-biting lemma, weighted space, Muckenhoupt weights
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Mathematical Subject Classification 2010
Primary: 35D99, 35J57, 35J60, 35A01
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Milestones
Received: 21 October 2015
Revised: 11 February 2016
Accepted: 30 March 2016
Published: 29 July 2016
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Authors |
| Miroslav Bulíček | |
| Mathematical Institute, Faculty of
Mathematics and Physics Charles University in Prague Sokolovská 83 186 75 Prague Czech Republic |
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| Lars Diening | |
| Institut für Mathematik Universität Osnabrück Albrechtstraße 28a D-49076 Osnabrück Germany |
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| Sebastian Schwarzacher | |
| Department of Mathematical Analysis,
Faculty of Mathematics and Physics Charles University in Prague Sokolovská 83 186 75 Prague Czech Republic |
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