Vol. 9, No. 5, 2016

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems

Miroslav Bulíček, Lars Diening and Sebastian Schwarzacher

Vol. 9 (2016), No. 5, 1115–1151
Abstract

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
Mathematical Subject Classification 2010
Primary: 35D99, 35J57, 35J60, 35A01
Milestones
Received: 21 October 2015
Revised: 11 February 2016
Accepted: 30 March 2016
Published: 29 July 2016
Authors
Miroslav Bulíček
Mathematical Institute, Faculty of Mathematics and Physics
Charles University in Prague
Sokolovská 83
186 75 Prague
Czech Republic
Lars Diening
Institut für Mathematik
Universität Osnabrück
Albrechtstraße 28a
D-49076 Osnabrück
Germany
Sebastian Schwarzacher
Department of Mathematical Analysis, Faculty of Mathematics and Physics
Charles University in Prague
Sokolovská 83
186 75 Prague
Czech Republic