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2016 Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems
Miroslav Bulíček, Lars Diening, Sebastian Schwarzacher
Anal. PDE 9(5): 1115-1151 (2016). DOI: 10.2140/apde.2016.9.1115

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.

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Miroslav Bulíček. Lars Diening. Sebastian Schwarzacher. "Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems." Anal. PDE 9 (5) 1115 - 1151, 2016. https://doi.org/10.2140/apde.2016.9.1115

Information

Received: 21 October 2015; Revised: 11 February 2016; Accepted: 30 March 2016; Published: 2016
First available in Project Euclid: 12 December 2017

zbMATH: 1347.35117
MathSciNet: MR3531368
Digital Object Identifier: 10.2140/apde.2016.9.1115

Subjects:
Primary: 35A01, 35D99, 35J57, 35J60

Rights: Copyright © 2016 Mathematical Sciences Publishers

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