EXISTENCE, UNIQUENESS AND OPTIMAL REGULARITY RESULTS FOR VERY WEAK SOLUTIONS TO NONLINEAR ELLIPTIC SYSTEMS

Autor(en): Bulicek, Miroslav
Diening, Lars 
Schwarzacher, Sebastian
Stichwörter: div-curl-biting lemma; EQUATIONS; existence; FLUIDS; Mathematics; Mathematics, Applied; monotone operator; Muckenhoupt weights; nonlinear elliptic systems; SOLENOIDAL LIPSCHITZ TRUNCATION; uniqueness; very weak solution; weighted estimates; WEIGHTED NORM INEQUALITIES; weighted space
Erscheinungsdatum: 2016
Herausgeber: MATHEMATICAL SCIENCE PUBL
Journal: ANALYSIS & PDE
Volumen: 9
Ausgabe: 5
Startseite: 1115
Seitenende: 1151
Zusammenfassung: 
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.
ISSN: 1948206X
DOI: 10.2140/apde.2016.9.1115

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