Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems

By Miroslav Bul'ivcek, Lars Diening, Sebastian Schwarzacher. 2016

50 citations recorded by the source corpus.

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 that one available for linear elliptic problems with continuous coeffcients, 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/f2c0c324753a0fab63caba99eff67dbdff31ec31

Citations in the corpus

• Second-Order Two-Sided Estimates in Nonlinear Elliptic Problems (2017, unknown): “This includes the classics [17,27,28,33,37,38,43,44,58,60–62], and the newer advances [11,13,13,14,22,31,41].”