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

TL;DR

This paper develops a comprehensive theory for very weak solutions to nonlinear elliptic systems, establishing existence, uniqueness, and optimal regularity under Uhlenbeck-type conditions, with novel analytical tools.

Contribution

It introduces a unified approach and new tools for analyzing very weak solutions to nonlinear elliptic boundary value problems, extending linear theory to nonlinear cases.

Findings

Existence and uniqueness of very weak solutions established.

Optimal regularity results obtained under structural assumptions.

Development of new estimates in weighted Lebesgue spaces and Lipschitz approximation techniques.

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 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.