Existence of distributional solutions to degenerate elliptic systems for locally integrable forcing

TL;DR

This paper establishes the existence of distributional solutions and regularity estimates for degenerate elliptic systems with locally integrable forcing, even when forcing terms are not in the dual of the energy space.

Contribution

It extends existence results to cases with locally integrable forcing that are not necessarily in the dual space of the energy space, using a novel combination of local energy estimates and truncation techniques.

Findings

Proves existence of distributional solutions under minimal forcing assumptions.

Provides maximal regularity estimates for solutions.

Extends previous results to unbounded Lipschitz domains.

Abstract

This paper presents an existence result and maximal regularity estimates for distributional solutions to degenerate/singular elliptic systems of $p$-Laplacian type with absorption and (prescribed) locally integrable forcing posed in (possibly unbounded) Lipschitz domains. In particular, the forcing terms may not belong to the dual space of an energy space, e.g., $W^{1,p}_{\rm loc}$, which is necessary for the existence of weak (or energy) solutions of class $W^{1,p}_{\rm loc}$. The method of a proof relies on both local energy estimates and a relative truncation technique developed by Bul\'{i}\v{c}ek and Schwarzacher (Calc. Var. PDEs in 2016), where the bounded domain case is studied for (globally) integrable forcing.