Gradient $L^q$ theory for a class of non-diagonal nonlinear elliptic systems

TL;DR

This paper develops a new $L^q$ regularity theory for solutions of certain nonlinear elliptic systems, extending the range of integrability results by leveraging asymptotic operator properties and the splitting condition.

Contribution

It introduces a novel approach that relies solely on the asymptotic operator and the splitting condition to significantly extend higher integrability results for nonlinear elliptic systems.

Findings

Established $L^q$ integrability for solutions with $q$ in $[p,p+2]$.

Extended higher integrability range beyond classical results for certain operators.

Provided new regularity results under minimal assumptions on the operator asymptotics.

Abstract

We consider a class of nonlinear non-diagonal elliptic systems with $p$-growth and establish the $L^q$-integrability for all $q\in [p,p+2]$ of any weak solution provided the corresponding right hand side belongs to the corresponding Lebesgue space and the involved elliptic operator asymptotically satisfies the $p$-uniform ellipticity, the so-called splitting condition and it is continuous with respect to the spatial variable. For operators satisfying the uniform $p$-ellipticity condition the higher integrability is known for $q\in[p,dp/(d-2)]$ and for operators having the so-called Uhlenbeck structure, the theory is valid for all $q\in [p,\infty)$. The key novelty of the paper is twofold. First, the statement uses only the information coming from the asymptotic operator and second, and more importantly, by using the splitting condition, we are able to extend the range of possible $q$'s significantly whenever $p<d-2$.