Partial continuity for nonlinear systems with nonstandard growth and discontinuous coefficients

TL;DR

This paper establishes new partial H"older continuity results for solutions to divergence form elliptic systems with discontinuous coefficients and nonstandard growth conditions, using advanced approximation methods.

Contribution

It introduces novel partial regularity results for elliptic systems with nonstandard growth and discontinuous coefficients, extending previous theories to more general settings.

Findings

Proves partial H"older continuity under VMO-discontinuous coefficients

Allows minimal log-H"older regularity on the exponent function

Recovers a local quantisation phenomenon for discontinuous coefficients

Abstract

We obtain new partial H\"older continuity results for solutions to divergence form elliptic systems with discontinuous coefficients, obeying $p(x)$-type nonstandard growth conditions. By an application of the method of $\mathcal{A}$-harmonic approximation, we are able to allow for both VMO-discontinuities in the coefficients, and the minimal $\log$-H\"older regularity assumption on the exponent function. In doing so, we recover a local version of the quantisation phenomenon characteristic continuous coefficient case.