Partial regularity for degenerate systems of double phase type

TL;DR

This paper proves that solutions to certain degenerate elliptic systems with double-phase growth are mostly smooth, with irregularities confined to a measure-zero set, under specific growth and regularity conditions.

Contribution

It establishes partial regularity results for double-phase elliptic systems with new conditions linking growth exponents and coefficient regularity.

Findings

Gradient of solutions is locally Hölder continuous outside a measure-zero set.

Regularity holds when the ratio q/p is bounded by 1 + α/n.

Results extend understanding of regularity in degenerate systems.

Abstract

We study partial regularity for degenerate elliptic systems of double-phase type, where the growth function is given by $H(x,t)=t^p+a(x)t^q$ with $1<p\leq q$ and $a(x)$ a nonnegative $C^{0,\alpha}$-continuous function. Our main result proves that if $\frac{q}{p}\leq 1+\frac{\alpha}{n}$, the gradient of any weak solution is locally H\"older continuous, except on a set of measure zero.