Partial regularity result for non-autonomous elliptic systems with general growth

TL;DR

This paper establishes a partial regularity (Hölder continuity) result for weak solutions to non-autonomous elliptic systems with general growth conditions, even when the operator has weak regularity and the inhomogeneity is controlled.

Contribution

It proves a Hölder partial regularity theorem for solutions to complex elliptic systems with minimal regularity assumptions on the operator and controlled growth of the inhomogeneity.

Findings

Weak solutions are Hölder continuous on a subset of the domain.

The operator's weak regularity does not prevent partial regularity.

Controlled growth of the inhomogeneity allows for regularity results.

Abstract

In this paper we prove a H\"older partial regularity result for weak solutions $u:\Omega\to \mathbb{R}^N$, $N\geq 2$, to non-autonomous elliptic systems with general growth of the type: \begin{equation*} -\rm{div}\, a(x, u, Du)= b(x, u, Du) \quad \mbox{ in } \Omega. \end{equation*} The crucial point is that the operator $a$ satisfies very weak regularity properties and a general growth, while the inhomogeneity $b$ has a controllable growth.