$\mathbb{C}$-elliptic operators and $\mathrm{W}^{1,1}$-regularity for linear growth functionals

TL;DR

This paper establishes higher Sobolev regularity for minimizers of convex integral functionals involving linear differential operators, extending previous results to the broader class of $ abla$-like operators including trace-free symmetric gradients in higher dimensions.

Contribution

It generalizes the regularity theory from full and symmetric gradients to all $ abla$-like $ ext{C}$-elliptic operators, including trace-free symmetric gradients, in dimensions $n \, \geq \, 3$.

Findings

Proves higher Sobolev regularity for minimizers with $ abla$-like operators.

Extends regularity results to trace-free symmetric gradients.

Applicable in dimensions $n \geq 3$.

Abstract

In this paper we prove the higher Sobolev regularity of minimisers for convex integral functionals evaluated on linear differential operators of order one. This intends to generalise the already existing theory for the cases of full and symmetric gradients to the entire class of $\mathbb{C}$-elliptic operators therein including the trace-free symmetric gradient for dimension $n \geq 3$.