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

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 $n\ge 3$.