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

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