Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients

We are interested in the local higher integrability of solutions to elliptic equations with linear growth of the form $$-\text{ div}A(x,Du)=f(x). $$ Under a Besov regularity on the partial map $x \mapsto A(x,ξ)$ and a summability condition on the datum $f$ in the scale of Lebesgue spaces, we prove