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 assumption both on the partial map $x \mapsto A(x,\xi)$ and the datum $f$, we prove that the solutions are almost Lipschitz continuous, i.e. their gradients belong locally to $L^q$, for any finite exponent $q$. In turn, solutions are locally $\gamma$-Hölder continuous, for every $\gamma \in (0,1)$. The difficulty arising from the lack of an explicit second variation for the problem is overcome by testing the equation with a function proportional to a power of the finite difference quotient of the solution. To the best of our knowledge, this technique is used in this context for the first time. We also provide an example showing the sharpness of our result in the scale of Lebesgue spaces.