On certain anisotropic elliptic equations arising in congested optimal transport: Local gradient bounds

Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic elliptic equations, exhibiting a wide range of degeneracy. The model case we have in mind is the following: \[ \partial_x \left[(|u_{x}|-\delta_1)_+^{q-1}\, \frac{u_{x}}{|u_{x}|}\right]+\partial_y \left[(|u_{y}|-\delta_2)_+^{q-1}\, \frac{u_{y}}{|u_{y}|}\right]=f, \] for $2\le q<\infty$ and some non negative parameters $\delta_1,\delta_2$. Here $(\,\cdot\,)_+$ stands for the positive part. We prove that if $f\in L^\infty_{loc}$, then $\nabla u\in L^r_{loc}$ for every $r\ge 1$