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

Summary 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: $${\partial }_x[(|u_x|-{\delta }_1{)}_{+}^{q-1}\frac{u_x}{|u_x|}]+{\partial }_y[(|u_y|-{\delta }_2{)}_{+}^{q-1}\frac{u_y}{|u_y|}]=f,$$ 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 $r\ge 1$.