Abstract

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 $\ abla u\\in L^r_{loc}$ for every $r\\ge 1$.