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 derivative(x) [(vertical bar u(x)vertical bar - delta(1))(+)(q-1) u(x)/vertical bar u(x vertical bar)] + partial derivative(y) [(vertical bar u(y)vertical bar - delta(2))(+)(q-1) u(y)/vertical bar u(y vertical bar)] = f, for 2 <= q < infinity and non-negative delta(1), delta(2). Here (center dot)(+) stands for the positive part. We prove that if f is an element of L-loc(infinity), then del(u) is an element of L-loc(r) for every r >= 1.