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: ∂x(|ux|-δ1)+q-1ux|ux|+∂y(|uy|-δ2)+q-1uy|uy|=f, for 2≤q<∞ and some non-negative parameters δ1,δ2. Here (·)+ stands for the positive part. We prove that if f∈L loc ∞, then ∇u∈L loc r for every r ≥ 1.

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 &lt;= q &lt; 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 &gt;= 1.