On a system of partial differential equations of Monge–Kantorovich type

We consider a system of PDEs of Monge–Kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors. The existence of a solution of such system (in a regular open domain Ω⊂Rn), whose construction is based on an asymmetric Minkowski distance from the boundary of Ω, was already established in [G. Crasta, A. Malusa, The distance function from the boundary in a Minkowski space, Trans. Amer. Math. Soc., submitted for publication]. In this paper we prove that this solution is essentially unique. A fundamental tool in our analysis is a new regularity result for an elliptic nonlinear equation in divergence form, which is of some interest by itself.