Inserted: 7 mar 2008
Journal: J. Differential Equations
We consider a system of PDEs of Monge-Kantorovich type arising from problems of shape optimization and models in granular matter theory. The existence of a solution of such system (in a regular open domain $\Omega\subset R^n$), whose construction is based on an asymmetric Minkowski distance from the boundary of $\Omega$, was already established. 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.