*Published Paper*

**Inserted:** 7 mar 2008

**Journal:** J. Differential Equations

**Volume:** 243

**Pages:** 427-447

**Year:** 2007

**Abstract:**

We consider the minimization problem for an integral functional $J$, possibly nonconvex and noncoercive in $W^{1,1}_0(\Omega)$, where $\Omega\subset R^n$ is a bounded smooth set. We prove sufficient conditions in order to guarantee that a suitable Minkowski distance is a minimizer of $J$. The main result is a necessary and sufficient condition in order to have the uniqueness of the minimizer. We show some application to the uniqueness of solution of a system of PDEs of Monge-Kantorovich type arising in problems of mass transfer theory.