A sharp uniqueness result for a class of variational problems solved by a distance function

Abstract

We consider the minimization problem for an integral functional $J$, possibly non-convex and non-coercive in $W_0^{1,1}(\Omega )$, where $\Omega \subset {\mathbb{R}}^n$ 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.

17 pages

References in corpus (3)

• The distance function from the boundary in a Minkowski space
• On a system of partial differential equations of Monge-Kantorovich type
• A variational approach to the macroscopic electrodynamics of anisotropic hard superconductors

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• An existence result for the sandpile problem on flat tables with walls
• A nonhomogeneous boundary value problem in mass transfer theory
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