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

G. Crasta·A. Malusa
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Abstract

We consider the minimization problem for an integral functional $J$, possibly non-convex and non-coercive 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.

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G. Crasta, A. Malusa. 2006-12-20. A sharp uniqueness result for a class of variational problems solved by a distance function. https://doi.org/10.1016/j.jde.2007.05.026

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