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

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.Comment: 17 page

A C1C^1 regularity result for the inhomogeneous normalized infinity Laplacian

We prove that the unique solution to the Dirichlet problem with constant source term for the inhomogeneous normalized infinity Laplacian on a convex domain of $\mathbb{R}^N$ is of class $C^1$. The result is obtained by showing as an intermediate step the power-concavity (of exponent $1/2$) of the solution.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1410.611

The distance function from the boundary in a Minkowski space

Let the space $\mathbb{R}^n$ be endowed with a Minkowski structure $M$ (that is $M\colon \mathbb{R}^n \to [0,+\infty)$ is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class $C^2$), and let $d^M(x,y)$ be the (asymmetric) distance associated to $M$. Given an open domain $\Omega\subset\mathbb{R}^n$ of class $C^2$, let $d_{\Omega}(x) := \inf\{d^M(x,y); y\in\partial\Omega\}$ be the Minkowski distance of a point $x\in\Omega$ from the boundary of $\Omega$. We prove that a suitable extension of $d_{\Omega}$ to $\mathbb{R}^n$ (which plays the r\"ole of a signed Minkowski distance to $\partial \Omega$) is of class $C^2$ in a tubular neighborhood of $\partial \Omega$, and that $d_{\Omega}$ is of class $C^2$ outside the cut locus of $\partial\Omega$ (that is the closure of the set of points of non--differentiability of $d_{\Omega}$ in $\Omega$). In addition, we prove that the cut locus of $\partial \Omega$ has Lebesgue measure zero, and that $\Omega$ can be decomposed, up to this set of vanishing measure, into geodesics starting from $\partial\Omega$ and going into $\Omega$ along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point $x\in \Omega$ outside the cut locus the pair $(p(x), d_{\Omega}(x))$, where $p(x)$ denotes the (unique) projection of $x$ on $\partial\Omega$, and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.Comment: 34 page

Characterization of stadium-like domains via boundary value problems for the infinity Laplacian

We give a complete characterization, as "stadium-like domains", of convex subsets $\Omega$ of $\mathbb{R}^n$ where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity Laplacian or its normalized version. In case of the not-normalized operator, our results extend those obtained in a previous work, where the problem was solved under some geometrical restrictions on $\Omega$. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in $\mathbb{R}^n$ where the solution to a Dirichlet problem is of class $C^{1,1} (\Omega)$.Comment: 21 page