arXiv2006
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 $Ω\subset\mathbb{R}^n$ of class $C^2$, let $d_Ω(x) := \inf\{d^M(x,y); y\in\partialΩ\}$ be the Minkowski distance of a point $x\inΩ$ from the boundary of $Ω$. We prove that a suitable extension of $d_Ω$ to $\mathbb{R}^n$ (which plays the röle of a signed Minkowski distance to $\partial Ω$) is of class $C^2$ in a tubular neighborhood of $\partial Ω$, and that $d_Ω$ is of class $C^2$ outside the cut locus of $\partialΩ$ (that is the closure of the set of points of non--differentiability of $d_Ω$ in $Ω$). In addition, we prove that the cut locus of $\partial Ω$ has Lebesgue measure zero, and that $Ω$ can be decomposed, up to this set of vanishing measure, into geodesics starting from $\partialΩ$ and going into $Ω$ 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 $x\in Ω$ outside the cut locus the pair $(p(x), d_Ω(x))$, where $p(x)$ denotes the (unique) projection of $x$ on $\partialΩ$, and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.