The distance function from the boundary in a Minkowski space

Crasta, Graziano; Malusa, Annalisa

doi:10.1090/S0002-9947-07-04260-2

The distance function from the boundary in a Minkowski space
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by Graziano Crasta and Annalisa Malusa PDF

Trans. Amer. Math. Soc. 359 (2007), 5725-5759 Request permission

Abstract:

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 rôle 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 nondifferentiability 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.

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