Calculus of Variations and Geometric Measure Theory

G. Crasta - A. Malusa

The distance function from the boundary in a Minkowski space.

created by malusa on 07 Mar 2008


Published Paper

Inserted: 7 mar 2008

Journal: Trans. Amer. Math. Soc.
Volume: 359
Pages: 5725-5759
Year: 2007


Let the space $R^n$ be endowed with a Minkowski structure $M$ (that is $M\colon \R^n \to [0,+\infty)$ 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 $\Omega\subset R^n$ of class $C^2$, let $d(x) := \inf\{d^M(x,y);\ y\in\partial\Omega\}$ be the Minkowski distance of a point $x\in\Omega$ from the boundary of $\Omega$. We prove that a suitable extension of $\dist$ to $R^n$ (which plays the röle of a signed Minkowski distance to $\partial \Omega$) is of class $C^2$ in a tubular neighborhood of $\partial \Omega$, and that $d$ 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 $\dist$ 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 $(p(x), d(x))$, 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.