The distance function from the boundary in a Minkowski space TeX Export

@misc{citeulike:1861602, 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 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.}, archivePrefix = {arXiv}, author = {Crasta, G. and Malusa, A.}, citeulike-article-id = {1861602}, citeulike-linkout-0 = {http://arxiv.org/abs/math/0612226}, citeulike-linkout-1 = {http://arxiv.org/pdf/math/0612226}, day = {9}, eprint = {math/0612226}, keywords = {convex, minkowski-space, pde, transport, viscosity-solution}, month = {Dec}, posted-at = {2008-02-17 12:53:06}, priority = {2}, title = {The distance function from the boundary in a Minkowski space}, url = {http://arxiv.org/abs/math/0612226}, year = {2006} }

TY - ELEC ID - citeulike:1861602 L3 - citeulike-article-id:1861602 N2 - 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. TI - The distance function from the boundary in a Minkowski space KW - convex KW - minkowski-space KW - pde KW - transport KW - viscosity-solution AU - Crasta, G AU - Malusa, A PY - 2006/Dec/09/ UR - http://arxiv.org/abs/math/0612226 ER -