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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The distance function from the boundary in a Minkowski space


Authors: Graziano Crasta and Annalisa Malusa
Journal: Trans. Amer. Math. Soc. 359 (2007), 5725-5759
MSC (2000): Primary 35A30; Secondary 26B05, 32F45, 35C05, 49L25, 58J60
Published electronically: July 3, 2007
MathSciNet review: 2336304
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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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Additional Information

Graziano Crasta
Affiliation: Dipartimento di Matematica “G. Castelnuovo”, Univ. di Roma I, P.le A. Moro 2 – 00185 Roma, Italy
Email: crasta@mat.uniroma1.it

Annalisa Malusa
Affiliation: Dipartimento di Matematica “G. Castelnuovo”, Univ. di Roma I, P.le A. Moro 2 – 00185 Roma, Italy
Email: malusa@mat.uniroma1.it

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04260-2
PII: S 0002-9947(07)04260-2
Keywords: Distance function, Minkowski structure, cut locus, Hamilton-Jacobi equations
Received by editor(s): May 31, 2005
Published electronically: July 3, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.