Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry

Marco Castelpietra; Ludovic Rifford

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 3, page 695-718
  • ISSN: 1292-8119

Abstract

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Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40], is due to Li and Nirenberg [Comm. Pure Appl. Math. 58 (2005) 85–146]. Finally, we give applications of our results in Riemannian geometry. Namely, we show that the distance function to the conjugate locus on a Riemannian manifold is locally semiconcave. Then, we show that if a Riemannian manifold is a C4-deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex.

How to cite

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Castelpietra, Marco, and Rifford, Ludovic. "Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 695-718. <http://eudml.org/doc/250736>.

@article{Castelpietra2010,
abstract = { Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40], is due to Li and Nirenberg [Comm. Pure Appl. Math. 58 (2005) 85–146]. Finally, we give applications of our results in Riemannian geometry. Namely, we show that the distance function to the conjugate locus on a Riemannian manifold is locally semiconcave. Then, we show that if a Riemannian manifold is a C4-deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex. },
author = {Castelpietra, Marco, Rifford, Ludovic},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Viscosity solution; Hamilton-Jacobi equation; regularity; cut locus; conjugate locus; Riemannian geometry; viscosity solution; Dirichlet-type Hamilton-Jacobi equation; locally semiconcave},
language = {eng},
month = {7},
number = {3},
pages = {695-718},
publisher = {EDP Sciences},
title = {Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry},
url = {http://eudml.org/doc/250736},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Castelpietra, Marco
AU - Rifford, Ludovic
TI - Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/7//
PB - EDP Sciences
VL - 16
IS - 3
SP - 695
EP - 718
AB - Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40], is due to Li and Nirenberg [Comm. Pure Appl. Math. 58 (2005) 85–146]. Finally, we give applications of our results in Riemannian geometry. Namely, we show that the distance function to the conjugate locus on a Riemannian manifold is locally semiconcave. Then, we show that if a Riemannian manifold is a C4-deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex.
LA - eng
KW - Viscosity solution; Hamilton-Jacobi equation; regularity; cut locus; conjugate locus; Riemannian geometry; viscosity solution; Dirichlet-type Hamilton-Jacobi equation; locally semiconcave
UR - http://eudml.org/doc/250736
ER -

References

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