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.
Mots clés : viscosity solution, Hamilton-Jacobi equation, regularity, cut locus, conjugate locus, riemannian geometry
@article{COCV_2010__16_3_695_0,
author = {Castelpietra, Marco and Rifford, Ludovic},
title = {Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of {Hamilton-Jacobi} equations and applications in riemannian geometry},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {695--718},
publisher = {EDP-Sciences},
volume = {16},
number = {3},
year = {2010},
doi = {10.1051/cocv/2009020},
mrnumber = {2674633},
zbl = {1201.35087},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2009020/}
}
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 PY - 2010 SP - 695 EP - 718 VL - 16 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009020/ DO - 10.1051/cocv/2009020 LA - en ID - COCV_2010__16_3_695_0 ER -
%0 Journal Article %A Castelpietra, Marco %A Rifford, Ludovic %T Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in riemannian geometry %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 695-718 %V 16 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2009020/ %R 10.1051/cocv/2009020 %G en %F COCV_2010__16_3_695_0
Castelpietra, Marco; 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, Tome 16 (2010) no. 3, pp. 695-718. doi : 10.1051/cocv/2009020. http://www.numdam.org/articles/10.1051/cocv/2009020/
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