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Zbl 1201.35087
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. (English)
[J] ESAIM, Control Optim. Calc. Var. 16, No. 3, 695-718 (2010). ISSN 1292-8119; ISSN 1262-3377/e

Summary: 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 {\it J.-i. Itoh} and {\it M. Tanaka} [Trans. Am. Math. Soc. 353, No.~1, 21--40 (2001; Zbl 0971.53031)], is due to {\it Y. Li} and {\it L. Nirenberg} [Commun. Pure Appl. Math. 58, No.~1, 85--146 (2005; Zbl 1062.49021)]. 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 $C^4$-deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex.

MSC 2000:: *35F20 General theory of first order nonlinear PDE
49L25 Viscosity solutions
53C22 Geodesics

Keywords: viscosity solution; Hamilton-Jacobi equation; regularity; cut locus; conjugate locus; Riemannian geometry; Dirichlet-type Hamilton-Jacobi equation; locally semiconcave

Citations: Zbl 0971.53031; Zbl 1062.49021

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