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, Volume 16 (2010) no. 3, pp. 695-718.

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.

DOI: 10.1051/cocv/2009020
Classification: 35F20, 49L25, 53C22
Keywords: 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, Volume 16 (2010) no. 3, pp. 695-718. doi : 10.1051/cocv/2009020. http://www.numdam.org/articles/10.1051/cocv/2009020/

[1] R. Abraham and J.E. Marsden, Foundations of Mechanics. Benjamin, London (1978). | Zbl

[2] A.A. Agrachev, Geometry of optimal control problems and Hamiltonian systems, in Nonlinear and optimal control theory, Lectures Notes in Mathematics 1932, Springer, Berlin (2008) 1-59. | Zbl

[3] G. Alberti, L. Ambrosio and P. Cannarsa, On the singularities of convex functions. Manuscripta Math. 76 (1992) 421-435. | Zbl

[4] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques et Applications 17. Springer-Verlag (1994). | Zbl

[5] G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133-1148. | Zbl

[6] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Birkhauser, Boston (2004). | Zbl

[7] A. Cannas Da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics 1764. Springer-Verlag, Berlin (2001). | Zbl

[8] F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). | Zbl

[9] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178. Springer-Verlag, New York (1998). | Zbl

[10] A. Fathi, Weak KAM Theorem and Lagrangian Dynamics. Cambridge University Press (to appear).

[11] A. Figalli, L. Rifford and C. Villani, Continuity of optimal transport on Riemannian manifolds in presence of focalization. Preprint (2009).

[12] A. Figalli, L. Rifford and C. Villani, On the Ma-Trudinger-Wang curvature on surfaces. Preprint (2009). | Zbl

[13] A. Figalli, L. Rifford and C. Villani, Nearly round spheres look convex. Preprint (2009).

[14] A. Figalli, L. Rifford and C. Villani, On the stability of Ma-Trudinger-Wang curvature conditions. Comm. Pure Appl. Math. (to appear).

[15] H. Ishii, A simple direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of Eikonal type. Proc. Amer. Math. Soc. 100 (1987) 247-251. | Zbl

[16] J. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus. Trans. Amer. Math. Soc. 353 (2001) 21-40. | Zbl

[17] Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations. Comm. Pure Appl. Math. 58 (2005) 85-146. | Zbl

[18] P.-L. Lions, Generalized solutions of Hamilton-Jacobi equations. Pitman, Boston (1982). | Zbl

[19] G. Loeper and C. Villani, Regularity of optimal transport in curved geometry: the nonfocal case. Duke Math. Journal (to appear). | Zbl

[20] C. Pignotti, Rectifiability results for singular and conjugate points of optimal exit time problems. J. Math. An. Appl. 270 (2002) 681-708. | Zbl

[21] L. Rifford, A Morse-Sard theorem for the distance function on Riemannian manifolds. Manuscripta Math. 113 (2004) 251-265. | Zbl

[22] L. Rifford, On viscosity solutions of certain Hamilton-Jacobi equations: Regularity results and generalized Sard's Theorems. Comm. Partial Differ. Equ. 33 (2008) 517-559. | Zbl

[23] L. Rifford, Nonholonomic Variations: An Introduction to Subriemannian Geometry. Monograph (in preparation).

[24] T. Sakai, Riemannian geometry, Translations of Mathematical Monographs 149. American Mathematical Society, Providence, USA (1996). | Zbl

[25] C. Villani, Optimal transport, old and new, Grundlehren des mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin (2009). | Zbl

[26] L. Zajicek, On the points of multiplicity of monotone operators. Comment. Math. Univ. Carolinae 19 (1978) 179-189. | Zbl

Cited by Sources: