Article
Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry
Publications Mathématiques de l'IHÉS, Tome 133 (2021), pp. 327-366

If U:[0,+∞[×M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation

∂ t U+H(x,∂ x U)=0,
where M is a not necessarily compact manifold, and H is a Tonelli Hamiltonian, we prove the set Σ(U), of points in ]0,+∞[×M where U is not differentiable, is locally contractible. Moreover, we study the homotopy type of Σ(U). We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

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DOI : 10.1007/s10240-021-00125-5 
@article{PMIHES_2021__133__327_0,
     author = {Cannarsa, Piermarco and Cheng, Wei and Fathi, Albert},
     title = {Singularities of solutions of time dependent {Hamilton-Jacobi} equations. {Applications} to {Riemannian} geometry},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {327--366},
     year = {2021},
     publisher = {Springer International Publishing},
     address = {Cham},
     volume = {133},
     doi = {10.1007/s10240-021-00125-5},
     zbl = {1473.35104},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-021-00125-5/}
}
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Cannarsa, Piermarco; Cheng, Wei; Fathi, Albert. Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry. Publications Mathématiques de l'IHÉS, Tome 133 (2021), pp. 327-366. doi: 10.1007/s10240-021-00125-5

[1.] Albano, P.; Cannarsa, P.; Nguyen, K. T.; Sinestrari, C. Singular gradient flow of the distance function and homotopy equivalence, Math. Ann., Volume 356 (2013), pp. 23-43 | MR | DOI | Zbl

[2.] Bardi, M.; Capuzzo-Dolcetta, I. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser, Boston, 1997 | DOI | Zbl

[3.] Barles, G. Solutions de viscosité des équations de Hamilton-Jacobi, Springer, Paris, 1994 | Zbl

[4.] Berger, M. A Panoramic View of Riemannian Geometry, Springer, Berlin, 2003 | Zbl | DOI

[5.] Bernard, P. Existence of C 1,1 critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. Éc. Norm. Supér., Volume 40 (2007), pp. 445-452 | DOI | Zbl | Numdam

[6.] Buttazzo, G.; Giaquinta, M.; Hildebrandt, S. One-Dimensional Variational Problems, 15, Oxford Univ. Press, New York, 1998 | Zbl

[7.] Cannarsa, P.; Peirone, R. Unbounded components of the singular set of the metric projection in 𝐑 n , Trans. Am. Math. Soc., Volume 353 (2001), pp. 4567-4581 | DOI | Zbl

[8.] Cannarsa, P.; Sinestrari, C. Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, 58, Birkhäuser, Boston, 2004 | Zbl | DOI

[9.] Cannarsa, P.; Cheng, W.; Fathi, A. On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation (Sur la topologie des singularités d’une solution de l’équation de Hamilton–Jacobi), C. R. Acad. Sci. Paris, Ser. I, Volume 355 (2017), pp. 176-180 | DOI | Zbl

[10.] Clarke, F. H. Methods of Dynamic and Nonsmooth Optimization, 57, SIAM, Philadelphia, 1989 | DOI | Zbl

[11.] Evans, L. C. Partial Differential Equations, 19, Am. Math. Soc., Providence, 1998 | Zbl

[12.] Fathi, A. Regularity of C 1 solutions of Hamilton-Jacobin equation, Ann. Fac. Sci. Toulouse Math., Volume 12 (2003), pp. 479-516 | MR | Numdam | Zbl | DOI

[13.] Fathi, A. Weak KAM Theorem in Lagrangian Dynamics, 2005 (Preliminary Version 10)

[14.] Fathi, A. Weak KAM from a PDE point of view: viscosity solutions of the Hamilton-Jacobi equation and Aubry set, Proc. R. Soc. Edinb., Sect. A, Volume 120 (2012), pp. 1193-1236 | MR | Zbl | DOI

[15.] A. Fathi, Viscosity solutions of the Hamilton-Jacobi equation on a noncompact manifold, 2020.

[16.] Fathi, A.; Figalli, A. Optimal transportation on non-compact manifolds, Isr. J. Math., Volume 175 (2010), pp. 1-59 | MR | DOI | Zbl

[17.] Fathi, A.; Figalli, A.; Rifford, L. On the Hausdorff dimension of the Mather quotient, Commun. Pure Appl. Math., Volume 62 (2009), pp. 445-500 | MR | DOI | Zbl

[18.] Gluck, H.; Singer, D. Scattering of geodesic fields, Ann. Math., Volume 108 (1978), pp. 347-372 | MR | DOI | Zbl

[19.] Ishii, H. A short introduction to viscosity solutions and the large time behavior of solutions of Hamilton–Jacobi equations, Y. Achdou et al., Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, 2074, Springer, Berlin, 2013 | Zbl

[20.] Klingenberg, W. P. A. Riemannian Geometry, 1, de Gruyter, Berlin, 1995 | DOI | Zbl

[21.] Lieutier, A. Any open bounded subset of 𝐑 k has the same homotopy type as its medial axis, Comput. Aided Des., Volume 36 (2004), pp. 1029-1046 | DOI

[22.] Zavidovique, M. Strict sub-solutions and Mañé potential in discrete weak KAM theory, Comment. Math. Helv., Volume 87 (2012), pp. 1-39 | MR | DOI | Zbl

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