Revista Matemática Iberoamericana

Almost classical solutions of Hamilton-Jacobi equations

Robert Deville and Jesús A. Jaramillo

Full-text: Open access


We study the existence of everywhere differentiable functions which are almost everywhere solutions of quite general Hamilton-Jacobi equations on open subsets of $\mathbb R^d$ or on $d$-dimensional manifolds whenever $d\geq 2$. In particular, when $M$ is a Riemannian manifold, we prove the existence of a differentiable function $u$ on $M$ which satisfies the Eikonal equation $\Vert \nabla u(x) \Vert_{x}=1$ almost everywhere on $M$.

Article information

Rev. Mat. Iberoamericana Volume 24, Number 3 (2008), 989-1010.

First available in Project Euclid: 9 December 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26B05: Continuity and differentiation questions 35B65: Smoothness and regularity of solutions 58J32: Boundary value problems on manifolds

Hamilton-Jacobi equations eikonal equation on manifolds almost everywhere solutions


Deville , Robert; Jaramillo , Jesús A. Almost classical solutions of Hamilton-Jacobi equations. Rev. Mat. Iberoamericana 24 (2008), no. 3, 989--1010.

Export citation


  • Azagra, D., Ferrera, J. and López-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220 (2005), no. 2, 304-361.
  • Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques el Applications 17. Springer-Verlag, Paris, 1994.
  • Benameur, M. T.: Triangulations and the stability theorem for foliations. Pacific J. Math. 179 (1997), no. 2, 221-239.
  • Buczolich, Z.: Solution to the gradient problem of C. E. Weil. Rev. Mat. Iberoamericana 21 (2005), no. 3, 889-910.
  • Crandall, M. G., Ishii, H. and Lions, P. L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67.
  • Deville, R. and Matheron, É.: Infinite games, Banach space geometry and the eikonal equation. Proc. Lond. Math. Soc. (3) 95 (2007), no. 1, 49-68.
  • Fathi, A. and Siconolfi, A.: Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155 (2004), no. 2, 363-388.
  • Malý, J. and Zelený, M.: A note on Buczolich's solution of the Weil gradient problem: a construction based on an infinite game. Acta Math. Hungar. 113 (2006), no. 1-2, 145-158.
  • Mantegazza, C. and Mennucci, A. C.: Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds. Appl. Math. Optim. 47 (2003), no. 1, 1-25.
  • Weil, C. E.: On properties of derivatives. Trans. Amer. Math. Soc. 114 (1965), 363-376.
  • Whitney, H.: Geometric integration theory. Princeton University Press. Princeton, N.J., 1957.