Revista Matemática Iberoamericana

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Volume 24, Issue 3, 2008, pp. 989–1010
DOI: 10.4171/RMI/564

Published online: 2008-12-31

Almost classical solutions of Hamilton-Jacobi equations

Robert Deville[1] and Jesús A. Jaramillo[2]

(1) Université de Bordeaux I, Talence, France
(2) Universidad Complutense de Madrid, Spain

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 $M$.

Keywords: Hamilton-Jacobi equations, eikonal equation on manifolds, almost everywhere solutions

Deville Robert, Jaramillo Jesús: Almost classical solutions of Hamilton-Jacobi equations. Rev. Mat. Iberoam. 24 (2008), 989-1010. doi: 10.4171/RMI/564