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Regularity and variationality of solutionsto Hamilton-Jacobi equations. Part I: Regularity

Published online by Cambridge University Press:  15 June 2004

Andrea C.G. Mennucci
Andrea C.G. Mennucci*
Affiliation:
Scuola Normale Superiore Piazza dei Cavalieri 7, 56126 Pisa, Italy; a.mennuci@sns.it.
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Abstract

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*Mas a symplectic manifold.The definition of “minsol solution” is suited to proving regularity results about u; in particular, we provein the first part that the closure of the set where u is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a ${{\mathcal H}}^{n-1}$ negligeable subset.These results can be applied to the cutlocus of a C 2 submanifold of a Finsler manifold.

Keywords

Hamilton-Jacobi equationsconjugate points.

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Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 3 , July 2004, pp. 426 - 451
Copyright
© EDP Sciences, SMAI, 2004

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