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Applied Mathematics & Optimization

, Volume 63, Issue 2, pp 191–216 | Cite as

Regularity and Variationality of Solutions to Hamilton—Jacobi Equations. Part II: Variationality, Existence, Uniqueness

  • Andrea C. G. Mennucci
Article

Abstract

We formulate an Hamilton–Jacobi partial differential equation
$$H(x,Du(x))=0$$
on a n dimensional manifold M, with assumptions of convexity of the sets \(\{p:H(x,p)\le 0\}\subset T^{*}_{x}M\), for all x.

We reduce the above problem to a simpler problem; this shows that u may be built using an asymmetric distance (this is a generalization of the “distance function” in Finsler geometry); this brings forth a ‘completeness’ condition, and a Hopf–Rinow theorem adapted to Hamilton–Jacobi problems. The ‘completeness’ condition implies that u is the unique viscosity solution to the above problem.

Keywords

Hamilton–Jacobi equation Differentiable manifold Viscosity solution Kuratowski convergence Asymmetric metric space Finsler metric Hopf–Rinow theorem Backward completeness Uniqueness of solution 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Scuola Normale Superiore Piazza dei Cavalieri 7PisaItaly

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