Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations

Details Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations

2000-06-02

arxiv.org/abs/math/0006015v1

Mathematics

Analysis of PDEs

14 pages

Scientific paper

We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual convexity hypothesis. Under the same assumptions we prove also the uniqueness, in a class of lower semicontinuous functions, of a slightly different notion of solution, where classical derivatives are replaced only by subdifferentials. These results follow from a new comparison theorem for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi equation, that is proved in the general case of lower semicontinuous Lagrangians.

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