Calculus of Variations and Geometric Measure Theory

G. Dal Maso - H. Frankowska

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

created on 13 Dec 2002


Conference proceedings

Inserted: 13 dec 2002

Journal: Optimal Control and Partial Differential Equations. In honour of Professor Alain Bensoussan 60th Birthday (Paris, 2000), J.L. Menaldi, E. Rofman, A. Sulem ed., IOS Press, Amsterdam
Pages: 335-345
Year: 2001


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.