  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.
