Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities

We investigate the value function of the Bolza problem of the Calculus of V ariations V (t, x) = inf {integral (t)(0) L(y(s), y'(s))ds + phi (y(t)) : y is an ele ment of W-1,W-1 (0, t; R-n); y(0) = x}, with a lower semicontinuous Lagrangian L and a final cost phi, and show tha t it is locally Lipschitz for t > 0 whenever L is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lag rangian is continuous, then the value function is the unique lower semicont inuous solution to the corresponding Hamilton-Jacobi equation, while for di scontinuous Lagrangian we characterize the value function by using the so c alled contingent inequalities.