Bolza Problems with Discontinuous Lagrangians and Lipschitz-Continuity of the Value Function

We study the local Lipschitz-continuity of the value function v associated with a Bolza problem in the presence of a Lagrangian $L(x,q)$, convex and uniformly superlinear in q, but only Borel-measurable in x. Under these assumptions, the associated integral functional is not lower semicontinuous with respect to the suitable topology which ensures the existence of minimizers, so all results known in the literature fail to apply. Yet, the Lipschitz regularity of v does not depend on the existence of minimizers. In fact, it is enough to control the derivatives of quasi-minimal curves, but the problem is nontrivial due to the general growth conditions assumed here on $L(x,\cdot)$. We propose a new approach, based on suitable reparameterization arguments, to obtain suitable a priori estimates on the Lipschitz constants of quasi minimizers. As a consequence of our analysis, we derive the Lipschitz-continuity of v and a compactness result for value functions associated with sequences of locally equibounded disco...