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 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 assures the existence of minimizers, so all results known in 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 non–trivial due to the general growth conditions assumed here on L(x, ·). 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 equi–bounded discontinuous Lagrangians.