VALUE FUNCTION FOR NONAUTONOMOUS PROBLEMS IN THE CALCULUS OF VARIATIONS

We consider the Hamilton-Jacobi equation associated with a calculus of variations problem in which the functional to minimize comprises an end-point cost function and an integral term involving a nonautonomous Lagrangian. We assume that the Lagrangian is merely Borel measurable, has a bounded variation behaviour w.r.t. the time variable, and satisfies growth conditions which are weaker than superlinearity. For this class of problems we extend (to the nonautonomous case) the regularity and existence (of generalized solution to the Hamilton-Jacobi equation, in terms of Dini/contingent derivatives) results obtained by Dal Maso and Frankowska in earlier work for autonomous Lagrangians, and we show that the value function is also a proximal solution. Imposing some additional assumptions (such as convexity and superlinearity in v), we also provide a uniqueness result for Dini/contingent solutions.