Hamilton-Jacobi Theory in the Calculus of Variations Under Partial Convexity Assumptions on the Lagrangian

Using general results about subdifferentials of value functions, under partial convexity assumptions on the Lagrangian, we derive the main result about the Hamilton-Jacobi theory in the calculus of variations obtained by R. T. Rockafellar and P. Wolenski [Convexity in Hamilton-Jacobi theory, SIAM J. Control Optimization 39/5 (2000) 1323--1372] under full convexity of the Lagrangian. Since a number of results in the calculus of variations are known to be valid under such an assumption, it is tempting to tackle such an aim, even if the nice duality theory presented in the work cited above seems to be out of reach.