  In this article we develop a new primal dual variational formulation suitable
for a large class of non-convex problems in the calculus of variations. The
results are obtained through basic tools of convex analysis, duality theory,
the Legendre transform concept and the respective relations between the primal
and dual variables. The novelty here is that the dual formulation is
established also for the primal variables, however with a large domain region
of concavity about a critical point. Finally, we formally prove there is no
duality gap between the primal and dual formulations in a local extremal
context.
