Duality for non-convex variational problems

We consider classical problems of the calculus of variations of the kind (1) I ( Ω ) : = inf ⁡ { ∫ Ω f ( u , ∇ u ) d x + ∫ Γ 1 γ ( u ) d H N − 1 , u = u 0 on Γ 0 } where Ω is an open bounded subset of R N , ( Γ 0 , Γ 1 ) is a partition of ∂Ω, γ is a Lipschitz function, and f = f ( t , z ) is an l.s.c. function satisfying suitable growth conditions, which is convex in z , but possibly not in t . We present a new duality theory in which the dual problem reads quite nicely as a linear programming problem. The solvability of such a dual problem is a major issue. It can be achieved in the one-dimensional case, and in higher dimensions under special assumptions on f . Our results apply to phase transition and free-boundary problems.