Duality for non-convex variational problems (Q2018279)

A new duality theory is proposed for a classical problem of the calculus of variations where the integrand is a lower semicontinuous function (satisfying suitable growth conditions) which is not necessarily convex in the first variable. Examples of such problems are provided, too. The newly introduced dual problem can be reformulated as a linear programming problem and its solvability can be directly achieved in the one-dimensional case and, under additional hypotheses, also in higher dimensions. Optimality conditions are delivered, too. Hints towards applications to phase transition and free-boundary problems are given as well.