Exact relaxations of non-convex variational problems

Here, we solve non-convex, variational problems given in the form min(u) I (u) = integral(1)(0) f (u' (x))dx s.t. u(0) = 0, u(1) = a, (1) where u is an element of (W-1,W-infinity(0,1))(k) and f : R-k -> R is a non-convex, coercive polynomial. To solve ( 1) we analyse the convex hull of the integrand at the point a, so that we can find vectors a(1),...,a(N) is an element of R-k and positive values lambda(1),...,lambda(N) satisfying the non-linear equation (1, a, f(c)(a)) = Sigma(N)(i=1)lambda(i)(1, a(i), f(a(i))). (2) Thus, we can calculate minimizers of (1) by following a proposal of Dacorogna in (Direct Methods in the Calculus of Variations. Springer, Heidelberg, 1989). Indeed, we can solve (2) by using a semidefinite program based on multidimensional moments.