Exact relaxations of non-convex variational problems

Here, we solve non-convex, variational problems given in the form Equation is presented where u ε(W ^1,∞(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 a¹,...,a^N ε ℝ^k and positive values λ₁, . . . , λ _N satisfying the non-linear equation (1, a, f_c(a)) = ∑_i=1M ^Nλ₁(1, aⁱ, f(aⁱ)).(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.