Exact relaxations of non-convex variational problems

Here, we solve non-convex, variational problems given in the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min_{u} I(u) = \int\limits_{0}^{1} f(u'(x))dx \quad {\rm s.t.} \quad u(0) = 0, u(1) = a, \quad(1)$$\end{document}where u ∈ (W1,∞(0, 1))k and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f : {\mathbb{R}}^k \rightarrow {\mathbb{R}}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a^1,\ldots,a^N \in {\mathbb{R}}^k$$\end{document} and positive values λ1, . . . , λN satisfying the non-linear equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1, a, f_c(a)) = \sum\limits_{i=1}^{N}\lambda_i(1, a^i, f(a^i)). \quad (2)$$\end{document}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.