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Exact relaxations of non-convex variational problems

Authors:

René Meziat Diego Patiño

Institution:

(1) Departamento de Matemáticas, Universidad de los Andes, Carrera 1 No. 18A-10, Bogotá, Colombia

Abstract:

Here, we solve non-convex, variational problems given in the form

$\min_{u} I(u) = \int\limits_{0}^{1} f(u'(x))dx \quad {\rm s.t.} \quad u(0) = 0, u(1) = a, \quad(1)$

where u ∈ (W ^1,∞(0, 1))^k and $f : {\mathbb{R}}^k \rightarrow {\mathbb{R}}$ 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 $a^1,\ldots,a^N \in {\mathbb{R}}^k$ and positive values λ₁, . . . , λ_N satisfying the non-linear equation

$(1, a, f_c(a)) = \sum\limits_{i=1}^{N}\lambda_i(1, a^i, f(a^i)). \quad (2)$

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. We dedicate this work to our colleague Jesús Bermejo.

Keywords:

Calculus of variations Convex analysis Semidefinite programming Multidimensional moment problem

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