Numerische Mathematik

, Volume 88, Issue 2, pp 299–318

A numerical approach to variational problems subject to convexity constraint

Authors

  • G. Carlier
    • Université Paris IX Dauphine, Ceremade, France; e-mail: carlier@ceremade.dauphine.fr
  • T. Lachand-Robert
    • Université Pierre et Marie Curie, Laboratoire d'Analyse Numérique, 75252 Paris Cedex 05, France; e-mail: {lachand,maury}@ann.jussieu.fr, http://www.ann.jussieu.fr
  • B. Maury
    • Université Pierre et Marie Curie, Laboratoire d'Analyse Numérique, 75252 Paris Cedex 05, France; e-mail: {lachand,maury}@ann.jussieu.fr, http://www.ann.jussieu.fr
Orignial article

DOI: 10.1007/PL00005446

Cite this article as:
Carlier, G., Lachand-Robert, T. & Maury, B. Numer. Math. (2001) 88: 299. doi:10.1007/PL00005446

Summary.

We describe an algorithm to approximate the minimizer of an elliptic functional in the form \(\int_\Omega j(x, u, \nabla u)\) on the set \({\cal C}\) of convex functions u in an appropriate functional space X. Such problems arise for instance in mathematical economics [4]. A special case gives the convex envelope \(u_0^{**}\) of a given function \(u_0\). Let \((T_n)\) be any quasiuniform sequence of meshes whose diameter goes to zero, and \(I_n\) the corresponding affine interpolation operators. We prove that the minimizer over \({\cal C}\) is the limit of the sequence \((u_n)\), where \(u_n\) minimizes the functional over \(I_n({\cal C})\). We give an implementable characterization of \(I_n({\cal C})\). Then the finite dimensional problem turns out to be a minimization problem with linear constraints.

Mathematics Subject Classification (1991): 65K10
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© Springer-Verlag Berlin Heidelberg 2000