An example of a <i>C</i><sup>1,1</sup> polyconvex function with no differentiable convex representative

Jonathan Bevan

doi:10.1016/S1631-073X(02)00015-8

Mathematical Analysis

An example of a C^1,1 polyconvex function with no differentiable convex representative
[Un exemple de fonction C^1,1 polyconvexe sans reprèsentant convexe differentiable]

Présenté par : Phillipe G. Ciarlet

Jonathan Bevan ¹

¹ Mathematical Institute, University of Oxford, OX1 3LB Oxford, UK

Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 11-14.

Résumé
Abstract

On construit une fonction C^1,1 polyconvexe W tel qu'il existe une matrice 2×2 Y satisfaisant la propriété suivante : tous les representants convexes de W ont au moins deux sousgradients distincts (et ne sont donc pas differentiable) au point (Y,detY). Ceci montre, en particulier, qu'une fonction polyconvexe peut être plus differentiable que tous ses representants convex.

We construct a C^1,1 polyconvex function W such that there exists a fixed 2×2 matrix Y with the property that all convex representatives of W have at least two distinct subgradients (and are hence not differentiable) at the point (Y,detY), showing in particular that a polyconvex function can be smoother than any of its convex representatives.

Reçu le : 2002-10-13
Accepté le : 2002-10-31
Publié le : 2003-01-01

DOI : 10.1016/S1631-073X(02)00015-8

Affiliations des auteurs :

Jonathan Bevan ¹

¹ Mathematical Institute, University of Oxford, OX1 3LB Oxford, UK

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Jonathan Bevan. An example of a C^1,1 polyconvex function with no differentiable convex representative. Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 11-14. doi : 10.1016/S1631-073X(02)00015-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)00015-8/

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