Comptes Rendus
no. 1
Mathematical Analysis
An example of a C1,1 polyconvex function with no differentiable convex representative
[Un exemple de fonction C1,1 polyconvexe sans reprèsentant convexe differentiable]

Présenté par : Phillipe G. Ciarlet

Jonathan Bevan1
1 Mathematical Institute, University of Oxford, OX1 3LB Oxford, UK
Comptes Rendus. Mathématique, Volume 336 (2003) no. 1, pp. 11-14.

On construit une fonction C1,1 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 C1,1 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 :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)00015-8
Jonathan Bevan 1

1 Mathematical Institute, University of Oxford, OX1 3LB Oxford, UK 
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Jonathan Bevan. An example of a C1,1 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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