[Un exemple de fonction C1,1 polyconvexe sans reprèsentant convexe differentiable]
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.
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@article{CRMATH_2003__336_1_11_0,
author = {Bevan, Jonathan},
title = {An example of a {\protect\emph{C}\protect\textsuperscript{1,1}} polyconvex function with no differentiable convex representative},
journal = {Comptes Rendus. Math\'ematique},
pages = {11--14},
publisher = {Elsevier},
volume = {336},
number = {1},
year = {2003},
doi = {10.1016/S1631-073X(02)00015-8},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)00015-8/}
}
TY - JOUR AU - Bevan, Jonathan TI - An example of a C1,1 polyconvex function with no differentiable convex representative JO - Comptes Rendus. Mathématique PY - 2003 SP - 11 EP - 14 VL - 336 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)00015-8/ DO - 10.1016/S1631-073X(02)00015-8 LA - en ID - CRMATH_2003__336_1_11_0 ER -
%0 Journal Article %A Bevan, Jonathan %T An example of a C1,1 polyconvex function with no differentiable convex representative %J Comptes Rendus. Mathématique %D 2003 %P 11-14 %V 336 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(02)00015-8/ %R 10.1016/S1631-073X(02)00015-8 %G en %F CRMATH_2003__336_1_11_0
Bevan, Jonathan. An example of a C1,1 polyconvex function with no differentiable convex representative. Comptes Rendus. Mathématique, Tome 336 (2003) no. 1, pp. 11-14. doi : 10.1016/S1631-073X(02)00015-8. http://www.numdam.org/articles/10.1016/S1631-073X(02)00015-8/
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