Comptes Rendus
no. 5
Calculus of Variations
Simple proof of two-well rigidity
[Une démonstration simple d'une estimation de rigidité pour deux puits]

Présenté par : John M. Ball

Camillo De Lellis1 ; László Székelyhidi2
1 Institut für Mathematik, Universität Zürich, CH-8057 Zürich
2 Departement Mathematik, ETH Zürich, CH-8092 Zürich
Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 367-370.

Nous donnons une démonstration simple d'une estimation de rigidité de Müller et Chaudhuri pour deux puits fortement incompatibles. Nous employons un argument de Ball et James pour montrer que l'incompatibilité pour les mesures de Young engendrées par des gradients permet de réduire les estimations de rigidité pour plusieurs puits à celles pour un puit.

We give a short proof of the rigidity estimate of Müller and Chaudhuri for two strongly incompatible wells. Making strong use of the arguments of Ball and James our approach shows that incompatibility for gradient Young measures can be used to reduce rigidity estimates for several wells to one-well rigidity.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.07.008
Camillo De Lellis 1 ; László Székelyhidi 2

1 Institut für Mathematik, Universität Zürich, CH-8057 Zürich
2 Departement Mathematik, ETH Zürich, CH-8092 Zürich 
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Camillo De Lellis; László Székelyhidi. Simple proof of two-well rigidity. Comptes Rendus. Mathématique, Volume 343 (2006) no. 5, pp. 367-370. doi : 10.1016/j.crma.2006.07.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.07.008/

[1] J.M. Ball, R.D. James, Incompatible sets of gradients and metastability, in preparation

[2] N. Chaudhuri; S. Müller Rigidity estimate for two incompatible wells, Calc. Var. Partial Differential Equations, Volume 19 (2004) no. 4, pp. 379-390

[3] N. Chaudhuri, S. Müller, Scaling of the energy for thin martensitic films, Preprint MPI-MIS Leipzig 59, 2004

[4] S. Conti; D. Faraco; F. Maggi A new approach to counterexamples to L1 estimates: Korn's inequality, geometric rigidity, and regularity for gradients of separately convex functions, Arch. Ration. Mech. Anal., Volume 175 (2005), pp. 287-300

[5] S. Conti; B. Schweizer Rigidity and Γ-convergence for solid-solid phase transitions with SO(2)-invariance, Comm. Pure Appl. Math., Volume 59 (2006), pp. 830-868

[6] G. Friesecke; R.D. James; S. Müller A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., Volume 55 (2002) no. 11, pp. 1461-1506

[7] M. de Guzmán Differentiation of Integrals in Rn, Lecture Notes in Mathematics, vol. 481, Springer-Verlag, 1975

[8] J.P. Matos Young measures and the absence of fine microstructures in a class of phase transitions, Eur. J. Appl. Math., Volume 3 (1992) no. 1, pp. 31-54

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