Simple proof of two-well rigidity

Camillo De Lellis; László Székelyhidi

doi:10.1016/j.crma.2006.07.008

Calculus of Variations

Simple proof of two-well rigidity

Presented by: John M. Ball

Camillo De Lellis ¹; László Székelyhidi ²

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.

Abstract (Original language)
Résumé

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.

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.

Received: 2006-03-13
Accepted: 2006-07-04
Published online: 2006-07-25

DOI: 10.1016/j.crma.2006.07.008

Author's affiliations:

Camillo De Lellis ¹; László Székelyhidi ²

¹ Institut für Mathematik, Universität Zürich, CH-8057 Zürich
² Departement Mathematik, ETH Zürich, CH-8092 Zürich

@article{CRMATH_2006__343_5_367_0,
     author = {Camillo De Lellis and L\'aszl\'o Sz\'ekelyhidi},
     title = {Simple proof of two-well rigidity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {367--370},
     publisher = {Elsevier},
     volume = {343},
     number = {5},
     year = {2006},
     doi = {10.1016/j.crma.2006.07.008},
     language = {en},
}

TY  - JOUR
AU  - Camillo De Lellis
AU  - László Székelyhidi
TI  - Simple proof of two-well rigidity
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 367
EP  - 370
VL  - 343
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2006.07.008
LA  - en
ID  - CRMATH_2006__343_5_367_0
ER  -

%0 Journal Article
%A Camillo De Lellis
%A László Székelyhidi
%T Simple proof of two-well rigidity
%J Comptes Rendus. Mathématique
%D 2006
%P 367-370
%V 343
%N 5
%I Elsevier
%R 10.1016/j.crma.2006.07.008
%G en
%F CRMATH_2006__343_5_367_0

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/

References
Cited by

[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 $L^{1}$ 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 $R^{n}$ , 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

Cited by Sources:

Comments - Policy