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Comptes Rendus Mathématique
Volume 351, n° 5-6
pages 247-250 (mars 2013)
Doi : 10.1016/j.crma.2013.01.017
Received : 18 September 2012 ;  accepted : 23 January 2013
Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity
Continuation unique pour des systèmes du premier ordre avec des coefficients intégrables et applications à lʼélasticité et à la plasticité
 

Johannes Lankeit , Patrizio Neff , Dirk Pauly
 Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany 

Abstract

Let   be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ∂Ω . We show that the solution to the linear first-order system:
(1)∇ζ=Gζ,ζ|Γ=0, vanishes if   and  . In particular, square-integrable solutions ζ of ((1)) with   vanish. As a consequence, we prove that:⦀⋅⦀:C∘∞(Ω,Γ;R3)→[0,∞),u↦‖sym(∇uP−1)‖L2(Ω) is a norm if   with  ,   for some   with   as well as  . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let  ,  , satisfy   for some   with  . Then there exists a constant translation vector   and a constant skew-symmetric matrix  , such that  .

The full text of this article is available in PDF format.
Résumé

Soit   un domaine et   un sous-ensemble relativement ouvert de sa frontière ∂Ω , supposée lipschitzienne. Nous démontrons que la solution du système linéaire du premier ordre :
(1)∇ζ=Gζ,ζ|Γ=0, sʼannule si   et  . En particulier, les solutions de carré intégrable de ((1)) avec   sʼannulent. Comme conséquence, nous prouvons que :⦀⋅⦀:C∘∞(Ω,Γ;R3)→[0,∞),u↦‖sym(∇uP−1)‖L2(Ω) est une norme lorsque   avec  ,   pour  ,  , et  . Nous présentons aussi une nouvelle démonstration du lemme du déplacement rigide infinitésimal en coordonnées curvilignes : si   satisfait   pour certain  , avec  , il existe des constantes   et   telles que  .

The full text of this article is available in PDF format.


© 2013  Published by Elsevier Masson SAS de la part de Académie des sciences.
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