Mathematical Problems in Mechanics
Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity
Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 247-250.

Let ΩRN 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:

ζ=Gζ,ζ|Γ=0,(1)
vanishes if GL1(Ω;R(N×N)×N) and ζW1,1(Ω;RN). In particular, square-integrable solutions ζ of (1) with GL1L2(Ω;R(N×N)×N) vanish. As a consequence, we prove that:
:C(Ω,Γ;R3)[0,),usym(uP1)L2(Ω)
is a norm if PL(Ω;R3×3) with CurlPLp(Ω;R3×3), CurlP1Lq(Ω;R3×3) for some p,q>1 with 1/p+1/q=1 as well as detPc+>0. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ΦH1(Ω;R3), ΩR3, satisfy sym(ΦΨ)=0 for some ΨW1,(Ω;R3)H2(Ω;R3) with detΨc+>0. Then there exists a constant translation vector aR3 and a constant skew-symmetric matrix Aso(3), such that Φ=AΨ+a.

Soit ΩRN un domaine et ΓΩ un sous-ensemble relativement ouvert de sa frontière ∂Ω, supposée lipschitzienne. Nous démontrons que la solution du système linéaire du premier ordre :

ζ=Gζ,ζ|Γ=0,(1)
sʼannule si GL1(Ω;R(N×N)×N) et ζW1,1(Ω;RN). En particulier, les solutions de carré intégrable de (1) avec GL1L2(Ω;R(N×N)×N) sʼannulent. Comme conséquence, nous prouvons que :
:C(Ω,Γ;R3)[0,),usym(uP1)L2(Ω)
est une norme lorsque PL(Ω;R3×3) avec CurlPLp(Ω;R3×3), CurlP1Lq(Ω;R3×3) pour p,q>1, 1/p+1/q=1, et detPc+>0. Nous présentons aussi une nouvelle démonstration du lemme du déplacement rigide infinitésimal en coordonnées curvilignes : si ΦH1(Ω;R3) satisfait sym(ΦΨ)=0 pour certain ΨW1,(Ω;R3)H2(Ω;R3), avec detΨc+>0, il existe des constantes aR3 et Aso(3) telles que Φ=AΨ+a.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.01.017
Lankeit, Johannes 1; Neff, Patrizio 1; Pauly, Dirk 1

1 Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany
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Lankeit, Johannes; Neff, Patrizio; Pauly, Dirk. Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity. Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 247-250. doi : 10.1016/j.crma.2013.01.017. http://www.numdam.org/articles/10.1016/j.crma.2013.01.017/

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