Details
| Originalsprache | Mehrere Sprachen |
|---|---|
| Seiten (von - bis) | 247-250 |
| Seitenumfang | 4 |
| Fachzeitschrift | Comptes rendus mathematique |
| Jahrgang | 351 |
| Ausgabenummer | 5-6 |
| Publikationsstatus | Veröffentlicht - März 2013 |
| Extern publiziert | Ja |
Abstract
Let Ω⊂ℝN 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:, vanishes if G∈L1(Ω;R(N×N)×N) and ζ∈W1,1(Ω;RN). In particular, square-integrable solutions ζ of (1) with G∈L1∩L2(Ω;R(N×N)×N) vanish. As a consequence, we prove that: is a norm if P∈L∞(Ω;R3×3) with CurlP∈Lp(Ω;R3×3), CurlP-1∈Lq(Ω;R3×3) for some p, q>1 with 1/p+1/q=1 as well as detP≥c+>0. We also give a new and different proof for the so-called 'infinitesimal rigid displacement lemma' in curvilinear coordinates: Let Φ∈H1(Ω;R3), Ω⊂R3, satisfy sym(∇;Φ⊤∇;Ψ)=0 for some Ψ∈W1,∞(Ω;R3)∩H2(Ω;R3) with det∇;Ψ≥c+>0. Then there exists a constant translation vector a∈R3 and a constant skew-symmetric matrix A∈so(3), such that Φ=AΨ+a.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Allgemeine Mathematik
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in: Comptes rendus mathematique, Jahrgang 351, Nr. 5-6, 03.2013, S. 247-250.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity
T2 - Continuation unique pour des systèmes du premier ordre avec des coefficients intégrables et applications à l’élasticité et à la plasticité
AU - Lankeit, Johannes
AU - Neff, Patrizio
AU - Pauly, Dirk
PY - 2013/3
Y1 - 2013/3
N2 - Let Ω⊂ℝN 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:, vanishes if G∈L1(Ω;R(N×N)×N) and ζ∈W1,1(Ω;RN). In particular, square-integrable solutions ζ of (1) with G∈L1∩L2(Ω;R(N×N)×N) vanish. As a consequence, we prove that: is a norm if P∈L∞(Ω;R3×3) with CurlP∈Lp(Ω;R3×3), CurlP-1∈Lq(Ω;R3×3) for some p, q>1 with 1/p+1/q=1 as well as detP≥c+>0. We also give a new and different proof for the so-called 'infinitesimal rigid displacement lemma' in curvilinear coordinates: Let Φ∈H1(Ω;R3), Ω⊂R3, satisfy sym(∇;Φ⊤∇;Ψ)=0 for some Ψ∈W1,∞(Ω;R3)∩H2(Ω;R3) with det∇;Ψ≥c+>0. Then there exists a constant translation vector a∈R3 and a constant skew-symmetric matrix A∈so(3), such that Φ=AΨ+a.
AB - Let Ω⊂ℝN 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:, vanishes if G∈L1(Ω;R(N×N)×N) and ζ∈W1,1(Ω;RN). In particular, square-integrable solutions ζ of (1) with G∈L1∩L2(Ω;R(N×N)×N) vanish. As a consequence, we prove that: is a norm if P∈L∞(Ω;R3×3) with CurlP∈Lp(Ω;R3×3), CurlP-1∈Lq(Ω;R3×3) for some p, q>1 with 1/p+1/q=1 as well as detP≥c+>0. We also give a new and different proof for the so-called 'infinitesimal rigid displacement lemma' in curvilinear coordinates: Let Φ∈H1(Ω;R3), Ω⊂R3, satisfy sym(∇;Φ⊤∇;Ψ)=0 for some Ψ∈W1,∞(Ω;R3)∩H2(Ω;R3) with det∇;Ψ≥c+>0. Then there exists a constant translation vector a∈R3 and a constant skew-symmetric matrix A∈so(3), such that Φ=AΨ+a.
UR - http://www.scopus.com/inward/record.url?scp=84878119316&partnerID=8YFLogxK
U2 - 10.1016/j.crma.2013.01.017
DO - 10.1016/j.crma.2013.01.017
M3 - Article
AN - SCOPUS:84878119316
VL - 351
SP - 247
EP - 250
JO - Comptes rendus mathematique
JF - Comptes rendus mathematique
SN - 1631-073X
IS - 5-6
ER -