Details
| Original language | Multiple languages |
|---|---|
| Pages (from-to) | 247-250 |
| Number of pages | 4 |
| Journal | Comptes rendus mathematique |
| Volume | 351 |
| Issue number | 5-6 |
| Publication status | Published - Mar 2013 |
| Externally published | Yes |
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 subject areas
- Mathematics(all)
- General Mathematics
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Comptes rendus mathematique, Vol. 351, No. 5-6, 03.2013, p. 247-250.
Research output: Contribution to journal › Article › Research › 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 -