Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Johannes Lankeit; Patrizio Neff; Dirk Pauly

doi:10.1016/j.crma.2013.01.017

Mathematical Problems in Mechanics

Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

Philippe G. Ciarlet

Johannes Lankeit ¹; Patrizio Neff ¹; Dirk Pauly ¹

¹Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany

Comptes Rendus. Mathématique, Volume 351 (2013) no. 5-6, pp. 247-250

Abstract (Original language)
Résumé

Let $Ω \subset R^{N}$ 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:

\nabla ζ = G ζ, ζ |_{Γ} = 0,

(1)

vanishes if

G \in L^{1} (Ω; R^{(N \times N) \times N})

and

ζ \in W^{1, 1} (Ω; R^{N})

. In particular, square-integrable solutions ζ of (1) with

G \in L^{1} \cap L^{2} (Ω; R^{(N \times N) \times N})

vanish. As a consequence, we prove that:

⦀ \cdot ⦀ : C_{\circ}^{\infty} (Ω, Γ; R^{3}) \to [0, \infty), u \mapsto ‖ sym (\nabla u P^{- 1}) ‖_{L^{2} (Ω)}

is a norm if

P \in L^{\infty} (Ω; R^{3 \times 3})

with

Curl P \in L^{p} (Ω; R^{3 \times 3})

,

Curl P^{- 1} \in L^{q} (Ω; R^{3 \times 3})

for some

p, q > 1

with

1 / p + 1 / q = 1

as well as

\det P ⩾ c^{+} > 0

. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let

Φ \in H^{1} (Ω; R^{3})

,

Ω \subset R^{3}

, satisfy

sym (\nabla Φ^{⊤} \nabla Ψ) = 0

for some

Ψ \in W^{1, \infty} (Ω; R^{3}) \cap H^{2} (Ω; R^{3})

with

\det \nabla Ψ ⩾ c^{+} > 0

. Then there exists a constant translation vector

a \in R^{3}

and a constant skew-symmetric matrix

A \in so (3)

, such that

Φ = A Ψ + a

.

Soit $Ω \subset R^{N}$ un domaine et $\emptyset \neq Γ \subset \partial Ω$ un sous-ensemble relativement ouvert de sa frontière ∂Ω, supposée lipschitzienne. Nous démontrons que la solution du système linéaire du premier ordre :

\nabla ζ = G ζ, ζ |_{Γ} = 0,

(1)

sʼannule si

G \in L^{1} (Ω; R^{(N \times N) \times N})

et

ζ \in W^{1, 1} (Ω; R^{N})

. En particulier, les solutions de carré intégrable de (1) avec

G \in L^{1} \cap L^{2} (Ω; R^{(N \times N) \times N})

sʼannulent. Comme conséquence, nous prouvons que :

⦀ \cdot ⦀ : C_{\circ}^{\infty} (Ω, Γ; R^{3}) \to [0, \infty), u \mapsto ‖ sym (\nabla u P^{- 1}) ‖_{L^{2} (Ω)}

est une norme lorsque

P \in L^{\infty} (Ω; R^{3 \times 3})

avec

Curl P \in L^{p} (Ω; R^{3 \times 3})

,

Curl P^{- 1} \in L^{q} (Ω; R^{3 \times 3})

pour

p, q > 1

,

1 / p + 1 / q = 1

, et

\det P ⩾ c^{+} > 0

. Nous présentons aussi une nouvelle démonstration du lemme du déplacement rigide infinitésimal en coordonnées curvilignes : si

Φ \in H^{1} (Ω; R^{3})

satisfait

sym (\nabla Φ^{⊤} \nabla Ψ) = 0

pour certain

Ψ \in W^{1, \infty} (Ω; R^{3}) \cap H^{2} (Ω; R^{3})

, avec

\det \nabla Ψ ⩾ c^{+} > 0

, il existe des constantes

a \in R^{3}

et

A \in so (3)

telles que

Φ = A Ψ + a

.

Article information
Cite this article

Received: 2012-09-18
Accepted: 2013-01-23
Published online: 2013-03-01

DOI: 10.1016/j.crma.2013.01.017

Author's affiliations:

Johannes Lankeit ¹ ; Patrizio Neff ¹ ; Dirk Pauly ¹

¹ Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany

Johannes Lankeit; Patrizio Neff; Dirk Pauly. 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

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     doi = {10.1016/j.crma.2013.01.017},
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