# Uniqueness of integrable solutions to $${\nabla \zeta=G \zeta, \zeta|_\Gamma = 0}$$ for integrable tensor coefficients G and applications to elasticity

## Abstract

Let $${\Omega \subset \mathbb{R}^{N}}$$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary $${\partial\Omega}$$. We show that the solution to the linear first-order system

$$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$

is unique if $${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$$ and $${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$$. As a consequence, we prove

$$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$

to be a norm for $${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$$ with Curl $${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$$, Curl $${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$$ for some p, q > 1 with 1/p + 1/q = 1 as well as det $${P \geq c^+ > 0}$$. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let $${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$$ satisfy sym $${(\nabla\Phi^\top\nabla\Psi) = 0}$$ for some $${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$$ with det $${\nabla\Psi \geq c^+ > 0}$$. Then, there exist a constant translation vector $${a \in \mathbb{R}^{3}}$$ and a constant skew-symmetric matrix $${A \in \mathfrak{so}(3)}$$, such that $${\Phi = A\Psi + a}$$.

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Correspondence to Patrizio Neff.

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Lankeit, J., Neff, P. & Pauly, D. Uniqueness of integrable solutions to $${\nabla \zeta=G \zeta, \zeta|_\Gamma = 0}$$ for integrable tensor coefficients G and applications to elasticity. Z. Angew. Math. Phys. 64, 1679–1688 (2013). https://doi.org/10.1007/s00033-013-0314-4

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• 74B99
• 35Q74
• 35A02

### Keywords

• Korn’s inequality
• Generalized Korn’s first inequality
• First-order system of partial differential equations
• Uniqueness
• Infinitesimal rigid displacement lemma
• Korn’s inequality in curvilinear coordinates
• Unique continuation