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


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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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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Mathematics Subject Classification

  • 74B99
  • 35Q74
  • 35A02


  • 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