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 ${\Phi = A\Psi + a}$.