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

@article{Lankeit2013UniquenessOI,
  title={Uniqueness of integrable solutions to \$\$\{\nabla \zeta=G \zeta, \zeta|_\Gamma = 0\}\$\$ for integrable tensor coefficients G and applications to elasticity},
  author={Johannes Lankeit and Patrizio Neff and Dirk Pauly},
  journal={Zeitschrift f{\"u}r angewandte Mathematik und Physik},
  year={2013},
  volume={64},
  pages={1679-1688}
}
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… CONTINUE READING

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