Abstract
We prove that the displacement problem of inhomogeneous elastostatics in a two–dimensional exterior Lipschitz domain has a unique solution with finite Dirichlet integral \(\boldsymbol{u}\), vanishing uniformly at infinity if and only if the boundary datum satisfies a suitable compatibility condition (Stokes paradox). Moreover, we prove that it is unique under the sharp condition \(\boldsymbol{u}=o(\log r)\) and decays uniformly at infinity with a rate depending on the elasticities. In particular, if these last ones tend to a homogeneous state at large distance, then \(\boldsymbol{u}=O(r^{-\alpha })\), for every \(\alpha <1\).
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Notes
For constant \(\boldsymbol{\mathsf{C}}\) (homogeneous elasticity) it is sufficient to assume that \(\boldsymbol{\mathsf{C}}\) is strongly elliptic, i.e., there is \(\lambda _{0}>0\) such that \(\lambda _{0}|{\boldsymbol{a}}|^{2}|{\boldsymbol{b}}|^{2}\le { \boldsymbol{a}}\cdot \boldsymbol{\mathsf{C}}[{\boldsymbol{a}}\otimes { \boldsymbol{b}}]{\boldsymbol{b}}\), for all \({\boldsymbol{a}},{\boldsymbol{b}}\in {\mathbb{R}}^{2}\).
It is worth recalling that for \(q=2\) Hardy’s inequality takes the form
$$ \int \limits _{{\mathcal{I}}} \frac{|{\boldsymbol{u}}|^{2}}{r^{2}\log ^{2} r}\le 4\int \limits _{ \mathcal{I}}|\nabla {\boldsymbol{u}}|^{2}+\frac{2}{\log R_{0}}\int \limits _{0}^{2\pi } |{\boldsymbol{u}}|^{2}(R_{0},\theta ). $$By virtue of [14] \(\bar{q}\) cannot be too large.
By abuse of notation, when \({\boldsymbol{f}}\) is a distribution the last integral is understood as a duality pairing.
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Ferone, A., Russo, R. & Tartaglione, A. The Stokes Paradox in Inhomogeneous Elastostatics. J Elast 142, 35–52 (2020). https://doi.org/10.1007/s10659-020-09788-3
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DOI: https://doi.org/10.1007/s10659-020-09788-3
Keywords
- Inhomogeneous elasticity
- Two–dimensional exterior domains
- Existence and uniqueness theorems
- Stokes paradox