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Journal of Elasticity

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24-08-2020

The Stokes Paradox in Inhomogeneous Elastostatics

Authors: Adele Ferone, Remigio Russo, Alfonsina Tartaglione

Published in: Journal of Elasticity | Issue 1/2020

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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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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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Title: The Stokes Paradox in Inhomogeneous Elastostatics
Authors: Adele Ferone
Remigio Russo
Alfonsina Tartaglione
Publication date: 24-08-2020
Publisher: Springer Netherlands
Published in: Journal of Elasticity / Issue 1/2020
Print ISSN: 0374-3535
Electronic ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-020-09788-3

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