The Stokes Paradox in Inhomogeneous Elastostatics

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 \(\alpha <1\).