The Stokes paradox in inhomogeneous elastostatics

We prove that the displacement problem of inhomogeneous elastostatics in a two--dimensional exterior Lipschitz domain has a unique solution with finite Dirichlet integral $\u$, 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 $\u=o(\log r)$ 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 $\u=O(r^{-\alpha})$, for every $\alpha<1$.