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(-a)), for every alpha < 1.