Some Remarks on the Dirichlet Problem in Plane Exterior Domains

In this paper we deal with the Dirichlet problem for the Laplace equation in a plane exterior domain $\Omega$ with a Lipschitz boundary. We prove that, if the boundary datum $a$ is square summable, then the problem admits a solution which tends to $a$ in the sense of nontangential convergence, is unique in a suitable function class and vanishes at infinity as $r^{-k}$ if and only if $a$ satisfies $k$ compatibility conditions, which we are able to explicit when $\Omega$ is the exterior of an ellipse.