Necessary and sufficient conditions for unique determination of plane domains

We say that a domain $U\in\mathbb R^n$ is uniquely determined from the relative metric of its Hausdorff boundary (the relative metric is the extension by continuity of the intrinsic metric of the domain to the boundary) if every domain $V\in\mathbb R^n$ with the Hausdorff boundary isometric in the relative metric to the Hausdorff boundary of $U$ is isometric to $U$ too (in the Euclidean metrics). In this article we state some necessary and sufficient conditions for a plane domain to be uniquely determined from the relative metric of its Hausdorff boundary.