  We prove that any weakly differentiable function with square integrable
gradient can be extended to a capacitary boundary of any simply connected plane
domain $\Omega\ne\mathbb R^2$ except a set of a conformal capacity zero. For
locally connected at boundary points domains the capacitary boundary coincides
with the Euclidean one. A concept of a capacitary boundary was proposed by
V.~Gol'dshtein and S.~K.~Vodop'yanov in 1978 for a study of boundary behavior
of quasi-conformal homeomorphisms. We prove in details the main properties of
the capacitary boundary. An abstract version of the extension property for more
general classes of plane domains is discussed also.
