Boundary Values of Functions of Dirichlet Spaces $L^1_2$ on Capacitary Boundaries Academic Article uri icon

abstract

  • 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.

publication date

  • January 1, 2014