The theory of prime ends and spatial mappings

It is given a canonical representation of prime ends in regular spatial domains and, on this basis, it is studied the boundary behavior of the so-called lower Q-homeomorphisms that are the natural generalization of the quasiconformal mappings. In particular, it is found a series of effective conditions on the function Q(x) for a homeomorphic extension of the given mappings to the boundary by prime ends in domains with regular boundaries. The developed theory is applied, in particular, to mappings of the classes of Sobolev and Orlicz-Sobolev and also to finitely bi-Lipschitz mappings that a far-reaching extension of the well--known classes of isometric and quasiisometric mappings.