Sobolev homeomorphic extensions

Let X and Y be l-connected Jordan domains, l is an element of N, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism phi : partial derivative X (sic) partial derivative Y admits a Sobolev homeomorphic extension h:(X) over bar (sic) (Y) over bar in W-1,W-1(X, C). If instead X has s-hyperbolic growth with s > p - 1, we show the existence of such an extension in the Sobolev class W-1,W-p(X, C) for p is an element of(1, 2). Our examples show that the assumptions of rectifiable boundary and hyperbolic growth cannot be relaxed. We also consider the existence of W-1,W-2-homeomorphic extensions with given boundary data.