Sobolev homeomorphic extensions

Let X and Y be ℓ-connected Jordan domains, ℓ 2 N, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism ' W ∂X onto -→ ∂Y admits a Sobolev homeomorphic extension hW X onto -→Y inW1;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 classW1;p.X;C/ for p 2 .1; 2/. Our examples show that the assumptions of rectifiable boundary and hyperbolic growth cannot be relaxed. We also consider the existence of W1;2-homeomorphic extensions with given boundary data.