Sobolev homeomorphic extensions

Abstract

Let X and Y be ℓ-connected Jordan domains, ℓ∈N, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism φ:∂X→∂Y admits a Sobolev homeomorphic extension h:X¯→Y¯ in W1,1(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 W1,p(X,C) for p∈(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.