Sobolev homeomorphic extensions

2018/12/05 by Koski, Aleksis, Onninen, Jani · 2 citations
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.02085

Abstract

Let \mathbb X and \mathbb Y be ℓ-connected Jordan domains, ℓ ∈ \mathbb N, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism φ\colon ∂ \mathbb X → ∂ \mathbb Y admits a Sobolev homeomorphic extension h \colon \mathbb X → \mathbb Y in W^1,1 (\mathbb X, \mathbb C). If instead \mathbb X has s-hyperbolic growth with s>p-1, we show the existence of such an extension lies in the Sobolev class W^1,p (\mathbb X, \mathbb 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 W^1,2-homeomorphic extensions subject to a given boundary data.

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• Global Integrability of the Reciprocal of Jacobians for Homeomorphisms of Finite Distortion
• Monotone Sobolev extensions in metric surfaces and applications to uniformization

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