Sobolev homeomorphic extensions from two to three dimensions

2021/12/27 by Hencl, Stanislav, Koski, Aleksis, Onninen, Jani · 2 citations
#46E35 #58E20 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.14767

Abstract

We study the basic question of characterizing which boundary homeomorphisms of the unit sphere can be extended to a Sobolev homeomorphism of the interior in 3D space. While the planar variants of this problem are well-understood, completely new and direct ways of constructing an extension are required in 3D. We prove, among other things, that a Sobolev homeomorphism φ\colon \mathbb R² → \mathbb R² in W_loc^1,p (\mathbb R² , \mathbb R²) for some p∈ [1,∞ ) admits a homeomorphic extension h \colon \mathbb R³ → \mathbb R³ in W_loc^1,q (\mathbb R³, \mathbb R³) for 1≤ q < (3)/(2)p. Such an extension result is nearly sharp, as the bound q=(3)/(2)p cannot be improved due to the Hölder embedding. The case q=3 gains an additional interest as it also provides an L¹-variant of the celebrated Beurling-Ahlfors extension result.

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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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