Schoenflies solutions of conformal boundary values may fail to be Sobolev

TL;DR

This paper demonstrates that certain planar Jordan domains with fractal boundaries can cause conformal maps and their extensions to lack Sobolev regularity, challenging assumptions about boundary behavior.

Contribution

It constructs examples of Jordan domains with fractal boundaries where conformal maps cannot be extended to Sobolev functions, revealing limitations in boundary regularity theory.

Findings

Existence of Jordan domains with Hausdorff dimension 1 boundary

Conformal maps to these domains lack Sobolev regularity upon extension

Extensions are not in W^{1,1}_{loc} or BV_{loc}

Abstract

We show that there exists a planar Jordan domains $\Omega$ with boundary of Hausdorff dimension $1$ such that, for any conformal maps $\varphi \colon \mathbb D \to \Omega$, any homeomorphic extension of $\varphi$ or $\varphi^{-1}$ to the entire plane is not in $W^{1,\,1}_{\rm loc}$ (or even not in $BV_{\rm loc}$).