Schoenflies solutions of conformal boundary values may fail to be Sobolev

Abstract

We show that there exist planar Jordan domains $\Omega_1$ and $\Omega_2$ with boundaries of Hausdorff dimension $1$ such that any conformal maps $\varphi_1 \colon \mathbb D \to \Omega_1$ and $\varphi_2 \colon \Omega_2 \to \mathbb D $ cannot be extended as global homeomorphisms between the Riemann spheres of $W^{1,\,1}$ class (or even not in $BV$).