A planar bi-Lipschitz extension Theorem

TL;DR

This paper proves that a planar bi-Lipschitz homeomorphism on the boundary of a square can be extended to the entire square while maintaining bi-Lipschitz properties, with explicit bounds on the constants.

Contribution

It provides an explicit construction for extending boundary bi-Lipschitz maps to the interior with controlled bi-Lipschitz constants, improving upon Tukia's earlier existence result.

Findings

Extension preserves bi-Lipschitz property with explicit constant bounds

Construction allows smooth or piecewise affine extensions

Bound on the extension's bi-Lipschitz constant is proportional to the fourth power of the boundary constant

Abstract

We prove that, given a planar bi-Lipschitz homeomorphism $u$ defined on the boundary of the unit square, it is possible to extend it to a function $v$ of the whole square, in such a way that $v$ is still bi-Lipschitz. In particular, denoting by $L$ and $\widetilde L$ the bi-Lipschitz constants of $u$ and $v$, with our construction one has $\widetilde L \leq C L^4$ (being $C$ an explicit geometrical constant). The same result was proved in 1980 by Tukia (see \cite{Tukia}), using a completely different argument, but without any estimate on the constant $\widetilde L$. In particular, the function $v$ can be taken either smooth or (countably) piecewise affine.