Mappings of finite distortion: Formation of exponential cusp

TL;DR

This paper investigates the conditions under which homeomorphisms of finite distortion can map the plane onto domains with exponential cusps, revealing limitations and possibilities based on distortion integrability conditions.

Contribution

It establishes that no finite distortion homeomorphism with exponential integrability of distortion can map the plane onto a domain with an exponential cusp, but such mappings exist under weaker L^p conditions.

Findings

No finite distortion homeomorphism with exponential distortion integrability maps to exponential cusps.

Existence of finite distortion homeomorphisms under L^p distortion conditions.

Characterization of mappings related to exponential cusp formation.

Abstract

We consider a quasi-convex planar domain \Omega with a rectifiable boundary containing an exponential cusp and show that there is no homeomorphism f: \bR^2\to\bR^2 of finite distortion with \exp(\lambda K)\in L_{loc}^{1}(\bR^2) for some \lambda>0 such that f(B)=\Omega. On the other hand, if we only require that K_f(x)\in L_{loc}^{p}(\bR^2), then such an f exists.