Mappings of finite distortion : formation of cusps

Mappings of finite distortion: Formation of cusps II

For s > 0 s>0 given, we consider a planar domain  Ω s \Omega _s with a rectifiable boundary but containing a cusp of degree  s s , and show that there is no homeomorphism f : R 2 → R 2 f\colon \mathbb {R}^2\to \mathbb {R}^2 of finite distortion with exp ⁡ ( λ K ) ∈ L l o c 1 ( R 2 ) \exp (\lambda K)\in L^1_{\mathrm {loc}}(\mathbb {R}^2) so that f ( B ) = Ω s f(B)=\Omega _s when λ > 4 / s \lambda >4/s and  B B is the unit disc. On the other hand, for λ > 2 / s \lambda >2/s such an  f f exists. The critical value for λ \lambda remains open.

A note to “Mappings of finite distortion: formation of cusps II”

We consider planar homeomorphisms f : R 2 → R 2 f\colon \mathbb {R}^2\to \mathbb {R}^2 that are of finite distortion and map the unit disk onto a specific cusp domain  Ω s \Omega _s . We study the relation between the degree  s s of the cusp and the integrability of the distortion function  K f K_f by sharpening a previous result where  K f K_f is assumed to be locally exponentially integrable.