Pointwise rotation for homeomorphisms with integrable distortion and controlled compression

TL;DR

This paper establishes precise bounds on the rotation of certain homeomorphisms with integrable distortion, relevant to fluid dynamics, and demonstrates the sharpness of these bounds through explicit examples.

Contribution

It provides sharp rotation bounds for homeomorphisms with $L^p$-integrable distortion and controlled inverse modulus of continuity, resolving the borderline case $p=1$.

Findings

Sharp rotation bounds for homeomorphisms with integrable distortion

Examples demonstrating the bounds are optimal

Resolution of the borderline case $p=1$ in previous work

Abstract

We obtain sharp rotation bounds for homeomorphisms $f:\mathbb{C}\to\mathbb{C}$ whose distortion is in $L^p_{loc}$, $p\geq1$, and whose inverse have controlled modulus of continuity. The motivation to study this class of maps comes from so-called Yudovich solutions to planar Euler equations. Furthermore, we present examples proving sharpness in a strong sense, thereby settling the borderline case $p=1$ in \cite[Theorem 3]{CHS}.