Bi-Sobolev homeomorphisms $f$ with $Df$ and $Df^{-1}$ of low rank using laminates

TL;DR

This paper constructs bi-Sobolev homeomorphisms with prescribed low-rank derivatives and inverses using laminates and convex integration, demonstrating sharp integrability conditions.

Contribution

It introduces a method to explicitly construct homeomorphisms with low-rank derivatives and inverse derivatives, advancing the understanding of their regularity and integrability properties.

Findings

Constructed homeomorphisms with prescribed low-rank derivatives

Established sharp integrability conditions for derivatives and inverses

Applied convex integration and laminates in the construction

Abstract

Let $\Omega\subset \mathbb{R}^{n}$ be a bounded open set. Given $1\leq m_1,m_2\leq n-2$, we construct a homeomorphism $f :\Omega\to \Omega$ that is H\"older continuous, $f$ is the identity on $\partial \Omega$, the derivative $D f$ has rank $m_1$ a.e.\ in $\Omega$, the derivative $D f^{-1}$ of the inverse has rank $m_2$ a.e.\ in $\Omega$, $Df\in W^{1,p}$ and $Df^{-1}\in W^{1,q}$ for $p<\min\{m_1+1,n-m_2\}$, $q<\min\{m_2+1,n-m_1\}$. The proof is based on convex integration and laminates. We also show that the integrability of the function and the inverse is sharp.