$(INV)$ condition and regularity of the inverse

TL;DR

This paper proves that Sobolev mappings of finite distortion satisfying the $(INV)$ condition have generalized inverses that are also Sobolev mappings with finite distortion, extending results to higher dimensions and characterizing inverse energy properties.

Contribution

The paper establishes the existence of Sobolev inverse mappings of finite distortion under the $(INV)$ condition, including higher-dimensional analogues and energy characterizations.

Findings

Existence of Sobolev inverses with finite distortion for planar mappings.

Higher-dimensional inverse existence for mappings in $W^{1,p}$ with $p > n-1$.

Characterization of inverse mappings with finite $n$-harmonic energy.

Abstract

Let $f \colon \Omega \to \Omega' $ be a Sobolev mapping of finite distortion between planar domains $\Omega $ and $\Omega'$, satisfying the $(INV)$ condition and coinciding with a homeomorphism near $\partial\Omega $. We show that $f$ admits a generalized inverse mapping $h \colon \Omega' \to \Omega$, which is also a Sobolev mapping of finite distortion and satisfies the $(INV)$ condition. We also establish a higher-dimensional analogue of this result: if a mapping $f \colon \Omega \to \Omega' $ of finite distortion is in the Sobolev class $W^{1,p}(\Omega, \mathbb{R}^n)$ with $p > n-1$ and satisfies the $(INV)$ condition, then $f$ has an inverse in $W^{1,1}(\Omega', \mathbb{R}^n)$ that is also of finite distortion. Furthermore, we characterize Sobolev mappings satisfying $(INV)$ whose generalized inverses have finite $n$-harmonic energy.