$(INV)$ condition and regularity of the inverse

Abstract

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