$(INV)$ condition and regularity of the inverse

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.