(INV) condition and regularity of the inverse

Let $f:\Omega \to {\Omega }^{\prime }$ be a Sobolev mapping of finite distortion between planar domains Ω and ${\Omega }^{\prime }$, satisfying the $\left(INV\right)$ condition and coinciding with a homeomorphism near ∂Ω. We show that f admits a generalized inverse mapping $h:{\Omega }^{\prime }\to \Omega$, which is also a Sobolev mapping of finite distortion and satisfies the $\left(INV\right)$ condition.

We also establish a higher-dimensional analogue of this result: if a mapping $f:\Omega \to {\Omega }^{\prime }$ of finite distortion is in the Sobolev class $W^{1,p}(\Omega ,R^n)$ with $p>n-1$ and satisfies the $\left(INV\right)$ condition, then f has an inverse in $W^{1,1}({\Omega }^{\prime },R^n)$ that is also of finite distortion.

Furthermore, we characterize Sobolev mappings satisfying $\left(INV\right)$ whose generalized inverses have finite n-harmonic energy.