$(INV)$ condition and regularity of the inverse

Abstract

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

Corrected typos

References in corpus (4)

• Monotone Sobolev Mappings of planar domains and surfaces
• Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity
• Weak limit of homeomorphisms in $W^{1,n-1}$ and (INV) condition
• Weak limit of homeomorphisms in $W^{1,n-1}$: invertibility and lower semicontinuity of energy