On boundary behavior of spatial mappings

TL;DR

This paper demonstrates that certain classes of homeomorphisms with finite distortion in Euclidean spaces exhibit specific boundary behaviors, extending known classes to include finitely bi-Lipschitz mappings and linking them to lower Q-homeomorphisms.

Contribution

It establishes that homeomorphisms in Orlicz–Sobolev classes with finite distortion are lower Q-homeomorphisms, broadening the scope to include finitely bi-Lipschitz mappings.

Findings

Homeomorphisms in $W^{1, heta}_{loc}$ are lower Q-homeomorphisms.

Extension of boundary behavior theory to finitely bi-Lipschitz mappings.

Applicable to classes including isometric and quasiisometric mappings.

Abstract

We show that homeomorphisms $f$ in ${\Bbb R}^n$, $n\geqslant3$, of finite distortion in the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with a condition on $\varphi$ of the Calderon type and, in particular, in the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$ are the so-called lower $Q$-homeomorphisms, $Q(x)=K^{\frac{1}{n-1}}_I(x,f)$, where $K_I(x,f)$ is its inner dilatation. The statement is valid also for all finitely bi-Lipschitz mappings that a far--reaching extension of the well-known classes of isometric and quasiisometric mappings. This makes pos\-sib\-le to apply our theory of the boundary behavior of the lower $Q$-homeomorphisms to all given classes.