On boundary behavior of spatial mappings

We show that homeomorphisms $f$ in ${\mathbb{R}}^n$, $n⩾3$, of finite distortion in the Orlicz–Sobolev classes $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,\phi }$ with a condition on $\phi$ of the Calderon type and, in particular, in the Sobolev classes $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}$ for $p>n-1$ are the so-called lower $Q$-homeomorphisms, $Q(x)=K_I^{\frac{1}{n-1}}(x,f)$, 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.