Lower QQ-Homeomorphisms With Respect To PP-Modulus And Orlicz-Sobolev Classes

We show that under a condition of the Calderon type on $\varphi$ the homeomorphisms $f$ with finite distortion in $W^{1,\varphi}_{\rm loc}$ and, in particular, $f\in W^{1,s}_{\rm loc}$ for $s>n-1$ are the so-called lower $Q$-homeomorphisms with respect to $p$-modulus where $Q(x)$ is equal to its outer $p$-dilatation $K_{p,f}(x)$.Comment: arXiv admin note: text overlap with arXiv:1012.501

On equicontinuity of homeomorphisms with finite distortion in the plane

It is stated equicontinuity and normality of families $\frak{F}^{\Phi}$ of the so--called homeomorphisms with finite distortion on conditions that $K_{f}(z)$ has finite mean oscillation, singularities of logarithmic type or integral constraints of the type $\int\Phi\left(K_{f}(z)\right)dx\,dy<\infty$ in a domain D\subset{\C}. It is shown that the found conditions on the function $\Phi$ are not only sufficient but also necessary for equicontinuity and normality of such families of mappings.Comment: 18 page