Convergence and compactness of the Sobolev mappings

TL;DR

This paper establishes compactness and convergence properties of Sobolev and Orlicz--Sobolev mappings, providing criteria for the compactness of classes of Sobolev homeomorphisms with fixed points and controlled dilatations.

Contribution

It introduces new compactness and convergence theorems for Sobolev and Orlicz--Sobolev mappings, including criteria for classes of Sobolev homeomorphisms with fixed points.

Findings

Proved compactness of certain Sobolev mappings with fixed points.

Established convergence theorems for Sobolev homeomorphisms.

Derived criteria for compactness of classes of Sobolev homeomorphisms.

Abstract

First of all, we establish compactness of continuous mappings of the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with the Calderon type condition on $\varphi$ and, in particular, of the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$ in ${\Bbb R}^n\,,$ $n\ge 3\,,$ with one fixed point. Then we give a series of theorems on convergence of the Orlicz--Sobolev homeomorphisms and on semicontinuity in the mean of dilatations of the Sobolev homeomorphisms. These results lead us to closeness of the corresponding classes of homeomorpisms. Finally, we come on this basis to criteria of compactness of classes of Sobolev's homeomorphisms with the corresponding conditions on dilatations and two fixed points.