# Composition operators on Sobolev spaces and Ball's classes

## TL;DR

This paper explores the geometric properties of Ball's classes within nonlinear elasticity by characterizing them through composition operators on Sobolev spaces, leading to volume distortion estimates and continuity results.

## Contribution

It introduces a novel characterization of Ball's classes via composition operators, enabling new volume distortion estimates and continuity properties for Sobolev homeomorphisms.

## Key findings

- Derived volume compression estimates for mappings in Ball's classes.
- Proved Sobolev homeomorphisms with Luzin N-property are absolutely continuous w.r.t. capacity.
- Connected geometric properties of Ball's classes to nonlinear elasticity applications.

## Abstract

In this paper we study geometric aspects of Ball's classes in the context of nonlinear elasticity problems. The suggested approach is based on the characterization of Ball's classes $A_{q,q'}(\Omega)$ in terms of composition operators on Sobolev spaces. This characterization allows us to obtain the volume compression (distortion) estimates of topological mappings of Ball's classes. We prove also that Sobolev homeomorphisms of Ball's classes which possess the Luzin $N$-property are absolutely continuous with respect to capacity.