Composition operators on Sobolev spaces, capacity estimates and Ball's classes

In this paper we study an interplay between composition operators on Sobolev spaces, corresponding capacitary estimates of so-called "singular" sets and Ball’s classes in context of nonlinear elasticity problems. The suggested approach is based on characterization of Ball’s classes Aq,r(Ω) in terms of composition operators on Sobolev spaces. On this base we prove that topological mappings of Ball’s classes which possess the Luzin N-property are absolutely continuous with respect to capacitory measure. The natural generalization of Ball’s classes in terms of the composition operators is given.