On defect of compactness for Sobolev spaces on manifolds

TL;DR

This paper investigates the defect of compactness in Sobolev space embeddings on Riemannian manifolds with bounded geometry, introducing a profile decomposition involving elementary concentrations on different manifolds.

Contribution

It develops a profile decomposition framework for Sobolev spaces on manifolds, extending the concept to include profiles on different limiting manifolds.

Findings

Profile decomposition involves elementary concentrations with disjoint supports.

Profiles satisfy a Plancherel-type inequality.

A Brezis-Lieb type relation holds for Lebesgue norms of profiles.

Abstract

Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on manifolds different from M, that are induced by a limiting procedure. The profiles satisfy an inequality of Plancherel type, and a similar relation, related to the Brezis-Lieb Lemma, holds for Lebesgue norms of profiles on the respective manifolds.