Uniform Continuity and Br\'ezis-Lieb Type Splitting for Superposition Operators in Sobolev Space

TL;DR

This paper proves a variant of the Brezis-Lieb-Lemma under weaker conditions using concentration-compactness, and establishes uniform continuity of superposition operators in Sobolev spaces, advancing understanding of nonlinear analysis.

Contribution

It introduces a new version of the Brezis-Lieb-Lemma with relaxed assumptions and demonstrates uniform continuity of superposition operators in Sobolev spaces.

Findings

A weaker assumption version of the Brezis-Lieb-Lemma is established.

Uniform continuity of superposition operators in Sobolev spaces is proven.

The results have implications for nonlinear analysis and PDEs.

Abstract

Using concentration-compactness arguments we prove a variant of the Brezis-Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.