Profile decomposition in Sobolev spaces and decomposition of integral functionals II: homogeneous case

TL;DR

This paper develops a profile decomposition theory for bounded sequences in homogeneous Sobolev spaces, allowing analysis of their lack of compactness and energy distribution, with applications to integral functional decompositions.

Contribution

It introduces a novel profile decomposition framework for homogeneous Sobolev spaces, including energy and integral functional decompositions, addressing the lack of compactness.

Findings

Decomposition of bounded sequences into profiles with dilations and translations.

Residual terms become arbitrarily small in lower-order Lebesgue or Sobolev spaces.

Provides strict decompositions of integral functionals with vanishing residuals.

Abstract

The present paper is devoted to a theory of profile decomposition for bounded sequences in \emph{homogeneous} Sobolev spaces, and it enables us to analyze the lack of compactness of bounded sequences. For every bounded sequence in homogeneous Sobolev spaces, the sequence is asymptotically decomposed into the sum of profiles with dilations and translations and a double suffixed residual term. One gets an energy decomposition in the homogeneous Sobolev norm. The residual term becomes arbitrarily small in the critical Lebesgue or Sobolev spaces of lower order, and then, the results of decomposition of integral functionals are obtained, which are important strict decompositions in the critical Lebesgue or Sobolev spaces where the residual term is vanishing.