Profile decomposition in Sobolev spaces and decomposition of integral functionals I: inhomogeneous case

TL;DR

This paper develops a profile decomposition method for bounded sequences in inhomogeneous Sobolev spaces, accounting for translation symmetries, and applies it to decompose integral functionals of subcritical order.

Contribution

It introduces a novel profile decomposition framework in inhomogeneous Sobolev spaces using group actions, extending previous results to the inhomogeneous case.

Findings

Profiles are characterized by bounded sequences and group actions.

Decomposition of the Sobolev norm into profiles is bounded by the sequence's norm.

Integral functionals of subcritical order can be decomposed using the profiles.

Abstract

The present paper is devoted to analysis of the lack of compactness of bounded sequences in \emph{inhomogeneous} Sobolev spaces, where bounded sequences might fail to be compact due to an isometric group action, that is, \emph{translation}. It will be proved that every bounded sequence $(u_n)$ has (possibly infinitely many) \emph{profiles}, and then the sequence is asymptotically decomposed into a sum of translated profiles and a double-suffixed residual term, where the residual term becomes arbitrarily small in appropriate Lebesgue or Sobolev spaces of lower order. To this end, functional analytic frameworks are established in an abstract way by making use of a group action $G$, in order to characterize profiles by $(u_n)$ and $G$. One also finds that a decomposition of the Sobolev norm into profiles is bounded by the supremum of the norm of $u_n$. Moreover, the profile decomposition leads to results of decomposition of integral functionals of subcritical order. It is noteworthy that the space where the decomposition of integral functionals holds is the same as that where the residual term is vanishing.