English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper is devoted to the description of the lack of compactness of theSobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$. Itturns out that up to cores our result is expressed in terms of theconcentration-type examples derived by J. Moser in \cite{M} as in the radialsetting investigated in \cite{BMM}. However, the analysis we used in this workis strikingly different from the one conducted in the radial case which isbased on an $L^ \infty$ estimate far away from the origin and which is nolonger valid in the general framework. Within the general framework of$H^1(\R^2)$, the strategy we adopted to build the profile decomposition interms of examples by Moser concentrated around cores is based on capacityarguments and relies on an extraction process of mass concentrations. Theessential ingredient to extract cores consists in proving by contradiction thatif the mass responsible for the lack of compactness of the Sobolev embedding inthe Orlicz space is scattered, then the energy used would exceed that of thestarting sequence.

Lack of compactness in the 2D critical Sobolev embedding, the general case