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Lack of compactness in the 2D critical Sobolev embedding, the general case
Analysis of PDEs · Submitted 13 Dec 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$.
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The abstract, sentence by sentence
This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$. It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of $H^1(\R^2)$, the strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. The essential ingredient to extract cores consists in proving by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$. | contribution 0.84 | 0.51 | 0.33 | asserted 0.91 |
| s1 | It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. | result 0.53 | 0.56 | 0.24 | asserted 0.97 |
| s2 | However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. | method 0.49 | 0.40 | 0.05 | asserted 1.00 |
| s3 | Within the general framework of $H^1(\R^2)$, the strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. | method 0.97 | 0.93 | 0.14 | asserted 1.00 |
| s4 | The essential ingredient to extract cores consists in proving by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence. | method 0.98 | 0.89 | 0.15 | asserted 1.00 |
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The paper
Original abstract
This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$. It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of $H^1(\R^2)$, the strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. The essential ingredient to extract cores consists in proving by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence.
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How the literature uses it
2 arXiv papers cite this in 4 sentences
Papers citing it · 2 sentences without a confident label
Uses its method or tool 1 paper
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The main idea to handle the second case which is the more challenging is the explicit description of that situation by means of the example by Moser settled in [ref] .
Scattering for the critical 2-D NLS with exponential growth · General scheme of the proof · 0.87
Cites it as background 1 paper
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In addition, in [ref] , [ref] and [ref] H.
Characterization of the lack of compactness of $H^2_{rad}(\R^4)$ into the Orlicz space · Development in critical Sobolev embedding · 0.84
What it cites · 1 sentence without a confident label
Cites as background 2 papers
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Tintarev in [ref] .
On compactness in the Trudinger-Moser inequality · Main results · 0.95
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Among others, one can mention [ref] .
On the stability in weak topology of the set of global solutions to the Navier-Stokes equations · Critical $2D$ Sobolev embedding · 0.86
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