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Eric A Carlen - One of the best experts on this subject based on the ideXlab platform.
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stability for a gns Inequality and the log hls Inequality with application to the critical mass keller segel equation
Duke Mathematical Journal, 2013Co-Authors: Eric A Carlen, Alessio FigalliAbstract:Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev Inequality, we deduce several dierent stability results for a Gagliardo-Nirenberg-Sobolev Inequality in the plane. Then, exploiting the connection between this Inequality and a fast diusion equation, we get stability for the Log-HLS Inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller-Segel system.
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stability for a gns Inequality and the log hls Inequality with application to the critical mass keller segel equation
arXiv: Analysis of PDEs, 2011Co-Authors: Eric A Carlen, Alessio FigalliAbstract:Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev Inequality, we deduce several different stability results for a Gagliardo-Nirenberg-Sobolev Inequality in the plane. Then, exploiting the connection between this Inequality and a fast diffusion equation, we get a quantitative stability for the Log-HLS Inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller-Segel system.
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hardy littlewood Sobolev inequalities via fast diffusion flows
Proceedings of the National Academy of Sciences of the United States of America, 2010Co-Authors: Eric A Carlen, Jose A Carrillo, Michael LossAbstract:We give a simple proof of the λ = d - 2 cases of the sharp Hardy-Littlewood-Sobolev Inequality for d≥3, and the sharp Logarithmic Hardy-Littlewood-Sobolev Inequality for d = 2 via a monotone flow governed by the fast diffusion equation.
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superadditivity of fisher s information and logarithmic Sobolev inequalities
Journal of Functional Analysis, 1991Co-Authors: Eric A CarlenAbstract:Abstract We prove a theorem characterizing Gaussian functions and we prove a strict superaddivity property of the Fisher information. We use these results to determine the cases of equality in the logarithmic Sobolev Inequality on R n equipped with Lebesgue measure and with Gauss measure. We also prove a strengthened form of Gross's logarithmic Sobolev Inequality with a “remainder term” added to the left side. Finally we show that the strict form of Gross's Inequality is a direct consequence of an Inequality due to Blachman and Stam, and that this in turn is a direct consequence of strict superadditivity of the Fisher information.
Alessio Figalli - One of the best experts on this subject based on the ideXlab platform.
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gradient stability for the Sobolev Inequality the case p geq 2
Journal of the European Mathematical Society, 2018Co-Authors: Alessio Figalli, Robin NeumayerAbstract:We show a strong form of the quantitative Sobolev Inequality in $\mathbb{R}^n$ for $p\geq 2$, where the deficit of a function $u\in \dot W^{1,p} $ controls $\| \nabla u -\nabla v\|_{L^p}$ for an extremal function $v$ in the Sobolev Inequality.
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gradient stability for the Sobolev Inequality the case p geq 2
arXiv: Analysis of PDEs, 2015Co-Authors: Alessio Figalli, Robin NeumayerAbstract:We prove a strong form of the quantitative Sobolev Inequality in $\mathbb{R}^n$ for $p\geq 2$, where the deficit of a function $u\in \dot W^{1,p} $ controls $\| \nabla u -\nabla v\|_{L^p}$ for an extremal function $v$ in the Sobolev Inequality.
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sharp stability theorems for the anisotropic Sobolev and log Sobolev inequalities on functions of bounded variation
Advances in Mathematics, 2013Co-Authors: Alessio Figalli, Francesco Maggi, Aldo PratelliAbstract:Abstract Combining rearrangement techniques with Gromov’s proof (via optimal mass transportation) of the 1-Sobolev Inequality, we prove a sharp quantitative version of the anisotropic Sobolev Inequality on B V ( R n ) . We also deduce, as a corollary of this result, a sharp stability estimate for the anisotropic 1-log-Sobolev Inequality.
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stability for a gns Inequality and the log hls Inequality with application to the critical mass keller segel equation
Duke Mathematical Journal, 2013Co-Authors: Eric A Carlen, Alessio FigalliAbstract:Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev Inequality, we deduce several dierent stability results for a Gagliardo-Nirenberg-Sobolev Inequality in the plane. Then, exploiting the connection between this Inequality and a fast diusion equation, we get stability for the Log-HLS Inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller-Segel system.
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stability for a gns Inequality and the log hls Inequality with application to the critical mass keller segel equation
arXiv: Analysis of PDEs, 2011Co-Authors: Eric A Carlen, Alessio FigalliAbstract:Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev Inequality, we deduce several different stability results for a Gagliardo-Nirenberg-Sobolev Inequality in the plane. Then, exploiting the connection between this Inequality and a fast diffusion equation, we get a quantitative stability for the Log-HLS Inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller-Segel system.
Boguslaw Zegarlinski - One of the best experts on this subject based on the ideXlab platform.
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the logarithmic Sobolev Inequality for discrete spin systems on a lattice
Communications in Mathematical Physics, 1992Co-Authors: Daniel W. Stroock, Boguslaw ZegarlinskiAbstract:For finite range lattice gases with a finite spin space, it is shown that the Dobrushin-Shlosman mixing condition is equivalent to the existence of a logarithmic Sobolev Inequality for the associated (unique) Gibbs state. In addition, implications of these considerations for the ergodic properties of the corresponding Glauber dynamics are examined.
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the logarithmic Sobolev Inequality for continuous spin systems on a lattice
Journal of Functional Analysis, 1992Co-Authors: Daniel W. Stroock, Boguslaw ZegarlinskiAbstract:Abstract In this article, we give (cf. Theorem 1.6) a sufficient condition on a local specification for the associated Gibbs state to satisfy a logarithmic Sobolev Inequality. In the final section (cf. Theorem 3.10) we relate our condition to the type of mixing condition introduced by Dobrushin and Shlosman in connection with their work on complete analyticity.
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The equivalence of the logarithmic Sobolev Inequality and the Dobrushin-Shlosman mixing condition
Communications in Mathematical Physics, 1992Co-Authors: Daniel W. Stroock, Boguslaw ZegarlinskiAbstract:Given a finite range lattice gas with a compact, continuous spin space, it is shown (cf. Theorem 1.2) that a uniform logarithmic Sobolev Inequality (cf. 1.4) holds if and only if the Dobrushin-Shlosman mixing condition (cf. 1.5) holds. As a consequence of our considerations, we also show (cf. Theorems 3.2 and 3.6) that these conditions are equivalent to a statement about the uniform rate at which the associated Glauber dynamics tends to equilibrium. In this same direction, we show (cf. Theorem 3.19) that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the Glauber dynamics.
Michael Loss - One of the best experts on this subject based on the ideXlab platform.
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hardy littlewood Sobolev inequalities via fast diffusion flows
Proceedings of the National Academy of Sciences of the United States of America, 2010Co-Authors: Eric A Carlen, Jose A Carrillo, Michael LossAbstract:We give a simple proof of the λ = d - 2 cases of the sharp Hardy-Littlewood-Sobolev Inequality for d≥3, and the sharp Logarithmic Hardy-Littlewood-Sobolev Inequality for d = 2 via a monotone flow governed by the fast diffusion equation.
Francis Seuffert - One of the best experts on this subject based on the ideXlab platform.
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an extension of the bianchi egnell stability estimate to bakry gentil and ledoux s generalization of the Sobolev Inequality to continuous dimensions
Journal of Functional Analysis, 2017Co-Authors: Francis SeuffertAbstract:Abstract This paper extends a stability estimate of the Sobolev Inequality established by Bianchi and Egnell in [3] . Bianchi and Egnell's Stability Estimate answers the question raised by H. Brezis and E.H. Lieb in [5] : “Is there a natural way to bound ‖ ∇ φ ‖ 2 2 − C N 2 ‖ φ ‖ 2 N N − 2 2 from below in terms of the ‘distance’ of φ from the manifold of optimizers in the Sobolev Inequality?” Establishing stability estimates – also known as quantitative versions of sharp inequalities – of other forms of the Sobolev Inequality, as well as other inequalities, is an active topic. See [9] , [11] , and [12] , for stability estimates involving Sobolev inequalities and [6] , [11] , and [14] for stability estimates on other inequalities. In this paper, we extend Bianchi and Egnell's Stability Estimate to a Sobolev Inequality for “continuous dimensions.” Bakry, Gentil, and Ledoux have recently proved a sharp extension of the Sobolev Inequality for functions on R + × R n , which can be considered as an extension to “continuous dimensions.” V.H. Nguyen determined all cases of equality. The present paper extends the Bianchi–Egnell stability analysis for the Sobolev Inequality to this “continuous dimensional” generalization.
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an extension of the bianchi egnell stability estimate to bakry gentil and ledoux s generalization of the Sobolev Inequality to continuous dimensions
arXiv: Analysis of PDEs, 2015Co-Authors: Francis SeuffertAbstract:This paper extends a stability estimate of the Sobolev Inequality established by G. Bianchi and H. Egnell in their paper "A note on the Sobolev Inequality." Bianchi and Egnell's Stability Estimate answers the question raised by H. Brezis and E. H. Lieb: "Is there a natural way to bound $\| \nabla \varphi \|_2^2 - C_N^2 \| \varphi \|_\frac{2N}{N-2}^2$ from below in terms of the 'distance' of $\varphi$ from the manifold of optimizers in the Sobolev Inequality?" Establishing stability estimates - also known as quantitative versions of sharp inequalities - of other forms of the Sobolev Inequality, as well as other sharp inequalities, is an active topic. In this paper, we extend Bianchi and Egnell's Stability Estimate to a Sobolev Inequality for "continuous dimensions." Bakry, Gentil, and Ledoux have recently proved a sharp extension of the Sobolev Inequality for functions on $\mathbb{R}_+ \times \mathbb{R}^n$, which can be considered as an extension to "continuous dimensions." V. H. Nguyen determined all cases of equality. The present paper extends the Bianchi-Egnell stability analysis for the Sobolev Inequality to this "continuous dimensional" generalization.
