Comptes Rendus
Partial differential equations/Functional analysis
Logarithmic Sobolev inequality revisited
Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 447-451.

We provide a new characterization of the logarithmic Sobolev inequality.

Nous donnons une nouvelle caractérisation de l'inégalité de Sobolev logarithmique.

Published online:
DOI: 10.1016/j.crma.2017.02.009

Hoai-Minh Nguyen 1; Marco Squassina 2

1 Department of Mathematics, EPFL SBCAMA, Station 8, CH-1015 Lausanne, Switzerland
2 Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121 Brescia, Italy
     author = {Hoai-Minh Nguyen and Marco Squassina},
     title = {Logarithmic {Sobolev} inequality revisited},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {447--451},
     publisher = {Elsevier},
     volume = {355},
     number = {4},
     year = {2017},
     doi = {10.1016/j.crma.2017.02.009},
     language = {en},
AU  - Hoai-Minh Nguyen
AU  - Marco Squassina
TI  - Logarithmic Sobolev inequality revisited
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 447
EP  - 451
VL  - 355
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2017.02.009
LA  - en
ID  - CRMATH_2017__355_4_447_0
ER  - 
%0 Journal Article
%A Hoai-Minh Nguyen
%A Marco Squassina
%T Logarithmic Sobolev inequality revisited
%J Comptes Rendus. Mathématique
%D 2017
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%V 355
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%I Elsevier
%R 10.1016/j.crma.2017.02.009
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Hoai-Minh Nguyen; Marco Squassina. Logarithmic Sobolev inequality revisited. Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 447-451. doi : 10.1016/j.crma.2017.02.009.

[1] R.A. Adams; F.H. Clarke Gross's logarithmic Sobolev inequality: a simple proof, Amer. J. Math., Volume 101 (1979), pp. 1265-1269

[2] J. Bourgain; H-M. Nguyen A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 75-80

[3] J. Bourgain; H. Brezis; P. Mironescu Another look at Sobolev spaces (J.L. Menaldi; E. Rofman; A. Sulem, eds.), Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan's 60th Birthday, IOS Press, Amsterdam, 2001, pp. 439-455

[4] J. Bourgain; H. Brezis; P. Mironescu Limiting embedding theorems for Ws,p when s1 and applications, J. Anal. Math., Volume 87 (2002), pp. 77-101

[5] H. Brezis How to recognize constant functions. Connections with Sobolev spaces, Russ. Math. Surv., Volume 57 (2002), pp. 693-708

[6] H. Brezis New approximations of the total variation and filters in imaging, Rend. Accad. Lincei, Volume 26 (2015), pp. 223-240

[7] H. Brezis; H-M. Nguyen The BBM formula revisited, Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl., Volume 27 (2016), pp. 515-533

[8] H. Brezis; H-M. Nguyen Two subtle convex nonlocal approximations of the BV-norm, Nonlinear Anal., Volume 137 (2016), pp. 222-245

[9] H. Brezis; H-M. Nguyen Non-local functionals related to the total variation and connections with image processing (preprint) | arXiv

[10] T. Cazenave Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., Volume 7 (1983), pp. 1127-1140

[11] T. Cazenave An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, vol. 26, Universidade Federal do Rio de Janeiro, 1996

[12] T. Cazenave; A. Haraux Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., Volume 2 (1980), pp. 21-51

[13] A. Cotsiolis; N. Tavoularis On logarithmic Sobolev inequalities for higher order fractional derivatives, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 205-208

[14] P. d'Avenia, M. Squassina, Ground states for fractional magnetic operators, ESAIM COCV, in press, . | DOI

[15] P. d'Avenia; E. Montefusco; M. Squassina On the logarithmic Schrödinger equation, Commun. Contemp. Math., Volume 16 (2014)

[16] M. Del Pino; J. Dolbeault The optimal Euclidean Lp-Sobolev logarithmic inequality, J. Funct. Anal., Volume 197 (2003), pp. 151-161

[17] L. Gross Logarithmic Sobolev Inequalities, Amer. J. Math., Volume 97 (1975), pp. 1061-1083

[18] E. Lieb; M. Loss Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, 2001

[19] H-M. Nguyen Some new characterizations of Sobolev spaces, J. Funct. Anal., Volume 237 (2006), pp. 689-720

[20] H-M. Nguyen Further characterizations of Sobolev spaces, J. Eur. Math. Soc., Volume 10 (2008), pp. 191-229

[21] H-M. Nguyen Some inequalities related to Sobolev norms, Calc. Var. Partial Differ. Equ., Volume 41 (2011), pp. 483-509

[22] H-M. Nguyen, A. Pinamonti, M. Squassina, E. Vecchi, A new characterization of magnetic Sobolev spaces, in preparation.

[23] A.J. Stam Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inf. Control, Volume 2 (1959), pp. 101-112

[24] W.C. Troy Uniqueness of positive ground state solutions of the logarithmic Schrödinger equation, Arch. Ration. Mech. Anal., Volume 222 (2016), pp. 1581-1600

[25] K.G. Zloshchastiev Logarithmic nonlinearity in theories of quantum gravity: origin of time and observational consequences, Gravit. Cosmol., Volume 16 (2010), pp. 288-297

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