Characterizations of the Sobolev norms and the total variation via nonlocal functionals, and related problems

Hoai-Minh Nguyen

doi:10.5802/crmath.802

Article de recherche - Analyse fonctionnelle

Characterizations of the Sobolev norms and the total variation via nonlocal functionals, and related problems
[Caractérisations des normes de Sobolev et de la variation totale par des fonctionnelles non locales, et problèmes liés]

Hoai-Minh Nguyen ¹

¹ Sorbonne Université, Universitsé Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1429-1455

Cet article fait partie du numéro thématique De génération en génération : l'héritage mathématique de Haïm Brezis coordonné par Henri Berestycki et al..

Abstract (VO)
Résumé

We briefly discuss the contribution of Haïm Brezis and his co-authors on the characterizations of the Sobolev norms and the total variation using non-local functionals. Some ideas of the analysis are given and new results are presented.

Nous discutons brièvement de la contribution de Haïm Brezis et de ses co-auteurs concernant les caractérisations des normes de Sobolev et de la variation totale en utilisant fonctionnelles non locales. Certaines idées de l’analyse sont présentées, ainsi que de nouveaux résultats.

Reçu le : 2025-07-01
Révisé le : 2025-09-15
Accepté le : 2025-10-09
Publié le : 2025-11-28

DOI : 10.5802/crmath.802

Classification : 26B15, 26B25, 26B30, 42B25
Keywords: Sobolev norms, total variations, Gamma-convergence, inequalities, non-local functionals
Mots-clés : Normes de Sobolev, variation totale, Gamma-convergence, inégalités, fonctionnelles non locales

Affiliations des auteurs :

Hoai-Minh Nguyen ¹

¹ Sorbonne Université, Universitsé Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, 75005 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Hoai-Minh Nguyen},
     title = {Characterizations of the {Sobolev} norms and the total variation via nonlocal functionals, and related problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1429--1455},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.802},
     language = {en},
}

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Hoai-Minh Nguyen. Characterizations of the Sobolev norms and the total variation via nonlocal functionals, and related problems. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1429-1455. doi: 10.5802/crmath.802

Bibliographie
Cité par

[1] Robert A. Adams Sobolev spaces, Pure and Applied Mathematics, Academic Press Inc., 1975 no. 65 | MR | Zbl

[2] Luigi Ambrosio; Guido De Philippis; Luca Martinazzi Gamma-convergence of nonlocal perimeter functionals, Manuscr. Math., Volume 134 (2011) no. 3–4, pp. 377-403 | DOI | MR | Zbl

[3] Clara Antonucci; Massimo Gobbino; Matteo Migliorini; Nicola Picenni Optimal constants for a nonlocal approximation of Sobolev norms and total variation, Anal. PDE, Volume 13 (2020) no. 2, pp. 595-625 | DOI | MR | Zbl

[4] Fabrice Bethuel; Haïm Brezis; Frédéric Hélein Ginzburg–Landau vortices, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, 1994 no. 13 | DOI | MR | Zbl

[5] Jean Bourgain; Haïm Brezis; Petru Mironescu Another look at Sobolev spaces, Optimal control and partial differential equations (José Luis Menaldi; Edmundo Rofman; Agnès Sulem, eds.), IOS Press, 2001, pp. 439-455 | MR | Zbl

[6] Jean Bourgain; Haïm Brezis; Petru Mironescu Complements to the paper: “Lifting, degree, and distributional Jacobian revisited” (2004) | HAL

[7] Jean Bourgain; Haïm Brezis; Petru Mironescu Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math., Volume 58 (2005) no. 4, pp. 529-551 | DOI | MR | Zbl

[8] Jean Bourgain; Haïm Brezis; Hoai-Minh Nguyen A new estimate for the topological degree, Comptes Rendus. Mathématique, Volume 340 (2005) no. 11, pp. 787-791 | DOI | MR | Numdam | Zbl

[9] Jean Bourgain; Hoai-Minh Nguyen A new characterization of Sobolev spaces, Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 75-80 | DOI | MR | Numdam | Zbl

[10] Haïm Brezis How to recognize constant functions. A connection with Sobolev spaces, Usp. Mat. Nauk, Volume 57 (2002) no. 4(346), pp. 59-74 | DOI | MR | Zbl

[11] Haïm Brezis Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, 2011 | MR | Zbl | DOI

[12] Haïm Brezis Some of my favorite open problems, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 34 (2023) no. 2, pp. 307-335 | DOI | MR | Zbl

[13] Haïm Brezis; Hoai-Minh Nguyen On a new class of functions related to VMO, Comptes Rendus. Mathématique, Volume 349 (2011) no. 3–4, pp. 157-160 | DOI | MR | Numdam

[14] Haïm Brezis; Hoai-Minh Nguyen The BBM formula revisited, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 27 (2016) no. 4, pp. 515-533 | DOI | MR | Zbl

[15] Haïm Brezis; Hoai-Minh Nguyen Non-local functionals related to the total variation and connections with image processing, Ann. PDE, Volume 4 (2018) no. 1, 9, 77 pages | DOI | MR | Zbl

[16] Haïm Brezis; Hoai-Minh Nguyen $Γ$ -convergence of non-local, non-convex functionals in one dimension, Commun. Contemp. Math., Volume 22 (2020) no. 7, 1950077, 27 pages | DOI | MR | Zbl

[17] Haïm Brezis; Andreas Seeger; Jean Van Schaftingen; Po-Lam Yung Sobolev spaces revisited, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 33 (2022) no. 2, pp. 413-437 | DOI | MR | Zbl

[18] Haïm Brezis; Andreas Seeger; Jean Van Schaftingen; Po-Lam Yung Families of functionals representing Sobolev norms, Anal. PDE, Volume 17 (2024) no. 3, pp. 943-979 | DOI | MR | Zbl

[19] Haïm Brezis; Jean Van Schaftingen; Po-Lam Yung A surprising formula for Sobolev norms, Proc. Natl. Acad. Sci. USA, Volume 118 (2021) no. 8, e2025254118, 6 pages | DOI | MR | Zbl

[20] Haïm Brezis; Jean Van Schaftingen; Po-Lam Yung Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail, Calc. Var. Partial Differ. Equ., Volume 60 (2021) no. 4, 129, 12 pages | DOI | MR | Zbl

[21] L. Caffarelli; R. Kohn; L. Nirenberg First order interpolation inequalities with weights, Compos. Math., Volume 53 (1984) no. 3, pp. 259-275 | MR | Numdam

[22] J. Dávila On an open question about functions of bounded variation, Calc. Var. Partial Differ. Equ., Volume 15 (2002) no. 4, pp. 519-527 | DOI | MR | Zbl

[23] Oscar Domínguez; Mario Milman New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators, Adv. Math., Volume 411 (2022), 108774, 76 pages | DOI | MR | Zbl

[24] Lawrence C. Evans; Ronald F. Gariepy Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, 1992 | MR | Zbl

[25] Massimo Gobbino; Nicola Picenni Gamma-liminf estimate for a class of non-local approximations of Sobolev and BV norms, J. Funct. Anal., Volume 289 (2025) no. 9, 111106, 25 pages | DOI | MR | Zbl

[26] F. John; L. Nirenberg On functions of bounded mean oscillation, Commun. Pure Appl. Math., Volume 14 (1961), pp. 415-426 | DOI | MR | Zbl

[27] Giovanni Leoni; Daniel Spector Characterization of Sobolev and $B V$ spaces, J. Funct. Anal., Volume 261 (2011) no. 10, pp. 2926-2958 | DOI | MR | Zbl

[28] Arka Mallick; Hoai-Minh Nguyen Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg interpolation inequalities associated with Coulomb–Sobolev spaces, J. Funct. Anal., Volume 283 (2022) no. 10, 109662, 33 pages | DOI | MR | Zbl

[29] Vladimir Maz’ya Sobolev spaces with applications to elliptic partial differential equations, Grundlehren der Mathematischen Wissenschaften, Springer, 2011 no. 342 | DOI | MR | Zbl

[30] Vladimir Maz’ya; T. Shaposhnikova On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Volume 195 (2002) no. 2, pp. 230-238 | DOI | MR | Zbl

[31] Hoai-Minh Nguyen Some new characterizations of Sobolev spaces, J. Funct. Anal., Volume 237 (2006) no. 2, pp. 689-720 | DOI | MR

[32] Hoai-Minh Nguyen $Γ$ -convergence and Sobolev norms, Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 679-684 | DOI | MR | Numdam

[33] Hoai-Minh Nguyen Optimal constant in a new estimate for the degree, J. Anal. Math., Volume 101 (2007), pp. 367-395 | DOI | MR

[34] Hoai-Minh Nguyen Further characterizations of Sobolev spaces, J. Eur. Math. Soc., Volume 10 (2008) no. 1, pp. 191-229 | MR | DOI

[35] Hoai-Minh Nguyen $Γ$ -convergence, Sobolev norms, and BV functions, Duke Math. J., Volume 157 (2011) no. 3, pp. 495-533 | DOI | MR

[36] Hoai-Minh Nguyen Some inequalities related to Sobolev norms, Calc. Var. Partial Differ. Equ., Volume 41 (2011) no. 3–4, pp. 483-509 | DOI | MR

[37] Hoai-Minh Nguyen A refined estimate for the topological degree, Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1046-1049 | DOI | MR | Numdam

[38] Hoai-Minh Nguyen; Andrea Pinamonti; Marco Squassina; Eugenio Vecchi New characterizations of magnetic Sobolev spaces, Adv. Nonlinear Anal., Volume 7 (2018) no. 2, pp. 227-245 | DOI | MR

[39] Hoai-Minh Nguyen; Marco Squassina Fractional Caffarelli–Kohn–Nirenberg inequalities, J. Funct. Anal., Volume 274 (2018) no. 9, pp. 2661-2672 | DOI | MR

[40] Hoai-Minh Nguyen; Marco Squassina On anisotropic Sobolev spaces, Commun. Contemp. Math., Volume 21 (2019) no. 1, 1850017, 13 pages | DOI | MR

[41] Hoai-Minh Nguyen; Marco Squassina On Hardy and Caffarelli–Kohn–Nirenberg inequalities, J. Anal. Math., Volume 139 (2019) no. 2, pp. 773-797 | DOI | MR

[42] Nicola Picenni New estimates for a class of non-local approximations of the total variation, J. Funct. Anal., Volume 287 (2024) no. 1, 110419, 21 pages | DOI | MR

[43] Andrea Pinamonti; Marco Squassina; Eugenio Vecchi Magnetic BV-functions and the Bourgain–Brezis–Mironescu formula, Adv. Calc. Var., Volume 12 (2019) no. 3, pp. 225-252 | DOI | MR

[44] Arkady Poliakovsky Some remarks on a formula for Sobolev norms due to Brezis, Van Schaftingen and Yung, J. Funct. Anal., Volume 282 (2022) no. 3, 109312, 47 pages | DOI | MR

[45] Augusto C. Ponce A new approach to Sobolev spaces and connections to $Γ$ -convergence, Calc. Var. Partial Differ. Equ., Volume 19 (2004) no. 3, pp. 229-255 | DOI | MR

[46] Augusto C. Ponce; Daniel Spector On formulae decoupling the total variation of BV functions, Nonlinear Anal., Theory Methods Appl., Volume 154 (2017), pp. 241-257 | DOI | MR

[47] Elias M. Stein Singular integrals and differentiability properties of functions, Princeton Mathematical Series, Princeton University Press, 1970 no. 30 | MR

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