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

Hoai-Minh Nguyen

doi:10.5802/crmath.802

Research article - Functional analysis

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

Hoai-Minh Nguyen ¹

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

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

This article is part of the thematic issue From Generation to Generation: The Mathematical Legacy of Haïm Brezis edited by Henri Berestycki et al..

Abstract (Original language)
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.

Received: 2025-07-01
Revised: 2025-09-15
Accepted: 2025-10-09
Published online: 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

Author's affiliations:

Hoai-Minh Nguyen ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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

References
Cited by

[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 | MR | DOI | 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 | MR | DOI | 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 | MR | DOI

[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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