no. 2
Potential theory/Calculus of variations
Optimal geometric estimates for fractional Sobolev capacities
[Estimées géométriques optimales pour les capacités fractionnelles de Sobolev]

Présenté par : Haïm Brézis

Xiao, Jie1
1 Department of Mathematics and Statistics, Memorial University, St. John's, NL A1C 5S7, Canada
Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 149-153.

Cette note révèle de nouvelles inégalités exactes mettant la capacité fractionnelle de Sobolev d'un ensemble en relation avec son volume standard et son périmètre fractionnel, respectivement, et démontre, par conséquence, que l'inégalité fractionnelle exacte de Sobolev est equivalente, soit à l'inégalité fractionnelle isocapacitaire exacte, soit à l'inégalité fractionnelle isopérimétrique exacte.

This note discovers new sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter, respectively, and consequently proves that the sharp fractional Sobolev inequality is equivalent to either the sharp fractional isocapacitary inequality or the sharp fractional isoperimetric inequality.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.014
Mots clés : Sharpness, Volume, Fractional Sobolev capacity, Fractional perimeter
Xiao, Jie 1

1 Department of Mathematics and Statistics, Memorial University, St. John's, NL A1C 5S7, Canada 
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Xiao, Jie. Optimal geometric estimates for fractional Sobolev capacities. Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 149-153. doi : 10.1016/j.crma.2015.10.014. http://www.numdam.org/articles/10.1016/j.crma.2015.10.014/

[1] Adams, D.R. Lectures on Lp-Potential Theory, University of Umea, Department of Mathematics, 1981 (2)

[2] Adams, D.R. Besov capacity redux, J. Math. Sci. (N.Y.), Volume 162 (2009), pp. 307-318

[3] Adams, D.R.; Xiao, J. Strong type estimates for homogeneous Besov capacities, Math. Ann., Volume 325 (2003), pp. 695-709

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

[5] Bourgain, J.; Brezis, H.; Mironescu, P. Limiting embedding theorems for Ws,p when s↑1 and applications, J. Anal. Math., Volume 87 (2002), pp. 77-101

[6] Federer, H.; Fleming, W.H. Normal and integral currents, Ann. Math., Volume 72 (1960), pp. 458-520

[7] Frank, R.L.; Seiringer, R. Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., Volume 252 (2008), pp. 3407-3430

[8] Fusco, N.; Millot, V.; Morini, M. A quantitative isoperimetric inequality for fractional perimeters, J. Funct. Anal., Volume 261 (2011), pp. 697-715

[9] Heinonen, J.; Kilpeläinen, T.; Martio, O. Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, UK, 1993

[10] Hurri-Syrjänen, R.; Vähäkangas, A.V. Characterizations to the fractional Sobolev inequality, 2013 | arXiv

[11] Ludwig, M. Anisotropic fractional Sobolev norms, Adv. Math., Volume 252 (2014), pp. 150-157

[12] Ludwig, M. Anisotropic fractional perimeters, J. Differ. Geom., Volume 96 (2014), pp. 77-93

[13] Maz'ya, V. Classes of domains and imbedding theorems for function spaces, Dokl. Akad. Nauk SSSR, Volume 3 (1960), pp. 527-530 (Russian); English transl. Sov. Math. Dokl., 1, 1961, pp. 882-885

[14] Maz'ya, V. Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer, 2011

[15] Maz'ya, V.; Shaposhnikova, T. On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Volume 195 (2002), pp. 230-238

[16] Maz'ya, V.; Shaposhnikova, T. Erratum to: “On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces” [J. Funct. Anal. 195 (2002) 230–238], J. Funct. Anal., Volume 201 (2003), pp. 298-300

[17] Silvestre, P. Capacities and embeddings via symmetrization and conductor inequalities, Proc. Amer. Math. Soc., Volume 142 (2014), pp. 497-505

[18] Visintin, A. Nonconvex functionals related to multiphase systems, SIAM J. Math. Anal., Volume 21 (1990), pp. 1281-1304

[19] Xiao, J. Homogeneous endpoint Besov space embeddings by Hausdorff capacity and heat equation, Adv. Math., Volume 207 (2006), pp. 828-846

[20] Xiao, J.; Xiao, J. The sharp Sobolev and isoperimetric inequalities split twice, Adv. Math., Volume 211 (2007), pp. 417-435 (Corrigendum)

[21] Xiao, J. The p-Faber–Krahn inequality noted, Around the Research of Vladimir Maz'ya I. Function Spaces, Springer, 2010, pp. 373-390

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