Mathematical analysis/Partial differential equations
An analytic proof of the planar quantitative isoperimetric inequality
[Une démonstration analytique de l'inégalité isopérimétrique quantitative dans le plan]

Présenté par : Haïm Brézis

Li, Guohua1 ; Zhao, Xinyu1 ; Ding, Zongqi1 ; Jiang, Renjin1, 2
1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, 100875 Beijing, People's Republic of China
2 Department of Mathematics, University Autònoma Barcelona, 08193 Bellaterra (Barcelona), Spain
Comptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 589-593.

On donne une démonstration analytique de l'inégalité isopérimétrique quantitative dans le plan, et on établit une estimation de la borne supérieure de la constante en maximisant la norme L∞ du gradient de la solution de l'équation de Poisson.

We give an analytic proof of the quantitative isoperimetric inequality in the plane and give an estimation of the upper bound of the constant via maximizing the L∞-norm of the gradient of solutions to the Poisson equation.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.04.006
Li, Guohua 1 ; Zhao, Xinyu 1 ; Ding, Zongqi 1 ; Jiang, Renjin 1, 2

1 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, 100875 Beijing, People's Republic of China
2 Department of Mathematics, University Autònoma Barcelona, 08193 Bellaterra (Barcelona), Spain 
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Li, Guohua; Zhao, Xinyu; Ding, Zongqi; Jiang, Renjin. An analytic proof of the planar quantitative isoperimetric inequality. Comptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 589-593. doi : 10.1016/j.crma.2015.04.006. http://www.numdam.org/articles/10.1016/j.crma.2015.04.006/

[1] Alvino, A.; Ferone, V.; Nitsch, C. A sharp isoperimetric inequality in the plane, J. Eur. Math. Soc., Volume 13 (2011), pp. 185-206

[2] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, 2000 (xviii+434 pp)

[3] Campi, S. Isoperimetric deficit and convex plane sets of maximum translative discrepancy, Geom. Dedic., Volume 43 (1992), pp. 71-81

[4] Chavel, I. Isoperimetric Inequalities, Cambridge Tracts in Math., vol. 145, Cambridge University Press, Cambridge, UK, 2001

[5] Cianchi, A. Maximizing the L∞-norm of the gradient of solutions to the Poisson equation, J. Geom. Anal., Volume 2 (1992), pp. 499-515

[6] M. Cicalese, G.P. Leonardi, On the absolute minimizers of the quantitative isoperimetric quotient in the plane, preprint.

[7] Cicalese, M.; Leonardi, G.P. A selection principle for the sharp quantitative isoperimetric inequality, Arch. Ration. Mech. Anal., Volume 206 (2012), pp. 617-643

[8] Cicalese, M.; Leonardi, G.P. Best constants for the isoperimetric inequality in quantitative form, J. Eur. Math. Soc., Volume 15 (2013), pp. 1101-1129

[9] De Giorgi, E. Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita, Atti Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat., Sez. I: Mat. Mecc. Astron. Geod. Geofis., Volume 8 (1958), pp. 33-44

[10] Evans, L.C.; Gariepy, R.F. Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, USA, 1992 (viii+268 p)

[11] Figalli, A.; Maggi, F.; Pratelli, A. A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., Volume 182 (2010), pp. 167-211

[12] Fusco, N. The classical isoperimetric theorem, Rend. Accad. Sci. Fis. Mat. Napoli, Volume 71 (2004), pp. 63-107

[13] Fusco, N.; Maggi, F.; Pratelli, A. The sharp quantitative isoperimetric inequality, Ann. of Math. (2), Volume 168 (2008), pp. 941-980

[14] Fusco, N.; Julin, V. A strong form of the quantitative isoperimetric inequality, Calc. Var. Partial Differ. Equ., Volume 50 (2014), pp. 925-937

[15] Hall, R.R. A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math., Volume 428 (1992), pp. 161-176

[16] Hall, R.R.; Hayman, W.K.; Weitsman, A.W. On asymmetry and capacity, J. Anal. Math., Volume 56 (1991), p. 87123

[17] Jiang, R.; Koskela, P. Isoperimetric inequality from the Poisson equation via curvature, Commun. Pure Appl. Math., Volume 65 (2012), pp. 1145-1168

[18] Maz'ya, V. Classes of regions and imbedding theorems for function spaces, Dokl. Akad. Nauk SSSR, Volume 133 (1960), pp. 527-530 (in Russian). English translation: Sov. Math. Dokl., 1, 1960, pp. 882-885

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