Comptes Rendus
Mathematical analysis/Partial differential equations
An analytic proof of the planar quantitative isoperimetric inequality
Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 589-593.

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.

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.

Published online:
DOI: 10.1016/j.crma.2015.04.006

Guohua Li 1; Xinyu Zhao 1; Zongqi Ding 1; Renjin Jiang 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
     author = {Guohua Li and Xinyu Zhao and Zongqi Ding and Renjin Jiang},
     title = {An analytic proof of the planar quantitative isoperimetric inequality},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {589--593},
     publisher = {Elsevier},
     volume = {353},
     number = {7},
     year = {2015},
     doi = {10.1016/j.crma.2015.04.006},
     language = {en},
AU  - Guohua Li
AU  - Xinyu Zhao
AU  - Zongqi Ding
AU  - Renjin Jiang
TI  - An analytic proof of the planar quantitative isoperimetric inequality
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 589
EP  - 593
VL  - 353
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2015.04.006
LA  - en
ID  - CRMATH_2015__353_7_589_0
ER  - 
%0 Journal Article
%A Guohua Li
%A Xinyu Zhao
%A Zongqi Ding
%A Renjin Jiang
%T An analytic proof of the planar quantitative isoperimetric inequality
%J Comptes Rendus. Mathématique
%D 2015
%P 589-593
%V 353
%N 7
%I Elsevier
%R 10.1016/j.crma.2015.04.006
%G en
%F CRMATH_2015__353_7_589_0
Guohua Li; Xinyu Zhao; Zongqi Ding; Renjin Jiang. An analytic proof of the planar quantitative isoperimetric inequality. Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 589-593. doi : 10.1016/j.crma.2015.04.006.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[18] V. Maz'ya 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

Cited by Sources:

Comments - Policy