Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, p. 51-71
The first eigenvalue of the p-Laplacian on an open set of given measure attains its minimum value if and only if the set is a ball. We provide a quantitative version of this statement by an argument that can be easily adapted to treat also certain isocapacitary and Cheeger inequalities.
Classification:  49R50,  35J20,  49J40,  26D20
@article{ASNSP_2009_5_8_1_51_0,
     author = {Fusco, Nicola and Maggi, Francesco and Pratelli, Aldo},
     title = {Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {1},
     year = {2009},
     pages = {51-71},
     zbl = {1176.49047},
     mrnumber = {2512200},
     language = {en},
     url = {http://http://www.numdam.org/item/ASNSP_2009_5_8_1_51_0}
}
Fusco, Nicola; Maggi, Francesco; Pratelli, Aldo. Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 51-71. http://www.numdam.org/item/ASNSP_2009_5_8_1_51_0/

[1] F. Alter, V. Caselles and A. Chambolle, A characterization of convex calibrable sets in n , Math. Ann. 332 (2005), 329–366. | MR 2178065 | Zbl 1108.35073

[2] M. Belloni and B. Kawohl, A direct uniqueness proof for equations involving the p-Laplace operator, Manuscripta Math. 109 (2002), 229–231. | MR 1935031 | Zbl 1100.35032

[3] T. Bhattacharya, Some observations on the first eigenvalue of the p-Laplacian and its connections with asymmetry, Electron. J. Differential Equations 35 (2001), 15 pp. | MR 1836803 | Zbl 0991.35032

[4] T. Bhattacharya and A. Weitsman, Bounds for capacities in terms of asymmetry, Rev. Mat. Iberoamericana 12 (1996), 593–639. | MR 1435477 | Zbl 0870.31001

[5] J. Brothers and W. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math. 384 (1988), 153–179. | MR 929981 | Zbl 0633.46030

[6] G. Buttazzo, G. Carlier and M. Comte, On the selection of maximal Cheeger sets, Differential Integral Equations 20 (2007) 991–1004. | MR 2349376 | Zbl 1212.49019

[7] G. Carlier and M. Comte, On a weighted total variation minimization problem, J. Funct. Anal. 250 (2007), 214–226. | MR 2345913 | Zbl 1120.49011

[8] V. Caselles, A. Chambolle and M. Novaga, Uniqueness of the Cheeger set of a convex body, Pacific J. Math. 232 (2007), 77–90. | MR 2358032 | Zbl 1221.35171

[9] A. Cianchi and N. Fusco, Functions of bounded variation and rearrangements, Arch. Ration. Mech. Anal. 165 (2002), 1–40. | MR 1947097 | Zbl 1028.49035

[10] L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. | MR 1158660 | Zbl 0804.28001

[11] C. Faber, Beweis dass unter allen homogen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. der Wiss. Math.-Phys., Munich (1923), 169–172. | JFM 49.0342.03

[12] W. H. Fleming and R. Rishel, An integral formula for total gradient variation, Arch. Math. (Basel) 11 (1960), 218–222. | MR 114892 | Zbl 0094.26301

[13] V. Fridman and B. Kawohl, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolin. 44 (2003), 659–667. | MR 2062882 | Zbl 1105.35029

[14] B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in n , Trans. Amer. Math. Soc. 314 (1989), 619–638. | MR 942426 | Zbl 0679.52007

[15] N. Fusco, F. Maggi and A. Pratelli, The quantitative sharp isoperimetric inequality, Ann. of Math. 168 (2008), 941–980. | MR 2456887 | Zbl 1187.52009

[16] R. R. Hall, A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math. 428 (1992), 161–176. | MR 1166511 | Zbl 0746.52012

[17] R. R. Hall, W.K. Hayman and A. W. Weitsman, On asymmetry and capacity, J. Anal. Math. 56 (1991), 87–123. | MR 1243100 | Zbl 0747.31004

[18] W. Hansen and N. Nadirashvili, Isoperimetric inequalities in potential theory, In: “Proceedings from the International Conference on Potential Theory” (Amersfoort, 1991), Potential Anal. 3 (1994), 1–14. | MR 1266215 | Zbl 0796.31003

[19] B. Kawohl and T. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plans, Pacific J. Math. 225 (2006), 103–118. | MR 2233727 | Zbl 1133.52002

[20] B. Kawohl and M. Novaga, The p-Laplace eigenvalue problem as p1 and Cheeger sets in a Finsler metric, J. Convex Anal. 15 (2008), 623–634. | MR 2431415 | Zbl 1186.35115

[21] E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), 97–100. | JFM 51.0356.05 | MR 1512244

[22] E. Krahn, Über Minimaleigenschaft des Kugel in drei und mehr Dimensionen, Acta Comment Univ. Tartu Math. (Dorpat) A9 (1926), 1–44. | JFM 52.0510.03

[23] E. H. Lieb and M. Loss, “Analysis”, Second edition, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001. | MR 1817225

[24] P. Lindqvist, On the equation div (|u| p-1 u)+λ|u| p-2 u=0, Proc. Amer. Math. Soc. 109 (1990), 157–164. | MR 1007505 | Zbl 0714.35029

[25] A. Melas, The stability of some eigenvalue estimates, J. Differential Geom. 36 (1992), 19–33. | MR 1168980 | Zbl 0770.35049

[26] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182–1238. | MR 500557 | Zbl 0411.52006

[27] R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), 1–29. | MR 519520 | Zbl 0404.52012

[28] G. Pólya and G. Szegö, “Isoperimetric Inequalities in Mathematical Physics”, Annals of Mathematics Studies, Vol. 27, Princeton University Press, Princeton, NJ, 1951. | MR 43486 | Zbl 0044.38301

[29] J. W. Strutt (Lord Rayleigh), “The Theory of Sound”, MacMillan, New York, 1877, 1894; Dover, New York, 1945. | MR 16009

[30] G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl. IV 110 (1976), 353–372. | MR 463908 | Zbl 0353.46018