A note on cheeger sets

Proceedings of the American Mathematical Society

Volumn 137, Issue 6, 2009, Pages 2057-2062

  Figalli, Alessio ^a,b   Maggi, Francesco ^c   Pratelli, Aldo ^d  

^a UNIVERSITÉ CÔTE D'AZUR   (France)

^b CENTRE DE MATHÉMATIQUES LAURENT SCHWARTZ   (France)

^c UNIVERSITY OF FLORENCE   (Italy)

^d UNIVERSITY OF PAVIA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 72949111457     PISSN: 00029939     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9939-09-09795-0     Document Type: Article

References (32)

1. N, Math. Ann., 332 (2005), no. 2, 329-366. MR2178065 (2006g:35091)
2. A. Alvino, V. Ferone and C. Nitsch, The quantitative isoperimetric inequality for convex domains in the plane, preprint.
3. L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292 (2003a:49002)
4. F. Bernstein, Über die isoperimetriche Eigenschaft des Kreises auf der Kugeloberflache und in der Ebene, Math. Ann., 60 (1905), 117-136. MR1511289
5. T. Bhattacharya, Some observations on the first eigenvalue of the p-Laplacian and its connections with asymmetry, Electron. J. Differential Equations (2001), No. 35, 15 pp. MR1836803 (2002e:35185)
6. T. Bhattacharya and A. Weitsman, Bounds for capacities in terms of asymmetry, Rev. Mat. Iberoamericana, 12 (1996), no. 3, 593-639. MR1435477 (99c:31004) (Pubitemid 126312683)
7. T. Bonnesen, Uber die isoperimetrische Defizit ebener Figuren, Math. Ann., 91 (1924), 252-268. MR1512192
8. H. Brezis and E.H. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal., 62 (1985), no. 1, 73-86. MR790771 (86i:46033)
9. G. Buttazzo, G. Carlier and M. Comte, On the selection of maximal Cheeger sets, Differential and Integral Equations, 20 (2007), no. 9, 991-1004. MR2349376 (2008i:49025)
10. G. Carlier and M. Comte, On a weighted total variation minimization problem, J. Funct. Anal., 250 (2007), 214-226. MR2345913
11. V. Caselles, A. Chambolle and M. Novaga, Uniqueness of the Cheeger set of a convex body, Pacific J. Math., 232 (2007), no. 1, 77-90. MR2358032 (2008j:49012)
12. A. Cianchi, A quantitative Sobolev inequality in BV, J.Funct. Anal., 237 (2006), no. 2, 466-481. MR2230346 (2007b:46053) (Pubitemid 43792917)
13. A. Cianchi, Sharp Morrey-Sobolev inequalities and the distance from extremals, Trans. Amer. Math. Soc., 360 (2008), no. 8, 4335-4347. MR2395175
14. A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, submitted paper. Available in preprint version on http://cvgmt.sns.it/.To appear in Journal of the European Mathematical Society.
15. A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, On the isoperimetric deicit in the Gauss space, submitted paper. Available in preprint version on http://cvgmt.sns.it/.
16. L. Esposito, N. Fusco and C. Trombetti, A quantitative version of the isoperimetric inequality: the anisotropic case, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 4 (2005), no. 4, 619-651. MR2207737 (2006k:52013) (Pubitemid 43853344)
17. A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperi- metric inequalities, submitted paper. Available in preprint version on http://cvgmt.sns.it/.
18. L. E. Fraenkel, On the increase of capacity with asymmetry, Computational Methods and Function Theory, 8 (2008), no. 1, 203-224. MR2419474
19. P. Freitas and D. Krejčiřík, A sharp upper bound for the first Dirichlet eigenvalue and the growth ofthe isoperimetric constant ofconvex domains, Proc. Amer. Math. Soc., 136 (2008), no. 8, 2997-3006. MR2399068
20. n, Trans. Amer. Math. Soc., 314 (1989), 619-638. MR942426 (89m:52016)
21. N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. 168 (2008).
22. N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative Sobolev inequality for functions of bounded variation, J. Funct. Anal. 244 (2007), no. 1, 315-341. MR2294486 (2008a:46033)
23. N. Fusco, F. Maggi and A. Pratelli, Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities, to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5).
24. R.R. Hall, A quantitative isoperimetric in ri-dimensional space, J. Reine Angew. Math., 428 (1992), 161-176. MR1166511 (93d:51041)
25. R.R. Hall, W.K. Hayman and A.W. Weitsman, On asymmetry and capacity, J. d'Analyse Math., 56 (1991), 87-123. MR1243100 (95h:31004)
26. W. Hansen and N. Nadirashvili, Isoperimetric inequalities in potential theory. Proceedings from the International Conference on Potential Theory (Amersfoort, 1991). Potential Anal., 3 (1994), no. 1, 1-14. MR1266215 (95c:31003)
27. B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carolin., 44 (2003), no. 4, 659-667. MR2062882 (2005g:35053)
28. B. Kawohl and T. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane, Pacific J. Math., 225 (2006), 103-118. MR2233727 (2007e:52002)
29. B.Kawohland M. Novaga, The p-Laplace eigenvalue problem as p - 1 and Cheeger sets in a Finsler metric, J. Convex Analysis, 15 (2008), 623-634. MR2431415
30. F. Maggi, Some methods for studying stability in isoperimetric type problems, Bull. Amer. Math. Soc., 45 (2008), 367-408. MR2402947
31. A. Melas, The stability of some eigenvalue estimates, J. Differential Geom., 36 (1992), no. 1, 19-33. MR1168980 (93d:58178)
32. R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly, 86 (1979), 1-29. MR519520 (80h:52013)