A note on Cheeger sets
HTML articles powered by AMS MathViewer
- by Alessio Figalli, Francesco Maggi and Aldo Pratelli PDF
- Proc. Amer. Math. Soc. 137 (2009), 2057-2062 Request permission
Abstract:
Starting from the quantitative isoperimetric inequality, we prove a sharp quantitative version of the Cheeger inequality.References
- F. Alter, V. Caselles, and A. Chambolle, A characterization of convex calibrable sets in $\Bbb R^N$, Math. Ann. 332 (2005), no. 2, 329–366. MR 2178065, DOI 10.1007/s00208-004-0628-9
- A. Alvino, V. Ferone and C. Nitsch, The quantitative isoperimetric inequality for convex domains in the plane, preprint.
- Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
- Felix Bernstein, Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene, Math. Ann. 60 (1905), no. 1, 117–136 (German). MR 1511289, DOI 10.1007/BF01447496
- Tilak Bhattacharya, Some observations on the first eigenvalue of the $p$-Laplacian and its connections with asymmetry, Electron. J. Differential Equations (2001), No. 35, 15. MR 1836803
- Tilak Bhattacharya and Allen Weitsman, Bounds for capacities in terms of asymmetry, Rev. Mat. Iberoamericana 12 (1996), no. 3, 593–639. MR 1435477, DOI 10.4171/RMI/208
- T. Bonnesen, Über das isoperimetrische Defizit ebener Figuren, Math. Ann. 91 (1924), no. 3-4, 252–268 (German). MR 1512192, DOI 10.1007/BF01556082
- Haïm Brezis and Elliott H. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), no. 1, 73–86. MR 790771, DOI 10.1016/0022-1236(85)90020-5
- Giuseppe Buttazzo, Guillaume Carlier, and Myriam Comte, On the selection of maximal Cheeger sets, Differential Integral Equations 20 (2007), no. 9, 991–1004. MR 2349376
- Guillaume Carlier and Myriam Comte, On a weighted total variation minimization problem, J. Funct. Anal. 250 (2007), no. 1, 214–226. MR 2345913, DOI 10.1016/j.jfa.2007.05.022
- Vicent Caselles, Antonin Chambolle, and Matteo Novaga, Uniqueness of the Cheeger set of a convex body, Pacific J. Math. 232 (2007), no. 1, 77–90. MR 2358032, DOI 10.2140/pjm.2007.232.77
- Andrea Cianchi, A quantitative Sobolev inequality in $BV$, J. Funct. Anal. 237 (2006), no. 2, 466–481. MR 2230346, DOI 10.1016/j.jfa.2005.12.008
- Andrea Cianchi, Sharp Morrey-Sobolev inequalities and the distance from extremals, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4335–4347. MR 2395175, DOI 10.1090/S0002-9947-08-04491-7
- 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.
- A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, On the isoperimetric deficit in the Gauss space, submitted paper. Available in preprint version on http://cvgmt.sns.it/.
- Luca Esposito, Nicola Fusco, and Cristina 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. MR 2207737
- A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, submitted paper. Available in preprint version on http://cvgmt.sns.it/.
- L. E. Fraenkel, On the increase of capacity with asymmetry, Comput. Methods Funct. Theory 8 (2008), no. 1-2, 203–224. MR 2419474, DOI 10.1007/BF03321684
- Pedro Freitas and David Krejčiřík, A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2997–3006. MR 2399068, DOI 10.1090/S0002-9939-08-09399-4
- Bent Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in $\textbf {R}^n$, Trans. Amer. Math. Soc. 314 (1989), no. 2, 619–638. MR 942426, DOI 10.1090/S0002-9947-1989-0942426-3
- N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. 168 (2008).
- 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. MR 2294486, DOI 10.1016/j.jfa.2006.10.015
- 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).
- R. R. Hall, A quantitative isoperimetric inequality in $n$-dimensional space, J. Reine Angew. Math. 428 (1992), 161–176. MR 1166511, DOI 10.1515/crll.1992.428.161
- R. R. Hall, W. K. Hayman, and A. W. Weitsman, On asymmetry and capacity, J. Analyse Math. 56 (1991), 87–123. MR 1243100, DOI 10.1007/BF02820461
- W. Hansen and N. Nadirashvili, Isoperimetric inequalities in potential theory, Proceedings from the International Conference on Potential Theory (Amersfoort, 1991), 1994, pp. 1–14. MR 1266215, DOI 10.1007/BF01047833
- 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. MR 2062882
- Bernd Kawohl and Thomas Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane, Pacific J. Math. 225 (2006), no. 1, 103–118. MR 2233727, DOI 10.2140/pjm.2006.225.103
- B. Kawohl and M. Novaga, The $p$-Laplace eigenvalue problem as $p\to 1$ and Cheeger sets in a Finsler metric, J. Convex Anal. 15 (2008), no. 3, 623–634. MR 2431415
- F. Maggi, Some methods for studying stability in isoperimetric type problems, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 3, 367–408. MR 2402947, DOI 10.1090/S0273-0979-08-01206-8
- Antonios D. Melas, The stability of some eigenvalue estimates, J. Differential Geom. 36 (1992), no. 1, 19–33. MR 1168980
- Robert Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1–29. MR 519520, DOI 10.2307/2320297
Additional Information
- Alessio Figalli
- Affiliation: Université de Nice-Sophia Antipolis, Labo. J.-A. Dieudonné, UMR 6621, Parc Valrose, 06108 Nice Cedex 02, France
- Address at time of publication: Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France
- Email: figalli@math.polytechnique.fr
- Francesco Maggi
- Affiliation: Dipartimento di Matematica, Università degli Studi di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
- Email: maggi@math.unifi.it
- Aldo Pratelli
- Affiliation: Dipartimento di Matematica, Università degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy
- Email: aldo.pratelli@unipv.it
- Received by editor(s): July 29, 2008
- Published electronically: January 26, 2009
- Additional Notes: The work of the second and third authors was partially supported by the GNAMPA through the 2008 research project Disuguaglianze geometrico-funzionali in forma ottimale e quantitativa
- Communicated by: Tatiana Toro
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 2057-2062
- MSC (2000): Primary 39B62
- DOI: https://doi.org/10.1090/S0002-9939-09-09795-0
- MathSciNet review: 2480287