Some methods for studying stability in isoperimetric type problems

Bulletin of the American Mathematical Society

Volumn 45, Issue 3, 2008, Pages 367-408

  Maggi, F ^a  

^a UNIVERSITY OF FLORENCE   (Italy)

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[No Author keywords available]

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EID: 46749140347     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-08-01206-8     Document Type: Article

References (35)

1. A. Alvino, V. Ferone, C. Nitsch, The quantitative isoperimetric inequality for convex domains in the plane, preprint.
2. L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000, xviii+434 pp. MR1857292 (2003a:49002)
3. F. Bernstein, Über die isoperimetriche Eigenschaft des Kreises auf der Kugeloberflache und in der Ebene, Math. Ann., 60 (1905), 117-136. MR1511289
4. G. Bianchi, H. Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), no. 1, 18-24. MR1124290 (92i:46033)
5. G.A. Bliss, An integral inequality, J. London Math. Soc. 5 (1930), 40-46.
6. T. Bonnesen, Uber die isoperimetrische Defizit ebener Figuren, Math. Ann. 91 (1924), 252-268. MR1512192
7. H. Brezis, E. H. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), no. 1, 73-86. MR790771 (86i:46033)
8. A. Cianchi, A quantitative Sobolev inequality in BV, J. Funct. Anal. 237 (2006), no. 2, 466-481. MR2230346 (2007b:46053)
9. A. Cianchi, Sharp Sobolev-Morrey inequalities and the distance from extremals, to appear in Trans. Amer. Math. Soc.
10. M. Chlebík, A. Cianchi, N. Fusco, The perimeter inequality under Steiner symmetrization: cases of equality. Ann. of Math. (2) 162 (2005), no. 1, 525-555. MR2178968 (2006m:49032)
11. A. Cianchi, L. Esposito, N. Fusco, C. Trombetti, A quantitative Pólya-Szegö principle, to appear in J. Reine Angew. Math.
12. A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, The sharp Sobolev inequality in quantitative form, submitted paper.
13. A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, On the isoperimetric deficit in the Gauss space, submitted paper.
14. D. Cordero-Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), no. 2, 307-332. MR2032031 (2005b:26023)
15. E. De Giorgi, Su una teoria generale della misura (r - 1)-dimensionale in uno spazio ad r dimensioni. (Italian) Ann. Mat. Pura Appl. (4) 36 (1954), 191-213. MR0062214 (15:945d)
16. E. De Giorgi, Nuovi teoremi relativi alle misure (r - 1)-dimensionali in uno spazio ad r dimensioni. (Italian) Ricerche Mat. 4 (1955), 95-113. MR0074499 (17:596a)
17. E. De Giorgi, Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita. (Italian) Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8) 5 (1958), 33-44. MR0098331 (20:4792)
18. L. Esposito, N. Fusco, C. Trombetti, A quantitative version of the isoperimetric in-equality: the anisotropic case. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 4, 619-651. MR2207737 (2006k:52013)
19. L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992, viii+268 pp. MR1158660 (93f:28001)
20. A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to isoperimetric type inequalities in quantitative form, in preparation.
21. W. H. Fleming, R. Rishel, An integral formula for total gradient variation. Arch. Math. 11 (1960), 218-222. MR0114892 (22:5710)
22. I. Fonseca, S. Müller, A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), no. 1-2, 125-136. MR1130601 (93c:49026)
23. n, Trans. Amer. Math. Soc. 314 (1989), 619-638. MR942426 (89m:52016)
24. N. Fusco, The classical isoperimetric theorem, Rend. Acad. Sci. Fis. Mat. Napoli (4) 71 (2004), 63-107. MR2147710 (2006a:49069)
25. N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality, to appear in Ann. of Math.
26. N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative Sobolev inequality for functions of bounded variation. J. Funct. Anal. 244 (2007), no. 1, 315-341. MR2294486
27. N. Fusco, F. Maggi, A. Pratelli, Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities, submitted paper.
28. R. R. Hall, A quantitative isoperimetric inequality in n-dimensional space, J. Reine Angew. Math. 428 (1992), 161-176. MR1166511 (93d:51041)
29. R. R. Hall, W. K. Hayman, A. W. Weitsman, On asymmetry and capacity, J. d'Analyse Math. 56 (1991), 87-123. MR1243100 (95h:31004)
30. B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Math. 1150, Springer-Verlag, Berlin, 1985. MR810619 (87a:35001)
31. V. D. Milman, G. Schechtman, Asymptotic theory of finite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, 1200. Springer-Verlag, Berlin, 1986, viii+156 pp. MR856576 (87m:46038)
32. R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), 1-29. MR519520 (80h:52013)
33. R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182-1238. MR0500557 (58:18161)
34. G. Pólya, G. Szegö, Isoperimetric inequalities in mathematical physics. Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, NJ, 1951. MR0043486 (13:270d)
35. G. Talenti, The standard isoperimetric theorem. Handbook of convex geometry, Vol. A, B, 73-123, North-Holland, Amsterdam, 1993. MR1242977 (94h:49065)