Some methods for studying stability in isoperimetric type problems
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Abstract:
We review the method of quantitative symmetrization inequalities introduced in Fusco, Maggi and Pratelli, “The sharp quantitative isoperimetric inequality”, Ann. of Math.References
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[AFN]AFN A. Alvino, V. Ferone, 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
- Gabriele Bianchi and Henrik Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), no. 1, 18–24. MR 1124290, DOI 10.1016/0022-1236(91)90099-Q [Bl]Bliss G.A. Bliss, An integral inequality, J. London Math. Soc. 5 (1930), 40–46.
- 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
- 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 [Ci2]C2 A. Cianchi, Sharp Sobolev-Morrey inequalities and the distance from extremals, to appear in Trans. Amer. Math. Soc.
- Miroslav Chlebík, Andrea Cianchi, and Nicola Fusco, The perimeter inequality under Steiner symmetrization: cases of equality, Ann. of Math. (2) 162 (2005), no. 1, 525–555. MR 2178968, DOI 10.4007/annals.2005.162.525 [CEFT]CEFT A. Cianchi, L. Esposito, N. Fusco, C. Trombetti, A quantitative Pólya–Szegö principle, to appear in J. Reine Angew. Math. [CFMP1]CFMP1 A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, The sharp Sobolev inequality in quantitative form, submitted paper. [CFMP2]CFMP2 A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, On the isoperimetric deficit in the Gauss space, submitted paper.
- D. Cordero-Erausquin, B. Nazaret, and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), no. 2, 307–332. MR 2032031, DOI 10.1016/S0001-8708(03)00080-X
- Ennio De Giorgi, Su una teoria generale della misura $(r-1)$-dimensionale in uno spazio ad $r$ dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191–213 (Italian). MR 62214, DOI 10.1007/BF02412838
- Ennio De Giorgi, Nuovi teoremi relativi alle misure $(r-1)$-dimensionali in uno spazio ad $r$ dimensioni, Ricerche Mat. 4 (1955), 95–113 (Italian). MR 74499
- Ennio 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. Natur. Sez. Ia (8) 5 (1958), 33–44 (Italian). MR 98331
- 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
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660 [FiMP]FiMP A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to isoperimetric type inequalities in quantitative form, in preparation.
- Wendell H. Fleming and Raymond Rishel, An integral formula for total gradient variation, Arch. Math. (Basel) 11 (1960), 218–222. MR 114892, DOI 10.1007/BF01236935
- Irene Fonseca and Stefan Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), no. 1-2, 125–136. MR 1130601, DOI 10.1017/S0308210500028365
- 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
- Nicola Fusco, The classical isoperimetric theorem, Rend. Accad. Sci. Fis. Mat. Napoli (4) 71 (2004), 63–107. MR 2147710 [FMP1]FMP1 N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality, to appear in Ann. of Math.
- 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 [FMP3]FMP3 N. Fusco, F. Maggi, A. Pratelli, Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities, submitted paper.
- 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
- Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619, DOI 10.1007/BFb0075060
- Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
- Robert Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1–29. MR 519520, DOI 10.2307/2320297
- Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4
- G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, No. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486
- Giorgio Talenti, The standard isoperimetric theorem, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 73–123. MR 1242977, DOI 10.1016/B978-0-444-89596-7.50008-0
Additional Information
- F. Maggi
- Affiliation: Dipartimento di Matematica, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
- Email: maggi@math.unifi.it
- Received by editor(s): August 29, 2007
- Published electronically: April 8, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Bull. Amer. Math. Soc. 45 (2008), 367-408
- MSC (2000): Primary 49Q20
- DOI: https://doi.org/10.1090/S0273-0979-08-01206-8
- MathSciNet review: 2402947