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Some methods for studying stability in isoperimetric type problems
Author:
F. Maggi
Journal:
Bull. Amer. Math. Soc. 45 (2008), 367-408
MSC (2000):
Primary 49Q20
Posted:
April 8, 2008
MathSciNet review:
2402947
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Additional Information
Abstract: We review the method of quantitative symmetrization inequalities introduced in Fusco, Maggi and Pratelli, ``The sharp quantitative isoperimetric inequality'', Ann. of Math.
- [AFN]
A. Alvino, V. Ferone, C. Nitsch, The quantitative isoperimetric inequality for convex domains in the plane, preprint.
- [AFP]
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
(2003a:49002)
- [Be]
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, http://dx.doi.org/10.1007/BF01447496
- [BE]
Gabriele
Bianchi and Henrik
Egnell, A note on the Sobolev inequality, J. Funct. Anal.
100 (1991), no. 1, 18–24. MR 1124290
(92i:46033), http://dx.doi.org/10.1016/0022-1236(91)90099-Q
- [Bl]
G.A. Bliss, An integral inequality, J. London Math. Soc. 5 (1930), 40-46.
- [Bo]
T.
Bonnesen, Über das isoperimetrische Defizit ebener
Figuren, Math. Ann. 91 (1924), no. 3-4,
252–268 (German). MR
1512192, http://dx.doi.org/10.1007/BF01556082
- [BL]
Haïm
Brezis and Elliott
H. Lieb, Sobolev inequalities with remainder terms, J. Funct.
Anal. 62 (1985), no. 1, 73–86. MR 790771
(86i:46033), http://dx.doi.org/10.1016/0022-1236(85)90020-5
- [Ci1]
Andrea
Cianchi, A quantitative Sobolev inequality in
𝐵𝑉, J. Funct. Anal. 237 (2006),
no. 2, 466–481. MR 2230346
(2007b:46053), http://dx.doi.org/10.1016/j.jfa.2005.12.008
- [Ci2]
A. Cianchi, Sharp Sobolev-Morrey inequalities and the distance from extremals, to appear in Trans. Amer. Math. Soc.
- [CCF]
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
(2006m:49032), http://dx.doi.org/10.4007/annals.2005.162.525
- [CEFT]
A. Cianchi, L. Esposito, N. Fusco, C. Trombetti, A quantitative Pólya-Szegö principle, to appear in J. Reine Angew. Math.
- [CFMP1]
A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, The sharp Sobolev inequality in quantitative form, submitted paper.
- [CFMP2]
A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, On the isoperimetric deficit in the Gauss space, submitted paper.
- [CNV]
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
(2005b:26023), http://dx.doi.org/10.1016/S0001-8708(03)00080-X
- [DG1]
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 0062214
(15,945d)
- [DG2]
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 0074499
(17,596a)
- [DG3]
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.
Nat. Sez. I (8) 5 (1958), 33–44 (Italian). MR 0098331
(20 #4792)
- [EFT]
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
(2006k:52013)
- [EG]
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
(93f:28001)
- [FiMP]
A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to isoperimetric type inequalities in quantitative form, in preparation.
- [FR]
Wendell
H. Fleming and Raymond
Rishel, An integral formula for total gradient variation,
Arch. Math. (Basel) 11 (1960), 218–222. MR 0114892
(22 #5710)
- [FM]
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
(93c:49026), http://dx.doi.org/10.1017/S0308210500028365
- [Fu]
Bent
Fuglede, Stability in the isoperimetric problem
for convex or nearly spherical domains in 𝑅ⁿ, Trans. Amer. Math. Soc. 314 (1989), no. 2, 619–638. MR 942426
(89m:52016), http://dx.doi.org/10.1090/S0002-9947-1989-0942426-3
- [Fs]
Nicola
Fusco, The classical isoperimetric theorem, Rend. Accad. Sci.
Fis. Mat. Napoli (4) 71 (2004), 63–107. MR 2147710
(2006a:49069)
- [FMP1]
N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality, to appear in Ann. of Math.
- [FMP2]
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
(2008a:46033), http://dx.doi.org/10.1016/j.jfa.2006.10.015
- [FMP3]
N. Fusco, F. Maggi, A. Pratelli, Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities, submitted paper.
- [Ha]
R.
R. Hall, A quantitative isoperimetric inequality in
𝑛-dimensional space, J. Reine Angew. Math.
428 (1992), 161–176. MR 1166511
(93d:51041), http://dx.doi.org/10.1515/crll.1992.428.161
- [HHW]
R.
R. Hall, W.
K. Hayman, and A.
W. Weitsman, On asymmetry and capacity, J. Analyse Math.
56 (1991), 87–123. MR 1243100
(95h:31004), http://dx.doi.org/10.1007/BF02820461
- [Ka]
Bernhard
Kawohl, Rearrangements and convexity of level sets in PDE,
Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin,
1985. MR
810619 (87a:35001)
- [MS]
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
(87m:46038)
- [Os1]
Robert
Osserman, Bonnesen-style isoperimetric inequalities, Amer.
Math. Monthly 86 (1979), no. 1, 1–29. MR 519520
(80h:52013), http://dx.doi.org/10.2307/2320297
- [Os2]
Robert
Osserman, The isoperimetric inequality,
Bull. Amer. Math. Soc. 84 (1978),
no. 6, 1182–1238. MR 0500557
(58 #18161), http://dx.doi.org/10.1090/S0002-9904-1978-14553-4
- [PS]
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
(13,270d)
- [Ta]
Giorgio
Talenti, The standard isoperimetric theorem, Handbook of
convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993,
pp. 73–123. MR 1242977
(94h:49065)
- [AFN]
- A. Alvino, V. Ferone, C. Nitsch, The quantitative isoperimetric inequality for convex domains in the plane, preprint.
- [AFP]
- 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. MR 1857292 (2003a:49002)
- [Be]
- F. Bernstein, Über die isoperimetriche Eigenschaft des Kreises auf der Kugeloberflache und in der Ebene, Math. Ann., 60 (1905), 117-136. MR 1511289
- [BE]
- G. Bianchi, H. Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), no. 1, 18-24. MR 1124290 (92i:46033)
- [Bl]
- G.A. Bliss, An integral inequality, J. London Math. Soc. 5 (1930), 40-46.
- [Bo]
- T. Bonnesen, Über die isoperimetrische Defizit ebener Figuren, Math. Ann. 91 (1924), 252-268. MR 1512192
- [BL]
- H. Brezis, E. H. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), no. 1, 73-86. MR 790771 (86i:46033)
- [Ci1]
- A. Cianchi, A quantitative Sobolev inequality in BV, J. Funct. Anal. 237 (2006), no. 2, 466-481. MR 2230346 (2007b:46053)
- [Ci2]
- A. Cianchi, Sharp Sobolev-Morrey inequalities and the distance from extremals, to appear in Trans. Amer. Math. Soc.
- [CCF]
- 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. MR 2178968 (2006m:49032)
- [CEFT]
- A. Cianchi, L. Esposito, N. Fusco, C. Trombetti, A quantitative Pólya-Szegö principle, to appear in J. Reine Angew. Math.
- [CFMP1]
- A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, The sharp Sobolev inequality in quantitative form, submitted paper.
- [CFMP2]
- A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, On the isoperimetric deficit in the Gauss space, submitted paper.
- [CNV]
- 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. MR 2032031 (2005b:26023)
- [DG1]
- E. De Giorgi, Su una teoria generale della misura
-dimensionale in uno spazio ad dimensioni. (Italian) Ann. Mat. Pura Appl. (4) 36 (1954), 191-213. MR 0062214 (15:945d)
- [DG2]
- E. De Giorgi, Nuovi teoremi relativi alle misure
-dimensionali in uno spazio ad dimensioni. (Italian) Ricerche Mat. 4 (1955), 95-113. MR 0074499 (17:596a)
- [DG3]
- 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. MR 0098331 (20:4792)
- [EFT]
- L. Esposito, N. Fusco, C. Trombetti, A quantitative version of the isoperimetric inequality: the anisotropic case. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 4, 619-651. MR 2207737 (2006k:52013)
- [EG]
- 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. MR 1158660 (93f:28001)
- [FiMP]
- A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to isoperimetric type inequalities in quantitative form, in preparation.
- [FR]
- W. H. Fleming, R. Rishel, An integral formula for total gradient variation. Arch. Math. 11 (1960), 218-222. MR 0114892 (22:5710)
- [FM]
- I. Fonseca, S. Müller, A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), no. 1-2, 125-136. MR 1130601 (93c:49026)
- [Fu]
- B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in
, Trans. Amer. Math. Soc. 314 (1989), 619-638. MR 942426 (89m:52016)
- [Fs]
- N. Fusco, The classical isoperimetric theorem, Rend. Acad. Sci. Fis. Mat. Napoli (4) 71 (2004), 63-107. MR 2147710 (2006a:49069)
- [FMP1]
- N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality, to appear in Ann. of Math.
- [FMP2]
- 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. MR 2294486
- [FMP3]
- N. Fusco, F. Maggi, A. Pratelli, Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities, submitted paper.
- [Ha]
- R. R. Hall, A quantitative isoperimetric inequality in
-dimensional space, J. Reine Angew. Math. 428 (1992), 161-176. MR 1166511 (93d:51041)
- [HHW]
- R. R. Hall, W. K. Hayman, A. W. Weitsman, On asymmetry and capacity, J. d'Analyse Math. 56 (1991), 87-123. MR 1243100 (95h:31004)
- [Ka]
- B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Math. 1150, Springer-Verlag, Berlin, 1985. MR 810619 (87a:35001)
- [MS]
- 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. MR 856576 (87m:46038)
- [Os1]
- R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), 1-29. MR 519520 (80h:52013)
- [Os2]
- R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182-1238. MR 0500557 (58:18161)
- [PS]
- G. Pólya, G. Szegö, Isoperimetric inequalities in mathematical physics. Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, NJ, 1951. MR 0043486 (13:270d)
- [Ta]
- G. Talenti, The standard isoperimetric theorem. Handbook of convex geometry, Vol. A, B, 73-123, North-Holland, Amsterdam, 1993. MR 1242977 (94h:49065)
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Additional Information
F. Maggi
Affiliation:
Dipartimento di Matematica, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
Email:
maggi@math.unifi.it
DOI:
http://dx.doi.org/10.1090/S0273-0979-08-01206-8
PII:
S 0273-0979(08)01206-8
Received by editor(s):
August 29, 2007
Posted:
April 8, 2008
Article copyright:
© Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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