Approximation and relaxation of perimeter in the Wiener space
Goldman, M. ; Novaga, M.
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 525-544 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

We characterize the relaxation of the perimeter in an infinite dimensional Wiener space, with respect to the weak L 2 -topology. We also show that the rescaled Allen–Cahn functionals approximate this relaxed functional in the sense of Γ-convergence.

@article{AIHPC_2012__29_4_525_0,
     author = {Goldman, M. and Novaga, M.},
     title = {Approximation and relaxation of perimeter in the Wiener space},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {29},
     year = {2012},
     pages = {525-544},
     doi = {10.1016/j.anihpc.2012.01.008},
     mrnumber = {2948287},
     zbl = {1244.49074},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_4_525_0}
}

Goldman, M.; Novaga, M. Approximation and relaxation of perimeter in the Wiener space. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 525-544. doi : 10.1016/j.anihpc.2012.01.008. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_4_525_0/

[1] L. Ambrosio, G. Da Prato, D. Pallara, BV functions in a Hilbert space with respect to a Gaussian measure, Rend. Accad. Lincei 21 no. 4 (2010), 405-414 | MR 2746092 | Zbl 1206.26014

[2] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Sci. Publ. (2000) | MR 1857292 | Zbl 0957.49001

[3] L. Ambrosio, A. Figalli, Surface measures and convergence of the Ornstein–Uhlenbeck semigroup in Wiener spaces, Ann. Fac. Sci. Toulouse Math. 20 no. 6 (2011), 407-438 | Numdam | MR 2847889 | Zbl 1228.60063

[4] L. Ambrosio, S. Maniglia, M. Miranda, D. Pallara, BV functions in abstract Wiener spaces, J. Funct. Anal. 258 no. 3 (2010), 785-813 | MR 2558177 | Zbl 1194.46066

[5] L. Ambrosio, M. Miranda, D. Pallara, Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability, Discrete Contin. Dyn. Syst. Ser. A 28 (2010), 591-606 | MR 2644758 | Zbl 1196.28023

[6] D. Bakry, M. Ledoux, Lévy–Gromovʼs isoperimetric inequality for an infinite dimensional diffusion generator, Invent. Math. 123 (1996), 259-281 | MR 1374200 | Zbl 0855.58011

[7] S.G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space, Ann. Probab. 25 no. 1 (1997), 206-214 | MR 1428506 | Zbl 0883.60031

[8] V.I. Bogachev, Gaussian Measures, Math. Surveys Monogr. vol. 62, Amer. Math. Soc., Providence (1998) | MR 1642391 | Zbl 0938.28010

[9] J. Brothers, W. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math. 384 (1988), 153-179 | MR 929981 | Zbl 0633.46030

[10] E. Carlen, C. Kerce, On the cases of equality in Bobkovʼs inequality and Gaussian rearrangement, Calc. Var. Partial Differential Equations 13 (2001), 1-18 | MR 1854254 | Zbl 1009.49029

[11] V. Caselles, A. Lunardi, M. Miranda, M. Novaga, Perimeter of sublevel sets in infinite dimensional spaces, Adv. Calc. Var. 5 no. 1 (2012), 59-76 | MR 2879567 | Zbl 1257.49006

[12] V. Caselles, M. Miranda, M. Novaga, Total variation and Cheeger sets in Gauss space, J. Funct. Anal. 259 no. 6 (2010), 1491-1516 | MR 2659769 | Zbl 1195.49054

[13] A. Chambolle, M. Goldman, M. Novaga, Representation, relaxation and convexity for variational problems in Wiener spaces, preprint, 2011. | MR 3035950

[14] A. Cianchi, N. Fusco, Functions of bounded variation and rearrangements, Arch. Ration. Mech. Anal. 165 (2002), 1-40 | MR 1947097 | Zbl 1028.49035

[15] A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, On the isoperimetric deficit in the Gauss space, Amer. J. Math. 133 no. 1 (2011), 131-186 | MR 2752937 | Zbl 1219.28005

[16] G. Dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston (1993) | MR 1201152

[17] F. Demengel, R. Temam, Convex functions of a measure and applications, Indiana Univ. Math. J. 33 (1984), 673-709 | MR 756154 | Zbl 0581.46036

[18] A. Ehrhard, Symétrisation dans lʼespace de Gauss, Math. Scand. 53 no. 2 (1983), 281-301 | MR 745081 | Zbl 0542.60003

[19] A. Ehrhard, Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes, Ann. Sci. Ec. Norm. Super. 17 (1984), 317-332 | Numdam | MR 760680 | Zbl 0546.49020

[20] L.C. Evans, G. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press (1992) | MR 1158660 | Zbl 0804.28001

[21] M. Fukushima, BV functions and distorted Ornstein–Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal. 174 (2000), 227-249 | MR 1761369 | Zbl 0978.60088

[22] M. Fukushima, M. Hino, On the space of BV functions and a related stochastic calculus in infinite dimensions, J. Funct. Anal. 183 (2001), 245-268 | MR 1837539 | Zbl 0993.60049

[23] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Springer Monogr. Math. vol. 80, Birkhäuser (1984) | MR 775682 | Zbl 0545.49018

[24] E. Lieb, M. Loss, Analysis, Grad. Stud. Math. vol. 14, Amer. Math. Soc., Providence (2001) | MR 1817225

[25] M. Ledoux, Isoperimetry and Gaussian Analysis, Lecture Notes in Math. vol. 1648, Springer (1996) | MR 1600888 | Zbl 0874.60005

[26] P. Malliavin, Stochastic Analysis, Grundlehren Math., Springer (1997) | MR 1450093

[27] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Ration. Mech. Anal. 98 (1987), 123-142 | MR 866718 | Zbl 0616.76004

[28] L. Modica, S. Mortola, Un esempio di Γ - -convergenza, Boll. Unione Mat. Ital. 14 no. 5 (1977), 285-299 | MR 445362 | Zbl 0356.49008