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May  2021, 20(5): 2101-2116. doi: 10.3934/cpaa.2021059

A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

1. 

Department of Mathematics, University of Maryland, 4176 Campus Dr, College Park, MD 20742, USA

2. 

Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, Finland

* Corresponding author

Received  November 2020 Revised  February 2021 Published  May 2021 Early access  April 2021

Fund Project: The first author is partially supported by the NSF DMS Grant No. 1906451. The second author has been partially supported by the Academy of Finland grant 314227

We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the non local Gaussian perimeter taken into consideration.

Citation: Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2101-2116. doi: 10.3934/cpaa.2021059
References:
[1]

L. AmbrosioG. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.  doi: 10.1007/s00229-010-0399-4.

[2]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.

[3]

M. BarchiesiA. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality, Ann. Probab., 45 (2017), 668-697.  doi: 10.1214/15-AOP1072.

[4]

C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent Math., 30 (1975), 202-216.  doi: 10.1007/BF01425510.

[5]

J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations J. L. Menaldi, E. Rofman and A. Sulem, eds., IOS Press (2001), 439–455.

[6]

L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Commun. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.

[7]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013) doi: 10.1016/j.aim.2013.08.007.

[8]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, 1993. doi: 10.1007/978-1-4612-0327-8.

[9]

E. De Giorgi, Nuovi teoremi relativi alle misure $ (r-1) $-dimensionali in uno spazio a $ r $ dimensioni, Ricerche Mat., 4 (1955), 95-113. 

[10]

E. De Giorgi and E. Letta, Une notion générale de convergence faible pour des fonctions croissantes d'ensemble, Ann. Scuola Norm. Sup. Pisa, 4 (1977), 61-99. 

[11]

A. De RosaS. Kolasinski and M. Santilli, Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets, Arch. Ration. Mech. Anal., 238 (2020), 1157-1198.  doi: 10.1007/s00205-020-01562-y.

[12]

A. De Rosa and S. Gioffrè, Quantitative stability for anisotropic nearly umbilical hypersurfaces, J. Geom. Anal., 29 (2019), 2318-2346.  doi: 10.1007/s12220-018-0079-2.

[13]

A. De Rosa, S. Gioffrè,, Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces,, (2018). arXiv: 1803.09118.

[14]

S. Di Pierro, A comparison between the nonlocal and the classical worlds: minimal surfaces, phase transitions, and geometric flows, Not. Am. Math. Soc., 67 (2020), 1324-1335.  doi: 10.1090/noti.

[15]

S. Dipierro, P. Miraglio, E. Valdinoci,, (Non)local $\Gamma$-convergence, Bruno Pini Mathematical Analysis Seminar, 11(1), 68–93.

[16]

I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in $L^1$, SIAM J. Math. Anal., 23 (1992), 1081-1098.  doi: 10.1137/0523060.

[17]

A. FigalliN. FuscoF. MaggiV. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Commun. Math. Phis., 336 (2014), 441-507.  doi: 10.1007/s00220-014-2244-1.

[18]

E. Giusti,, Minimal Surfaces and Functions of Bounded Variation, Brickhauser, Basel 1984. doi: 10.1007/978-1-4684-9486-0.

[19]

D.A. La Manna, Local minimality of the ball for the Gaussian perimeter, Adv. Calc. Var., 12 (2019), 193-210.  doi: 10.1515/acv-2017-0007.

[20]

V. Maz'ya and T. Shaposhnikova, Erratum to: "On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces", J. Funct. Anal., 201 (2003), 298-300.  doi: 10.1016/S0022-1236(03)00002-8.

[21]

A. P. Morse, Perfect blankets, Trans. Amer. Math. Soc., 61 (1947), 418-422.  doi: 10.2307/1990381.

[22]

M. NovagaD. Pallara and Y. Sire, A fractional isoperimetric problem in the Wiener space, J. Anal. Math., 134 (2018), 787-800.  doi: 10.1007/s11854-018-0026-y.

[23]

A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math., 8 (1991), 175-201.  doi: 10.1007/BF03167679.

show all references

References:
[1]

L. AmbrosioG. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.  doi: 10.1007/s00229-010-0399-4.

[2]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.

[3]

M. BarchiesiA. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality, Ann. Probab., 45 (2017), 668-697.  doi: 10.1214/15-AOP1072.

[4]

C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent Math., 30 (1975), 202-216.  doi: 10.1007/BF01425510.

[5]

J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations J. L. Menaldi, E. Rofman and A. Sulem, eds., IOS Press (2001), 439–455.

[6]

L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Commun. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.

[7]

L. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013) doi: 10.1016/j.aim.2013.08.007.

[8]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, 1993. doi: 10.1007/978-1-4612-0327-8.

[9]

E. De Giorgi, Nuovi teoremi relativi alle misure $ (r-1) $-dimensionali in uno spazio a $ r $ dimensioni, Ricerche Mat., 4 (1955), 95-113. 

[10]

E. De Giorgi and E. Letta, Une notion générale de convergence faible pour des fonctions croissantes d'ensemble, Ann. Scuola Norm. Sup. Pisa, 4 (1977), 61-99. 

[11]

A. De RosaS. Kolasinski and M. Santilli, Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets, Arch. Ration. Mech. Anal., 238 (2020), 1157-1198.  doi: 10.1007/s00205-020-01562-y.

[12]

A. De Rosa and S. Gioffrè, Quantitative stability for anisotropic nearly umbilical hypersurfaces, J. Geom. Anal., 29 (2019), 2318-2346.  doi: 10.1007/s12220-018-0079-2.

[13]

A. De Rosa, S. Gioffrè,, Absence of bubbling phenomena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces,, (2018). arXiv: 1803.09118.

[14]

S. Di Pierro, A comparison between the nonlocal and the classical worlds: minimal surfaces, phase transitions, and geometric flows, Not. Am. Math. Soc., 67 (2020), 1324-1335.  doi: 10.1090/noti.

[15]

S. Dipierro, P. Miraglio, E. Valdinoci,, (Non)local $\Gamma$-convergence, Bruno Pini Mathematical Analysis Seminar, 11(1), 68–93.

[16]

I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in $L^1$, SIAM J. Math. Anal., 23 (1992), 1081-1098.  doi: 10.1137/0523060.

[17]

A. FigalliN. FuscoF. MaggiV. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Commun. Math. Phis., 336 (2014), 441-507.  doi: 10.1007/s00220-014-2244-1.

[18]

E. Giusti,, Minimal Surfaces and Functions of Bounded Variation, Brickhauser, Basel 1984. doi: 10.1007/978-1-4684-9486-0.

[19]

D.A. La Manna, Local minimality of the ball for the Gaussian perimeter, Adv. Calc. Var., 12 (2019), 193-210.  doi: 10.1515/acv-2017-0007.

[20]

V. Maz'ya and T. Shaposhnikova, Erratum to: "On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces", J. Funct. Anal., 201 (2003), 298-300.  doi: 10.1016/S0022-1236(03)00002-8.

[21]

A. P. Morse, Perfect blankets, Trans. Amer. Math. Soc., 61 (1947), 418-422.  doi: 10.2307/1990381.

[22]

M. NovagaD. Pallara and Y. Sire, A fractional isoperimetric problem in the Wiener space, J. Anal. Math., 134 (2018), 787-800.  doi: 10.1007/s11854-018-0026-y.

[23]

A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math., 8 (1991), 175-201.  doi: 10.1007/BF03167679.

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