On the asymptotic behaviour of nonlocal perimeters
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 48.

We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense that has been recently introduced by A. Chambolle et al. [Archiv. Rational Mech. Anal. 218 (2015) 1263–1329]. Also, we establish existence of minimisers for the corresponding Plateau’s problem and, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions. A Γ-convergence result is discussed in the second part of the work. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel that has faster-than-L1 decay at infinity and we show that the Γ-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2018038
Classification : 49Q20, 53A10
Mots-clés : Nonlocal perimeters, nonlocal Plateau’s problem, Γ-convergence
Berendsen, Judith 1 ; Pagliari, Valerio 1

1 
@article{COCV_2019__25__A48_0,
     author = {Berendsen, Judith and Pagliari, Valerio},
     title = {On the asymptotic behaviour of nonlocal perimeters},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {25},
     year = {2019},
     doi = {10.1051/cocv/2018038},
     zbl = {1443.49052},
     mrnumber = {4011022},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2018038/}
}
TY  - JOUR
AU  - Berendsen, Judith
AU  - Pagliari, Valerio
TI  - On the asymptotic behaviour of nonlocal perimeters
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2019
VL  - 25
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2018038/
DO  - 10.1051/cocv/2018038
LA  - en
ID  - COCV_2019__25__A48_0
ER  - 
%0 Journal Article
%A Berendsen, Judith
%A Pagliari, Valerio
%T On the asymptotic behaviour of nonlocal perimeters
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2019
%V 25
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2018038/
%R 10.1051/cocv/2018038
%G en
%F COCV_2019__25__A48_0
Berendsen, Judith; Pagliari, Valerio. On the asymptotic behaviour of nonlocal perimeters. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 48. doi : 10.1051/cocv/2018038. http://www.numdam.org/articles/10.1051/cocv/2018038/

[1] G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies. Eur. J. Appl. Math. 3 (1998) 261–284. | DOI | MR | Zbl

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications, Clarendon Press, Oxford (2000). | DOI | MR | Zbl

[3] L. Ambrosio, G.D. Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals. Manuscr. Math. 134 (2011) 377–403. | DOI | MR | Zbl

[4] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, edited by J.L. Menaldi, E. Rofman and A. Sulem, IOS Press (2001) 439–455. | MR | Zbl

[5] L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. Partial Differ. Equ. 41 (2011) 203–240. | DOI | MR | Zbl

[6] L. Caffarelli, J-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63 (2010) 1111–1144. | DOI | MR | Zbl

[7] A. Cesaroni and M. Novaga, The isoperimetric problem for nonlocal perimeters. Dis. Contin. Dyn. Syst. Ser. S 11 (2018) 425–440. | MR | Zbl

[8] A. Chambolle, A. Giacomini and L. Lussardi, Continuous limits of discrete perimeters. ESAIM: M2AN 44 (2010) 207–230. | DOI | Numdam | MR | Zbl

[9] A. Chambolle, M. Morini and M. Ponsiglione, Nonlocal curvature flows. Arch. Ration. Mech. Anal. 218 (2015) 1263–1329. | DOI | MR | Zbl

[10] E. Cinti, J. Serra and E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces. J. Differ. Geom. 112 (2019) 447–504. | MR | Zbl

[11] J. Dávila, On an open question about functions of bounded variation. Calc. Var. Partial Differ. Equ. 15 (2002) 519–527. | DOI | MR | Zbl

[12] E.D. Giorgi, Nuovi teoremi relativi alle misure (r − 1)-dimensionali in uno spazio a r dimensioni. Ricerche Mat. 4 (1955) 95–113. | MR | Zbl

[13] H. Federer, A note on the Gauss Green theorem. Proc. Am. Math. Soc. 9 (1959) 447–451. | DOI | MR | Zbl

[14] I. Fonseca and S. Müller, Relaxation of quasiconvex functional in BV (;Rp) for integrands f(x; u;ru). Arch. Ration. Mech. Anal. 123 (1993) 1–49. | DOI | MR | Zbl

[15] M. Ludwig, Anisotropic fractional perimeters. J. Differ. Geom. 96 (2014) 77–93. | MR | Zbl

[16] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press (2012). | MR | Zbl

[17] J.M. Mazón, J.D. Rossi and J. Toledo, Nonlocal perimeter, curvature and minimal surfaces for measurable sets, in: Frontiers in Mathematics. Springer (2019). | MR | Zbl

[18] A.C. Ponce, A new approach to Sobolev spaces and connections to Γ-convergence. Calc. Var. Partial Differ. Equ. 19 (2004) 229–255. | DOI | MR

[19] E. Valdinoci, A fractional framework for perimeters and phase transitions. Milan J. Math. 8 (2013) 1–23. | DOI | MR | Zbl

[20] A. Visintin, Generalized coarea formula and fractal sets. Jpn. J. Indust. Appl. Math. 81 (1991) 175–201. | DOI | MR | Zbl

Cité par Sources :