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Second-order asymptotics of the fractional perimeter as s → 1

1 Dipartimento di Scienze Statistiche, Università di Padova, Via Cesare Battisti 241/243, 35121 Padova, Italy
2 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

This contribution is part of the Special Issue: Partial Differential Equations from theory to applications—Dedicated to Alberto Farina, on the occasion of his 50th birthday
Guest Editors: Serena Dipierro; Luca Lombardini
Link: http://www.aimspress.com/newsinfo/1396.html

Special Issues: Partial Differential Equations from theory to applications—Dedicated to Alberto Farina, on the occasion of his 50th birthday

In this note we provide a second-order asymptotic expansion of the fractional perimeter Ps(E), as $s\to 1^-$, in terms of the local perimeter and of a higher order nonlocal functional.
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References

1. Ambrosio L, De Philippis G, Martinazzi L (2011) Gamma-convergence of nonlocal perimeter functionals. Manuscripta Math 134: 377-403.    

2. Bourgain J, Brezis H, Mironescu P (2001) Another look at Sobolev spaces, In: Optimal Control and Partial Differential Equations, Amsterdam: IOS, 439-455.

3. Brasco L, Lindgren E, Parini E (2014) The fractional Cheeger problem. Interface Free Bound 16: 419-458.    

4. Caffarelli L, Roquejoffre JM, Savin O (2010) Nonlocal minimal surfaces. Commun Pure Appl Math 63: 1111-1144.

5. Caffarelli L, Valdinoci E (2013) Regularity properties of nonlocal minimal surfaces via limiting arguments. Adv Math 248: 843-871.    

6. Chambolle A, Novaga M, Pagliari V (2019) On the convergence rate of some nonlocal energies. arXiv:1907.06030.

7. Dávila J, del Pino M, Wei J (2018) Nonlocal s-minimal surfaces and Lawson cones. J. Differ Geom 109: 111-175.    

8. De Luca L, Novaga M, Ponsiglione M (2019) The 0-fractional perimeter between fractional perimeters and Riesz potentials. arXiv:1906.06303.

9. Di Castro A, Novaga M, Ruffini B, et al. (2015) Nonlocal quantitative isoperimetric inequalities. Calc Var Partial Dif 54: 2421-2464.    

10. Di Nezza E, Palatucci G, Valdinoci E (2012) Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math 136: 521-573.    

11. Dipierro S, Figalli A, Palatucci G, et al. (2013) Asymptotics of the s-perimeter as s → 0. Discrete Cont Dyn Syst 33: 2777-2790.    

12. Frank RL, Lieb E (2015) A compactness lemma and its application to the existence of minimizers for the liquid drop model. SIAM J Math Anal 47: 4436-4450.    

13. Knüpfer H, Muratov CB, Novaga M (2016) Low density phases in a uniformly charged liquid. Commun Math Phys 345: 141-183.    

14. Maggi F (2012) Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge: Cambridge University Press.

15. Mazýa V (2003) Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. Contemp Math 338: 307-340.    

16. Muratov CB, Simon T (2019) A nonlocal isoperimetric problem with dipolar repulsion. Commun Math Phys 372: 1059-1115.    

17. Valdinoci E (2013) A fractional framework for perimeters and phase transitions. Milan J Math 81: 1-23.    

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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