August  2017, 10(4): 715-727. doi: 10.3934/dcdss.2017036

Volume constrained minimizers of the fractional perimeter with a potential energy

1. 

Department of Statistical Sciences, University of Padova, Via Cesare Battisti 141,35121 Padova, Italy

2. 

Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5,56127 Pisa, Italy

* Corresponding author

Received  March 2016 Revised  May 2016 Published  April 2017

Fund Project: The authors were supported by the Italian GNAMPA and by the University of Pisa via grant PRA-2015-0017.

We consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume integral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove existence and regularity of minimizers under suitable assumptions on the potential energy, which cover the periodic case. In the small volume regime we show that minimizers are close to balls, with a quantitative estimate.

Citation: Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036
References:
[1]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815.  doi: 10.1080/01630563.2014.901837.

[2]

B. BarriosA. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609-639. 

[3]

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

[4]

L. A. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871.  doi: 10.1016/j.aim.2013.08.007.

[5]

M. C. Caputo and N. Guillen, Regularity for non-local almost minimal boundaries and applications, Preprint, (2011). Available from: arXiv: 1003.2470.

[6]

G. Ciraolo, A. Figalli, F. Maggi and M. Novaga, Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature, J. Reine Angew. Math. , (2016). Available from: arXiv: 1503.00653. doi: 10.1515/crelle-2015-0088.

[7]

J. DavilaM. del PinoS. Dipierro and E. Valdinoci, Nonlocal Delaunay surfaces, Nonlinear Analysis, 137 (2016), 357-380.  doi: 10.1016/j.na.2015.10.009.

[8]

A. Di CastroM. NovagaB. Ruffini and E. Valdinoci, Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Differential Equations, 54 (2015), 2421-2464.  doi: 10.1007/s00526-015-0870-x.

[9]

A. FigalliN. FuscoF. MaggiV. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys., 336 (2015), 441-507.  doi: 10.1007/s00220-014-2244-1.

[10]

A. Figalli and F. Maggi, On the shape of liquid drops and crystals in the small mass regime, Arch. Ration. Mech. Anal., 201 (2011), 143-207.  doi: 10.1007/s00205-010-0383-x.

[11]

A. ChambolleM. Goldman and M. Novaga, Existence and qualitative properties of isoperimetric sets in periodic media, In Geometric Partial Differential Equations, Edizioni della Normale, CRM Series, 15 (2013), 75-92.  doi: 10.1007/978-88-7642-473-1_3.

[12]

M. Goldman and M. Novaga, Volume-constrained minimizers for the prescribed curvature problem in periodic media, Calc. Var. Partial Differential Equations, 44 (2012), 297-318.  doi: 10.1007/s00526-011-0435-6.

[13]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, In: An Introduction to Geometric Measure Theory, Cambridge Studies in Adavanced Mathematics, Vol. 135, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139108133.

[14]

M. Novaga and E. Valdinoci, Closed curves of prescribed curvature and a pinning effect, Netw. Heterog. Media, 6 (2011), 77-88. 

[15]

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39.  doi: 10.1007/s00526-012-0539-7.

[16]

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]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815.  doi: 10.1080/01630563.2014.901837.

[2]

B. BarriosA. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609-639. 

[3]

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

[4]

L. A. Caffarelli and E. Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math., 248 (2013), 843-871.  doi: 10.1016/j.aim.2013.08.007.

[5]

M. C. Caputo and N. Guillen, Regularity for non-local almost minimal boundaries and applications, Preprint, (2011). Available from: arXiv: 1003.2470.

[6]

G. Ciraolo, A. Figalli, F. Maggi and M. Novaga, Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature, J. Reine Angew. Math. , (2016). Available from: arXiv: 1503.00653. doi: 10.1515/crelle-2015-0088.

[7]

J. DavilaM. del PinoS. Dipierro and E. Valdinoci, Nonlocal Delaunay surfaces, Nonlinear Analysis, 137 (2016), 357-380.  doi: 10.1016/j.na.2015.10.009.

[8]

A. Di CastroM. NovagaB. Ruffini and E. Valdinoci, Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Differential Equations, 54 (2015), 2421-2464.  doi: 10.1007/s00526-015-0870-x.

[9]

A. FigalliN. FuscoF. MaggiV. Millot and M. Morini, Isoperimetry and stability properties of balls with respect to nonlocal energies, Comm. Math. Phys., 336 (2015), 441-507.  doi: 10.1007/s00220-014-2244-1.

[10]

A. Figalli and F. Maggi, On the shape of liquid drops and crystals in the small mass regime, Arch. Ration. Mech. Anal., 201 (2011), 143-207.  doi: 10.1007/s00205-010-0383-x.

[11]

A. ChambolleM. Goldman and M. Novaga, Existence and qualitative properties of isoperimetric sets in periodic media, In Geometric Partial Differential Equations, Edizioni della Normale, CRM Series, 15 (2013), 75-92.  doi: 10.1007/978-88-7642-473-1_3.

[12]

M. Goldman and M. Novaga, Volume-constrained minimizers for the prescribed curvature problem in periodic media, Calc. Var. Partial Differential Equations, 44 (2012), 297-318.  doi: 10.1007/s00526-011-0435-6.

[13]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, In: An Introduction to Geometric Measure Theory, Cambridge Studies in Adavanced Mathematics, Vol. 135, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139108133.

[14]

M. Novaga and E. Valdinoci, Closed curves of prescribed curvature and a pinning effect, Netw. Heterog. Media, 6 (2011), 77-88. 

[15]

O. Savin and E. Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations, 48 (2013), 33-39.  doi: 10.1007/s00526-012-0539-7.

[16]

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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