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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[16]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[16]

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

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