# American Institute of Mathematical Sciences

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
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