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Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies

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Abstract

We obtain a sharp quantitative isoperimetric inequality for nonlocal s-perimeters, uniform with respect to s bounded away from 0. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal s-perimeter plus a non-local repulsive interaction term. In the particular case s = 1, the s-perimeter coincides with the classical perimeter, and our results improve the ones of Knuepfer and Muratov (Comm. Pure Appl. Math. 66(7):1129–1162, 2013; Comm. Pure Appl. Math., 2014) concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts.

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Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, 2515 Speedway Stop C1200, Austin, TX, 78712-1202, USA

    A. Figalli & F. Maggi

  2. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, via Cintia, 80126, Naples, Italy

    N. Fusco

  3. Laboratoire J.-L. Lions (CNRS UMR 7598), Université Paris Diderot, Paris 7, Paris, France

    V. Millot

  4. Dipartimento di Matematica, Università di Parma, viale G. P. Usberti 53/a, Campus, 43100, Parma, Italy

    M. Morini

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  1. A. Figalli
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  2. N. Fusco
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  3. F. Maggi
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  4. V. Millot
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  5. M. Morini
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Correspondence to A. Figalli.

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Communicated by L. Caffarelli

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Figalli, A., Fusco, N., Maggi, F. et al. Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies. Commun. Math. Phys. 336, 441–507 (2015). https://doi.org/10.1007/s00220-014-2244-1

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  • DOI: https://doi.org/10.1007/s00220-014-2244-1

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