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A nonlocal isoperimetric problem with density perimeter

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Abstract

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent \(\alpha \), under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter \(\gamma \). We show that for a wide class of density functions the energy admits a minimizer for any value of \(\gamma \). Moreover these minimizers are bounded. For monomial densities of the form \(|x|^p\) we prove that when \(\gamma \) is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the \(\gamma \rightarrow 0\) limit corresponds, under a suitable rescaling, to a small mass \(m=|\Omega |\rightarrow 0\) limit when \(p<d-\alpha +1\), but to a large mass \(m\rightarrow \infty \) for powers \(p>d-\alpha +1\).

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Acknowledgements

The authors would like to thank Marco Bonacini and Gian Paolo Leonardi for valuable discussions regarding the properties of density perimeters. We also thank the reviewer for carefully reading the paper and providing many useful suggestions. SA, LB, and AZ were supported via an NSERC (Canada) Discovery Grant. AZ was partially funded by ANID Chile under grants Becas Chile de Postdoctorado en el Extranjero N\(^{\circ }\) 74200091 and FONDECYT de Iniciación en Investigación N \(^{\circ }\) 11201259.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada

    Stan Alama, Lia Bronsard & Andres Zuniga

  2. Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA

    Ihsan Topaloglu

  3. Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins (UOH), Rancagua, Chile

    Andres Zuniga

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  1. Stan Alama
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Correspondence to Ihsan Topaloglu.

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Communicated by A. Malchiodi.

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Alama, S., Bronsard, L., Topaloglu, I. et al. A nonlocal isoperimetric problem with density perimeter. Calc. Var. 60, 1 (2021). https://doi.org/10.1007/s00526-020-01865-8

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