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Nonlocal quantitative isoperimetric inequalities

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Abstract

We show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the \(t\)-perimeter, up to multiplicative constants, controls from above that of the \(s\)-perimeter, with \(s\) smaller than \(t\). To do this we consider a problem of independent interest: we characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the \(t\)-perimeter and the \(s\)-perimeter. In particular, we show that balls are the unique minimizers if the volume is sufficiently small, depending on \(t-s\), while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all \(s,\,t\). When \(s=0\) this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers.

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Notes

  1. Using [10], one can actually show that the dimension is at most \(N-8\) even for \(t\) sufficiently close to \(1\).

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Acknowledgments

The authors are members of the Italian CNR-GNAMPA, whose support they gratefully acknowledge. M.N. and E.V. have been partially supported by the ERC grant \(\epsilon \) “Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities”. B.R. was partially supported by the project ANR-12-BS01-0014-01 “Geometrya”.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy

    Agnese Di Castro & Matteo Novaga

  2. CMAP, Ecole Polytechnique, Route de Saclay, 21128, Palaiseau, France

    Berardo Ruffini

  3. Weierstrass Institute, Mohrenstrasse 39, 10117, Berlin, Germany

    Enrico Valdinoci

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  1. Agnese Di Castro
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  2. Matteo Novaga
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  3. Berardo Ruffini
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  4. Enrico Valdinoci
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Correspondence to Matteo Novaga.

Additional information

Communicated by L. Ambrosio.

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Di Castro, A., Novaga, M., Ruffini, B. et al. Nonlocal quantitative isoperimetric inequalities. Calc. Var. 54, 2421–2464 (2015). https://doi.org/10.1007/s00526-015-0870-x

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