WIAS Preprint No. 1987, (2014)

Nonlocal isoperimetric problems


  • di Castro, Agnese
  • Novaga, Matteo
  • Ruffini, Berardo
  • Valdinoci, Enrico
    ORCID: 0000-0001-6222-2272

2010 Mathematics Subject Classification

  • 35R11 49Q05 53A10


  • fractional perimeter, isoperimetric problem, existence, rigidity and regularity results




We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of fractional perimeters. Exploiting the quantitative fractional isoperimetric inequality, we show that balls are the unique minimizers if the volume is sufficiently small, 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.

Appeared in

  • Calc. Var. Partial Differ. Equ., 54 (2015) pp. 2421--2464.

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