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arXiv · 1406.7545

Non-Local Isoperimetric Problems

Agnese Di Castro·Berardo Ruffini·Novaga Matteo·Enrico Valdinoci
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Abstract

We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $t$-perimeter and the $s$-perimeter, with $s$ smaller than $t$. Exploiting the quantitative fractional isoperimetric inequality, 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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Agnese Di Castro, Berardo Ruffini, Novaga Matteo, Enrico Valdinoci. 2014-06-29. Non-Local Isoperimetric Problems. https://arxiv.org/abs/1406.7545

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