*Published Paper*

**Inserted:** 28 jun 2014

**Last Updated:** 26 oct 2015

**Journal:** Calc. Var. Partial Differential Equations

**Volume:** 54

**Number:** 3

**Pages:** 2421-2464

**Year:** 2015

**Abstract:**

We show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the $t$-perimeter, up to moltiplicative 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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