Inserted: 28 jun 2014
Last Updated: 26 oct 2015
Journal: Calc. Var. Partial Differential Equations
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.