Nonlocal quantitative isoperimetric inequalities

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.