Nonlocal quantitative isoperimetric inequalities

Abstract

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.