Nonlocal quantitative isoperimetric inequalities

作者：Di Castro Agnese; Novaga Matteo^*; Ruffini Berardo; Valdinoci Enrico

来源：Calculus of Variations and Partial Differential Equations, 2015, 54(3): 2421-2464.

DOI：10.1007/s00526-015-0870-x

We show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the -perimeter, up to multiplicative constants, controls from above that of the -perimeter, with smaller than . 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 -perimeter and the -perimeter. In particular, we show that balls are the unique minimizers if the volume is sufficiently small, depending on , 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 . When this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers.

• 出版日期2015-11