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.
Nonlocal quantitative isoperimetric inequalities
NOVAGA, MATTEO;
2015-01-01
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
