TY - JOUR

T1 - Nonlocal quantitative isoperimetric inequalities

AU - Di Castro, Agnese

AU - Novaga, Matteo

AU - Ruffini, Berardo

AU - Valdinoci, Enrico

PY - 2015/11/1

Y1 - 2015/11/1

N2 - 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.

AB - 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.

KW - 35R11

KW - 49Q05

KW - 53A10

UR - http://www.scopus.com/inward/record.url?scp=84944355959&partnerID=8YFLogxK

U2 - 10.1007/s00526-015-0870-x

DO - 10.1007/s00526-015-0870-x

M3 - Article

AN - SCOPUS:84944355959

VL - 54

SP - 2421

EP - 2464

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 3

ER -