Non-Local Isoperimetric Problems

Agnese Di Castro (1) , Berardo Ruffini (2) , Novaga Matteo (1) , Enrico Valdinoci (3)

1 Dipartimento di Matematica [Pisa]
2 IF - Institut Fourier
3 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]

We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $t$-perimeter and the $s$-perimeter, with $s$ smaller than $t$. Exploiting the quantitative fractional isoperimetric inequality, 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.

Domaines: Equations aux dérivées partielles [math.AP]

Agnese Di Castro, Berardo Ruffini, Novaga Matteo, Enrico Valdinoci. Non-Local Isoperimetric Problems. 2014. ⟨hal-01016191⟩