Title: Optimal partition problems for the fractional laplacian,
Abstract: In this work, we prove an existence result for an optimal partition problem
of the form $$\min \{F_s(A_1,\dots,A_m)\colon A_i \in \mathcal{A}_s, \, A_i\cap
A_j =\emptyset \mbox{ for } i\neq j\},$$ where $F_s$ is a cost functional with
suitable assumptions of monotonicity and lowersemicontinuity, $\mathcal{A}_s$
is the class of admissible domains and the condition $A_i\cap A_j =\emptyset$
is understood in the sense of the Gagliardo $s$-capacity, where $0<s<1$.
Examples of this type of problem are related to the fractional eigenvalues. In
addition, we prove some type of convergence of the $s$-minimizers to the
minimizer of the problem with $s=1$, studied in \cite{Bucur-Buttazzo-Henrot}.