Optimal partition problems for the fractional laplacian

TL;DR

This paper proves the existence of solutions for optimal partition problems involving the fractional Laplacian and explores the convergence of these solutions as the fractional parameter approaches 1.

Contribution

It establishes the existence of optimal partitions for fractional Laplacian problems and analyzes their convergence to classical cases as the fractional parameter tends to 1.

Findings

Existence of optimal partitions for fractional Laplacian problems.

Convergence of fractional minimizers to classical minimizers as s approaches 1.

Application to fractional eigenvalue problems.

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}.