# On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets

## TL;DR

This paper investigates how removing small sets with zero fractional capacity from a domain affects the eigenvalues of the fractional Laplacian, providing asymptotic expansions related to eigenfunctions.

## Contribution

It introduces fractional capacity for compact sets and analyzes eigenvalue variations when removing sets concentrating to zero capacity points.

## Key findings

- Eigenvalues are unaffected by removal of zero fractional capacity sets.
- Asymptotic expansion of eigenvalue variation depends on the associated eigenfunction.
- Analysis of eigenvalue behavior when sets concentrate to a point.

## Abstract

We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are concentrating to a set of zero fractional capacity and we detect the asymptotic expansion of the eigenvalue variation; this expansion depends on the eigenfunction associated to the limit eigenvalue. Finally, we study the case in which the family of compact sets is concentrating to a point.