# A quantitative stability estimate for the fractional Faber-Krahn inequality

## TL;DR

This paper establishes a quantitative stability estimate for the fractional Faber-Krahn inequality, providing explicit constants and extending classical results to the nonlocal fractional Laplacian setting.

## Contribution

It introduces a new stability estimate for the fractional Faber-Krahn inequality using the Caffarelli-Silvestre extension and adapts existing techniques to the nonlocal context.

## Key findings

- Derived an explicit stability constant for the fractional Faber-Krahn inequality.
- Extended classical stability results to fractional Laplacian operators.
- Demonstrated stability as the fractional order approaches 1.

## Abstract

We prove a quantitative version of the Faber-Krahn inequality for the first eigenvalue of the fractional Dirichlet-Laplacian of order s. This is done by using the so-called Caffarelli-Silvestre extension and adapting to the nonlocal setting a trick by Hansen and Nadirashvili. The relevant stability estimate comes with an explicit constant, which is stable as the fractional order of differentiability goes to 1.