The quantitative Faber-Krahn inequality for the Robin Laplacian

TL;DR

This paper establishes a quantitative version of the Faber-Krahn inequality for the first Robin Laplacian eigenvalue, linking geometric asymmetry to spectral properties with explicit constants.

Contribution

It introduces a new quantitative inequality for the Robin Laplacian eigenvalue involving the Fraenkel asymmetry, extending classical results to Robin boundary conditions.

Findings

Proves a quantitative Faber-Krahn inequality for Robin Laplacian eigenvalues.

The asymmetry term involves the square of the Fraenkel asymmetry.

Constants depend on Robin parameter, dimension, and measure.

Abstract

We prove a quantitative Faber-Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.