Inserted: 16 jun 2020
Last Updated: 16 jun 2020
The aim of this paper is to prove a quantitative form of a reverse Faber-Krahn type inequality for the first Robin Laplacian eigenvalue $\lambda_\beta$ with negative boundary parameter among convex sets of prescribed perimeter. In that framework, the ball is the only maximizer for $\lambda_\beta$ and the distance from the optimal set is considered in terms of Hausdorff distance. The key point of our stategy is to prove a quantitative reverse Faber-Krahn inequality for the first eigenvalue of a Steklov-type problem related to the original Robin problem.