A Sharp estimate for the first Robin–Laplacian eigenvalue with negative boundary parameter
Dorin Bucur, Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti · Rendiconti Lincei Matematica e Applicazioni · 2019
In this paper we prove that the ball maximizes the first eigenvalue of the Robin Laplacian operator with negative boundary parameter, among all convex sets of Rn with prescribed perimeter. The key of the proof is a dearrangement procedure of the first eigenfunction of the ball on the level sets of the distance function to the boundary of the convex set, which controls the boundary and the volume energies of the Rayleigh quotient.