Estimates for Robin $p$-Laplacian eigenvalues of convex sets with prescribed perimeter

TL;DR

This paper establishes upper bounds and quantitative inequalities for the first Robin eigenvalue of the p-Laplacian on convex sets with fixed perimeter, extending classical spectral geometry results.

Contribution

It provides new bounds and a quantitative reverse Faber-Krahn inequality for Robin p-Laplacian eigenvalues on convex sets with prescribed perimeter.

Findings

Upper bound for the first Robin eigenvalue with positive boundary parameter

Quantitative reverse Faber-Krahn inequality for negative boundary parameter

Comparison method using inner sets for spectral estimates

Abstract

In this paper, we prove an upper bound for the first Robin eigenvalue of the $p$-Laplacian with a positive boundary parameter and a quantitative version of the reverse Faber-Krahn type inequality for the first Robin eigenvalue of the $p$-Laplacian with negative boundary parameter, among convex sets with prescribed perimeter. The proofs are based on a comparison argument obtained by means of inner sets, introduced by Payne, Weimberger and Polya.