On the stability of some isoperimetric inequalities for the fundamental tones of free plates

TL;DR

This paper establishes a quantitative isoperimetric inequality for the fundamental tone of biharmonic Neumann problems, extending previous second-order results and confirming their sharpness for related biharmonic problems.

Contribution

It introduces a new quantitative inequality for the biharmonic operator's fundamental tone, extending second-order results to fourth-order problems.

Findings

Quantitative inequality for biharmonic Neumann problem

Sharpness of the inequality demonstrated

Extension of second-order results to biharmonic operators

Abstract

We provide a quantitative version of the isoperimetric inequality for the fundamental tone of a biharmonic Neumann problem. Such an inequality has been recently established by Chasman adapting Weinberger's argument for the corresponding second order problem. Following a scheme introduced by Brasco and Pratelli for the second order case, we prove that a similar quantitative inequality holds also for the biharmonic operator. We also prove the sharpness of both such an inequality and the corresponding one for the biharmonic Steklov problem.