On some isoperimetric inequalities involving eigenvalues of fixed membranes

Abstract

In 1956, Hersch (1965) derived some isoperimetric inequalities for eigenvalues of a fixed membrane Ω , simply connected, with a center of symmetry O . In this note we are going to derive some sharper versions of Hersch’s results. More precisely, if Ω is symmetric of order 2 we show that we have λ 2 ( Ω ) + λ 3 ( Ω ) ≤ 2 λ 2 ( D r o ) , where D r o is a disc of radius r o ( Ω ) (the conformal radius of Ω at O ). Also, if Ω is symmetric of order 4, we have λ 4 ( Ω ) + λ 5 ( Ω ) ≤ 2 λ 4 ( D r o ) .