Universal bounds for eigenvalues of the biharmonic operator

Abstract

In this paper, we study the eigenvalues of the clamped plate problem: { Δ 2 u = λ u , in D , u | ∂ D = ∂ u ∂ ν | ∂ D = 0 , where D is a bounded connected domain in an n-dimensional complete minimal submanifold of a unit m-sphere S m ( 1 ) or of an m-dimensional Euclidean space R m . Let 0 < λ 1 < λ 2 ⩽ ⋯ ⩽ λ k ⩽ ⋯ be the eigenvalues of the above problem. We obtain universal bounds on λ k + 1 in terms the first k eigenvalues independent of the domains. For example, when D is contained in an n-dimensional complete minimal submanifold of S m ( 1 ) , we show that λ k + 1 − 1 k ∑ i = 1 k λ i ⩽ 1 k n { ∑ i = 1 k ( λ k + 1 − λ i ) 1 / 2 ( ( 2 n + 4 ) λ i 1 / 2 + n 2 ) } 1 / 2 ⋅ { ∑ i = 1 k ( λ k + 1 − λ i ) 1 / 2 ( 4 λ i 1 / 2 + n 2 ) } 1 / 2 , from which one can obtain a more explicit upper bound on λ k + 1 in terms of λ 1 , … , λ k (see Corollary 1). When D is contained in a complete n-dimensional minimal submanifold of R m , we prove the inequality λ k + 1 ⩽ 1 k ∑ i = 1 k λ k + ( 8 ( n + 2 ) n 2 ) 1 / 2 1 k ∑ i = 1 k ( λ i ( λ k + 1 − λ i ) ) 1 / 2 which generalizes the main theorem in Cheng and Yang (2006) [10] that states that the same estimate holds when D is a connected and bounded domain in R n .