Universal bounds for eigenvalues of the biharmonic operator

In this paper, we study the eigenvalues of the clamped plate problem:{Δ2u=λu,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 Sm(1)Sm(1) or of an m  -dimensional Euclidean space RmRm. Let 0<λ1<λ2⩽⋯⩽λk⩽⋯0<λ1<λ2⩽⋯⩽λk⩽⋯ be the eigenvalues of the above problem. We obtain universal bounds on λk+1λk+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 Sm(1)Sm(1), we show thatλk+1−1k∑i=1kλi⩽1kn{∑i=1k(λk+1−λi)1/2((2n+4)λi1/2+n2)}1/2⋅{∑i=1k(λk+1−λi)1/2(4λi1/2+n2)}1/2, from which one can obtain a more explicit upper bound on λk+1λk+1 in terms of λ1,…,λkλ1,…,λk (see Corollary 1). When D is contained in a complete n  -dimensional minimal submanifold of RmRm, we prove the inequalityλk+1⩽1k∑i=1kλk+(8(n+2)n2)1/21k∑i=1k(λi(λk+1−λi))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 RnRn.