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For classical Neumann eigenvalue, buckling eigenvalue and clamped plate eigenvalue, we give the corresponding Rellich type identities. As an application of these results, then, we obtain a new necessary and sufficient condition for a domain without the Pompeiu property.

Let Ω be a bounded domain with C 2 -smooth boundary in an n-dimensional oriented Riemannian manifold. It is well known that for the biharmonic equation Δ 2 u = 0 in Ω with the condition u = 0 on ∂Ω, there exists an infinite set { u k } of biharmonic functions in Ω with positive eigenvalues { λ k } satisfying Δ u k + λ k ϱ ∂ u k ∂ ν = 0 on ∂Ω. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues λ k .