The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds

Abstract

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 .