In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Steklov eigenvalues with Neumann boundary condition in a bounded domain of an n-dimensional Riemannian manifold.1991 Mathematics Subject Classification. 35P20, 58J50, 58C40, 65L15

For a given bounded domain with smooth boundary in a smooth Riemannian manifold (M, g), by decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferential operator, and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as t → 0 + . In particular, we explicitly give the first four coefficients of this asymptotic expansion. These coefficients provide precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Steklov problem.

For a compact Riemannian manifold (Ω, g) with smooth boundary ∂Ω, we explicitly give local representation and full symbol expression for the elastic Dirichlet-to-Neumann map Ξg by factorizing an equivalent elastic equation. We prove that for a strong convex or extendable real-analytic manifold Ω with boundary, the elastic Dirichlet-to-Neumann map Ξg uniquely determines metric g of Ω in the sense of isometry. We also give a procedure by which we can explicitly calculate all coefficients a 0 , a 1) as t → 0 + , where τ k is the k-th eigenvalue of the elastic Dirichlet-to-Neumann map (i.e., k-th elastic Steklov eigenvalue). The coefficients am are spectral invariants and provide precise information for the boundary volume vol(∂Ω) of the elastic medium Ω and the total mean curvature as well as other total curvatures of the boundary ∂Ω. These conclusions give answer to two open problems.

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