A sharp asymptotic remainder estimate for biharmonic Steklov eigenvalues on Riemannian manifolds

Let $\Omega$ be a bounded domain with $C^\infty$ boundary in an $n$-dimensional $C^\infty$ Riemannian manifold, and let $\varrho$ be a non-negative bounded function defined on $\partial \Omega$. It is well-known that for the biharmonic equation $\Delta^2 u=0$ in $\Omega$ with the 0-Dirichlet boundary condition, there exists an infinite set $\{u_k\}$ of biharmonic functions in $\Omega$ with positive eigenvalues $\{\lambda_k\}$ satisfying $\Delta u_k+ \lambda_k \varrho \frac{\partial u_k}{\partial \nu}=0$ on the boundary $\partial \Omega$. In this paper, we give the Weyl-type asymptotic formula with a sharp remainder estimate for the counting function of the biharmonic Steklov eigenvalues $\lambda_k$.