The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues with Dirichlet boundary condition on Riemannian manifolds

Let $\mathrm{\Omega }$ be a bounded domain with $C^2$-smooth boundary in an $n$-dimensional oriented Riemannian manifold. It is well-known that for the bi-harmonic equation ${\mathrm{\Delta }}^{2}u=0$ in $\mathrm{\Omega }$ with the $0$-Dirichlet boundary condition, there exists an infinite set $\{u_k\}$ of biharmonic functions in $\mathrm{\Omega }$ with positive eigenvalues $\{{\lambda }_k\}$ satisfying $\Delta u_k+ \lambda_k \varrho \frac{\partial u_k}{\partial \nu}=0$ontheboundary$\partial \Omega$. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Stekloff eigenvalues ${\lambda }_k$.