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

TL;DR

This paper establishes a Weyl-type asymptotic formula for the distribution of biharmonic Steklov eigenvalues on smooth bounded domains within Riemannian manifolds, advancing spectral theory for higher-order elliptic operators.

Contribution

The paper introduces a new method to derive the Weyl-type asymptotic formula for biharmonic Steklov eigenvalues, extending spectral asymptotics to higher-order problems on Riemannian manifolds.

Findings

Derived the Weyl-type asymptotic formula for biharmonic Steklov eigenvalues

Extended spectral asymptotics to biharmonic operators on Riemannian manifolds

Provided a new approach for analyzing higher-order elliptic eigenvalue problems

Abstract

Let $\Omega$ 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 $\Delta^2 u=0$ 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, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Stekloff eigenvalues $\lambda_k$.