Sharp Weyl-Type Formulas of the Spectral Functions for Biharmonic Steklov Eigenvalues

TL;DR

This paper derives precise Weyl-type asymptotic formulas with sharp remainder estimates for biharmonic Steklov eigenvalues, solving a longstanding problem in spectral geometry.

Contribution

It explicitly calculates principal symbols and applies Hörmander's spectral function theorem to obtain sharp asymptotics for biharmonic Steklov eigenvalues.

Findings

Weyl-type asymptotic formulas with sharp remainder estimates derived

Solved a longstanding open problem in spectral analysis

Provides explicit calculations of principal symbols for pseudodifferential operators

Abstract

In this paper, by explicitly calculating the principal symbols of pseudodifferential operators and by applying H\"omander's spectral function theorem, we obtain the Weyl-type asymptotic formulas with sharp remainder estimates for the counting functions of the two classes of biharmonic Steklov eigenvalues $\lambda_k$ and $\mu_k$ in a smooth bounded domain of a Riemannian manifold. This solves a longstanding challenging problem.