Upper bound of the counting function of Steklov eigenvalues

TL;DR

This paper establishes an upper bound for the counting function of Steklov eigenvalues on compact manifolds, linking it to Weyl's law, and explores eigenfunction decay near the boundary.

Contribution

It provides a new upper bound for Steklov eigenvalue counting function and insights into eigenfunction decay, extending understanding beyond classical results.

Findings

Upper bound involving Weyl's law for Steklov eigenvalues

Description of eigenfunction decay near the boundary

Relation to Pólya's conjecture in the Steklov context

Abstract

We study the counting function of Steklov eigenvalues on compact manifolds with boundary and obtain its upper bound involving the leading term of Weyl's law. Our estimate can be viewed as a weakened version of P\'{o}lya's Conjecture in the Steklov case on general manifolds. As a byproduct, we also obtain a description about the decay behavior of Steklov eigenfunctions near the boundary.