Doubling property and vanishing order of Steklov eigenfunctions

TL;DR

This paper investigates the doubling estimates and vanishing order of Steklov eigenfunctions on smooth boundary domains, providing improved bounds and showing the vanishing order is linearly bounded by the eigenvalue.

Contribution

It improves the doubling property estimates for Steklov eigenfunctions and establishes a linear bound on their vanishing order in terms of the eigenvalue.

Findings

Enhanced doubling estimates for Steklov eigenfunctions.

Proved vanishing order is less than C times the eigenvalue.

Results depend only on the domain, not on specific eigenfunctions.

Abstract

The paper is concerned with the doubling estimates and vanishing order of the Steklov eigenfunction on the boundary of a smooth boundary domain $\mathbb R^n$. The eigenfunction is given by a Dirichlet-to-Neumann map. We improve the doubling property shown by Lin and Bellova \cite{BL}. Furthermore, we show that the vanishing order of Steklov eigenfunction is everywhere less than $C\lambda$ where $\lambda$ is the Steklov eigenvalue and $C$ depends only on $\Omega$.