Steklov flows on trees and applications

TL;DR

This paper introduces Steklov flows on finite trees and uses them to analyze the monotonicity and rigidity of the first nonzero Steklov eigenvalues, establishing key inequalities and conditions for equality.

Contribution

The paper develops the concept of Steklov flows on trees and proves new monotonicity and rigidity results for Steklov eigenvalues on finite trees.

Findings

First nonzero Steklov eigenvalue increases with subgraph inclusion.

Established conditions for equality in eigenvalue inequalities.

Provided a framework for analyzing Steklov problems on tree structures.

Abstract

We introduce the Steklov flows on finite trees, i.e. the flows (or currents) associated with the Steklov problem. By constructing appropriate Steklov flows, we prove the monotonicity and rigidity of the first nonzero Steklov eigenvalues on trees: for finite trees $\g_1$ and $\g_2,$ the first nonzero Steklov eigenvalue of $\g_1$ is greater than or equal to that of $\g_2$, provided that $\g_1$ is a subgraph of $\g_2.$ Moreover, we give the sufficient and necessary condition in which the equality holds.