Homogenization of Steklov eigenvalues with rapidly oscillating weights

TL;DR

This paper investigates how Steklov eigenvalues behave when the boundary weights oscillate rapidly, providing homogenization rates and boundary oscillation analysis, with applications to spectral curve estimates.

Contribution

It introduces new homogenization rate results for Steklov eigenvalues with oscillating weights and analyzes boundary oscillations and eigenfunction bounds.

Findings

Derived homogenization rates for Steklov eigenvalues

Established boundary oscillation estimates for eigenfunctions

Provided spectral curve estimates for the Dancer-Fučík spectrum

Abstract

In this article we study the homogenization rates of eigenvalues of a Steklov problem with rapidly oscillating periodic weight functions. The results are obtained via a careful study of oscillating functions on the boundary and a precise estimate of the $L^\infty$ bound of eigenfunctions. As an application we provide some estimates on the first nontrivial curve of the Dancer-{F}u{\v{c}}{\'{\i}}k spectrum.