Homogenization of Steklov spectral problems with indefinite density function in perforated domains

TL;DR

This paper studies the asymptotic behavior of Steklov eigenvalue problems with indefinite density functions in perforated domains, revealing how the spectrum splits and depends on the average weight over the perforation surface.

Contribution

It provides a detailed homogenization analysis of Steklov problems with sign-changing densities in perforated domains using two-scale convergence.

Findings

Spectrum consists of two diverging sequences, one to -infinity and one to +infinity.

The asymptotic behavior depends on the sign of the average weight over the perforation surface.

The analysis covers cases where the average weight is positive, negative, or zero.

Abstract

The asymptotic behavior of second order self-adjoint elliptic Steklov eigenvalue problems with periodic rapidly oscillating coefficients and with indefinite (sign-changing) density function is investigated in periodically perforated domains. We prove that the spectrum of this problem is discrete and consists of two sequences, one tending to -{\infty} and another to +{\infty}. The limiting behavior of positive and negative eigencouples depends crucially on whether the average of the weight over the surface of the reference hole is positive, negative or equal to zero. By means of the two-scale convergence method, we investigate all three cases.