Boundary homogenization for a triharmonic intermediate problem

TL;DR

This paper investigates how oscillatory boundary perturbations affect the spectral properties of the triharmonic operator, identifying limit behaviors and a unique homogenization case with a strange boundary term.

Contribution

It introduces a detailed analysis of spectral homogenization for the triharmonic operator under oscillatory boundary perturbations, including the critical case with a novel boundary term.

Findings

Identification of limit problems depending on oscillation strength

Analysis of the critical oscillation case with a strange boundary term

Characterization of spectral behavior under boundary perturbations

Abstract

We consider the triharmonic operator subject to homogeneous boundary conditions of intermediate type on a bounded domain of the N-dimensional Euclidean space. We study its spectral behaviour when the boundary of the domain undergoes a perturbation of oscillatory type. We identify the appropriate limit problems which depend on whether the strength of the oscillation is above or below a critical threshold. We analyse in detail the critical case which provides a typical homogenization problem leading to a strange boundary term in the limit problem.