On a Babu\v{s}ka paradox for polyharmonic operators: spectral stability and boundary homogenization for intermediate problems

TL;DR

This paper investigates how high-order elliptic operators' spectra change under domain perturbations, revealing a polyharmonic Babuška paradox and establishing optimal boundary homogenization conditions.

Contribution

It introduces sharp assumptions for spectral convergence of polyharmonic operators under domain perturbations, extending known results and analyzing a boundary homogenization problem.

Findings

Identifies sharp conditions for spectral stability of polyharmonic operators.

Reveals a smooth version of the Babuška paradox for polyharmonic operators.

Provides detailed analysis of boundary homogenization in this context.

Abstract

We analyse the spectral convergence of high order elliptic differential operators subject to singular domain perturbations and homogeneous boundary conditions of intermediate type. We identify sharp assumptions on the domain perturbations improving, in the case of polyharmonic operators of higher order, conditions known to be sharp in the case of fourth order operators. The optimality is proved by analysing in detail a boundary homogenization problem, which provides a smooth version of a polyharmonic Babu\v{s}ka paradox.