Higher order elliptic operators on variable domains. Stability results and boundary oscillations for intermediate problems

TL;DR

This paper investigates how higher order elliptic operators' spectra change under domain perturbations, especially with boundary oscillations, revealing critical behaviors and boundary condition effects using homogenization techniques.

Contribution

It provides new stability results for spectral behavior of elliptic operators with various boundary conditions and analyzes the impact of boundary oscillations, including the emergence of strange terms.

Findings

Spectral stability results for Dirichlet, Neumann, and intermediate boundary conditions.

Identification of critical oscillatory behavior affecting the limit problem.

Discovery of a strange boundary term in the homogenized limit at the critical oscillation.

Abstract

We study the spectral behavior of higher order elliptic operators upon domain perturbation. We prove general spectral stability results for Dirichlet, Neumann and intermediate boundary conditions. Moreover, we consider the case of the bi-harmonic operator with those intermediate boundary conditions which appears in study of hinged plates. In this case, we analyze the spectral behavior when the boundary of the domain is subject to a periodic oscillatory perturbation. We will show that there is a critical oscillatory behavior and the limit problem depends on whether we are above, below or just sitting on this critical value. In particular, in the critical case we identify the strange term appearing in the limiting boundary conditions by using the unfolding method from homogenization theory.