[Paper Review] Two-scale convergence of elliptic spectral problems with indefinite density function in perforated domains

Hermann Douanla|arXiv (Cornell University)|Jun 20, 2011

Advanced Mathematical Modeling in Engineering23 references4 citations

TL;DR

This paper investigates the homogenization of elliptic spectral problems with indefinite (sign-changing) density functions in periodically perforated domains using two-scale convergence. It establishes concise homogenized limits for positive and negative eigencouples depending on whether the average of the density over the perforated part is positive, negative, or zero, revealing distinct asymptotic behaviors in each case.

ABSTRACT

Spectral asymptotics of linear periodic elliptic operators with indefinite (sign-changing) density function is investigated in perforated domains with the two-scale convergence method. The limiting behavior of positive and negative eigencouples depends crucially on whether the average of the weight over the solid part is positive, negative or equal to zero. We prove concise homogenization results in all three cases.

Motivation & Objective

• To analyze the spectral asymptotics of linear elliptic operators with indefinite density functions in periodically perforated domains.
• To determine how the homogenized limit of eigenvalues and eigenfunctions depends on the sign of the average of the density function over the perforated region.
• To extend two-scale convergence techniques to spectral problems with sign-changing weights in non-fixed, oscillating domains.
• To provide a complete classification of homogenization results in the three cases: positive, negative, and zero average of the density function.

Proposed method

• Application of two-scale convergence to handle the oscillatory behavior of the density and coefficient functions in the perforated domain.
• Use of periodic unfolding and corrector constructions to derive the homogenized limit equations.
• Derivation of two-scale asymptotic expansions for eigenfunctions and eigenvalues, including corrector terms involving fast and slow variables.
• Solution of cell problems in the periodicity cell $Y^*$ to determine effective coefficients and homogenized operators.
• Employment of variational formulations and weak convergence arguments to pass to the limit in the spectral problem.
• Use of energy methods and minimax principles to characterize eigenvalues and ensure convergence of eigencouples.

Experimental results

Research questions

• RQ1 How do the positive and negative eigencouples behave as the perforation size $\varepsilon \to 0$ when the density function changes sign?
• RQ2 What is the homogenized limit of the eigenvalue problem when the average of the density over the perforated region is positive, negative, or zero?
• RQ3 Can two-scale convergence be effectively applied to spectral problems with indefinite weights in periodically perforated domains?
• RQ4 How do the eigenfunctions and eigenvalues converge, and what is the structure of the corrector terms in the asymptotic expansion?
• RQ5 What are the effective operators and eigenvalue equations in the homogenized limit, and how do they depend on the average of the weight function?

Key findings

• When the average of the density over $Y^*$ is positive, the positive eigencouples converge to a homogenized problem with a positive definite effective operator.
• When the average is negative, the negative eigencouples converge to a homogenized problem with a negative definite effective operator.
• When the average is zero, the homogenized limit involves a nontrivial coupling between the slow and fast variables, leading to a modified eigenvalue problem with a non-degenerate effective operator.
• The eigenfunctions converge in $H^1$-norm with a corrector term involving the two-scale structure, ensuring strong convergence of gradients.
• The orthonormality of the homogenized eigenfunctions is characterized by a factor involving $\nu^2 = \int_{Y^*} \rho(y) \chi^0(y) dy$, which depends on the cell problem solution.
• For simple eigenvalues, the convergence holds for the full sequence and a corrector-type convergence result for the gradients is established.