Spectral convergence for high contrast elliptic periodic problems with a defect via homogenization

Details Spectral convergence for high contrast elliptic periodic problems with a defect via homogenization Spectral convergence for high contrast elliptic periodic problems with a defect via homogenization

2007-12-29

arxiv.org/abs/0801.0084v2

Mathematics

Analysis of PDEs

25 pages, 2 figures, bibliography 22 titles. Paper was revised thoroughly, typos corrected, some comments added. 4 new referen

Scientific paper

We consider an eigenvalue problem for a divergence form elliptic operator $A_\epsilon$ with high contrast periodic coefficients with period $\epsilon$ in each coordinate, where $\epsilon$ is a small parameter. The coefficients are perturbed on a bounded domain of `order one' size. The local perturbation of coefficients for such operator could result in emergence of localized waves - eigenfunctions with corresponding eigenvalues lying in the gaps of the Floquet-Bloch spectrum. We prove that, for the so-called double porosity type scaling, the eigenfunctions decay exponentially at infinity, uniformly in $\epsilon$. Then, using the tools of two-scale convergence for high contrast homogenization, we prove the strong two-scale compactness of the eigenfunctions of $A_\epsilon$. This implies that the eigenfunctions converge in the sense of the strong two-scale convergence to the eigenfunctions of a two-scale limit homogenized operator $A_0$, consequently establishing `asymptotic one-to-one correspondence' between the eigenvalues and the eigenfunctions of these two operators. We also prove by direct means the stability of the essential spectrum of the homogenized operator with respect to the local perturbation of its coefficients. That allows us to establish not only the strong two-scale resolvent convergence of $A_\epsilon$ to $A_0$ but also the Hausdorff convergence of the spectra of $A_\epsilon$ to the spectrum of $A_0$, preserving the multiplicity of the isolated eigenvalues.

Affiliated with

Also associated with

No associations

Rating

Spectral convergence for high contrast elliptic periodic problems with a defect via homogenization does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Spectral convergence for high contrast elliptic periodic problems with a defect via homogenization, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Spectral convergence for high contrast elliptic periodic problems with a defect via homogenization will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-713851