标题： 通过均质化方法对具有缺陷的高对比度介质的谱收敛 显示英文标题

标题： Spectral convergence for high contrast media with a defect via homogenization

摘要： 我们考虑一个散度形式椭圆算子$A_\epsilon$的特征值问题，该算子具有局部扰动的高对比度周期系数。周期性尺寸$\epsilon$是一个小参数。此类算子的系数的局部扰动可能导致局域波的出现——其对应的特征值位于 Floquet-Bloch 谱的间隙中。我们证明，在双孔隙类型尺度下，在一些自然假设下，特征函数在无穷远处指数衰减，且在$\epsilon$中一致。然后，利用高对比度均质化中的双尺度收敛工具，我们证明了$A_\epsilon$的归一化特征函数的双尺度紧性，即，至多子序列而言，它们双尺度收敛到双尺度极限均质化算子$A_0$的特征函数。这从而建立了$A_\epsilon$和$A_0$的特征值和特征函数之间的渐近一一对应关系。我们还通过直接方法证明了均质化算子的本质谱在系数的局部扰动下的稳定性。 这使我们不仅能够建立$A_\epsilon$到$A_0$的强双尺度预解收敛，而且能够保持孤立特征值的重数，使$A_\epsilon$的谱到$A_0$的谱的豪斯多夫收敛。 显示英文摘要

摘要： We consider an eigenvalue problem for a divergence form elliptic operator $A_\epsilon$ with locally perturbed high contrast periodic coefficients. Periodicity size $\epsilon$ is a small parameter. 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 a double porosity type scaling, under some natural assumptions the eigenfunctions decay exponentially at infinity, uniformly in $\epsilon$. Then, using the tools of two-scale convergence for high contrast homogenization, we prove the two-scale compactness of the normalized eigenfunctions of $A_\epsilon$, i.e. that, up to a subsequence, they two-scale converge to the eigenfunctions of a two-scale limit homogenized operator $A_0$. This consequently establishes asymptotic one-to-one correspondence between eigenvalues and eigenfunctions of $A_\epsilon$ and $A_0$. 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 strong two-scale resolvent convergence of $A_\epsilon$ to $A_0$ but also Hausdorff convergence of the spectra of $A_\epsilon$ to the spectrum of $A_0$, preserving the multiplicity of the isolated eigenvalues.