标题： 通过均质化方法对具有缺陷的高对比度周期椭圆问题的谱收敛性 显示英文标题

标题： Spectral convergence for high contrast elliptic periodic problems 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，该算子具有在每个坐标方向上周期为$\epsilon$的高对比度周期系数，其中$\epsilon$是一个小参数。 系数在一个“阶为一”的有界区域内被扰动。 对于此类算子的系数局部扰动可能导致局域波的出现——即对应的特征值位于 Floquet-Bloch 谱间隙中的特征函数。 我们证明，在所谓的双孔隙型标度下，特征函数在无穷远处指数衰减，且在$\epsilon$上一致。 然后，利用高对比度均匀化的双尺度收敛工具，我们证明了算子$A_\epsilon$的特征函数具有强双尺度紧性。 这表明特征函数在强双尺度收敛的意义下收敛到双尺度极限均匀化算子$A_0$的特征函数，从而在这些两个算子的特征值和特征函数之间建立了“渐近一一对应”关系。 我们还通过直接方法证明了均匀化算子的本质谱在系数局部扰动下的稳定性。 这使我们不仅能够建立$A_\epsilon$到$A_0$的强双尺度预解算子收敛，而且能够保证$A_\epsilon$的谱到$A_0$的谱的豪斯多夫收敛，并保持孤立特征值的重数。 显示英文摘要

摘要： 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.