高对比度随机介质中局域于缺陷的本征函数
Eigenfunctions localised on a defect in high-contrast random media
AI总结：
该研究针对高对比度随机介质中缺陷引发的本征问题，分析带随机双孔隙标度系数的散度型椭圆算子在缺陷扰动下的谱特性，证明了点谱的豪斯多夫收敛、本征函数的一致指数衰减及强随机双尺度收敛性。
摘要：
我们研究高对比度随机算子谱间隙中由缺陷产生的本征值及对应本征函数的性质。我们考虑一族散度型椭圆算子$\u2702^{\u03b5}$，其系数为随机的，具有双孔隙度类型的标度性，且在一个固定尺寸的紧域（缺陷）上受到扰动。在未受扰算子$\u00c2^{\u03b5}$的极限谱间隙中开展研究，我们证明当$\u03b5 \to 0$时，$\u2702^{\u03b5}$的点谱按豪斯多夫意义收敛于极限双尺度算子$\u2702^{\u2095\u1d6fm}$的点谱。此外，我们证明对于足够小的$\u03b5$，$\u2702^{\u03b5}$的本征函数在无穷远处一致指数衰减。这进而得到这类本征函数到$\u2702^{\u2095\u1d6fm}$本征函数的强随机双尺度收敛性。
英文摘要：
We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrum of a high-contrast random operator. We consider a family of elliptic operators $\mathcal{A}^\varepsilon$ in divergence form whose coefficients are random, possess double porosity type scaling, and are perturbed on a fixed-size compact domain (a defect). Working in the gaps of the limiting spectrum of the unperturbed operator $\hat{\mathcal{A}}^\varepsilon$, we show that the point spectrum of $\mathcal{A}^\varepsilon$ converges in the sense of Hausdorff to the point spectrum of the limiting two-scale operator $\mathcal{A}^\mathrm{hom}$ as $\varepsilon \to 0$. Furthermore, we prove that the eigenfunctions of $\mathcal{A}^\varepsilon$ decay exponentially at infinity uniformly for sufficiently small $\varepsilon$. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of $\mathcal{A}^\mathrm{hom}$.
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