具有局部缺陷的多尺度问题的局部精确逼近
Local precised approximation in multiscale problems with local defects
摘要：
我们在这里继续我们的系统研究，开始于[3]，多尺度问题的缺陷，在均匀化理论的背景下。这里考虑的情况下是一个扩散方程的扩散系数的形式的周期函数扰动的$L^r (R^d ) , 1 < r < +$ \infty $ $ , function modeling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3,4], allows to approximate the solution of the original multiscale equation with essentially the same accuracy as in the purely periodic case. The rates of convergence may however vary, and are made precise, depending upon the $ L^r$可积性的缺陷。概括到一个抽象的设置提到。我们的证明严格地遵循了Avellaneda和Lin在[1]中关于周期情形的原始证明的模式，并在Kenig及其合作者的工作[13]中进行了扩展，并从中借鉴了许多内容。
英文摘要：
We proceed here with our systematic study, initiated in [3], of multiscale problems with defects, within the context of homogenization theory. The case under consideration here is that of a diffusion equation with a diffusion coefficient of the form of a periodic function perturbed by an $L^r (R^d ) , 1 < r < +$\infty$$ , function modeling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3,4], allows to approximate the solution of the original multiscale equation with essentially the same accuracy as in the purely periodic case. The rates of convergence may however vary, and are made precise, depending upon the $L^r$ integrability of the defect. The generalization to an abstract setting is mentioned. Our proof exactly follows, step by step, the pattern of the original proof of Avellaneda and Lin in [1] in the periodic case, extended in the works of Kenig and collaborators [13], and borrows a lot from it. The details of the results announced in this Note are given in our forthcoming publications [2,12].
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