论文标题
随机均质化和几何奇异性：关于角落的研究
Stochastic homogenization and geometric singularities : a study on corners
论文作者
论文摘要
在这一贡献中，我们对具有均角的多边形域中的线性椭圆方程的定量均质化特性感兴趣。为了开始我们对这种情况的研究，我们考虑了在两个维度上的角度扇区的设置：与整个空间不同，在这样的扇区上存在非平滑谐波函数（这些取决于扇区的角度）。在这里，我们构建了与这些谐波函数相对应的扩展均化校正器，并证明了这些谐波功能的增长估计值，这些校正估计值是准最佳的，即最佳的，直到对数损失。我们对角校正器的构建依赖于该领域中A谐波功能的大规模规律性理论，我们也证明了该理论，并且作为副产品，它产生了liouville原理。我们还提出了一个非标准的2尺度扩展，该扩展适用于部门域并结合了角校正器。我们的最终结果是该改编的2尺度扩展的准最佳误差估计。
In this contribution we are interested in the quantitative homogenization properties of linear elliptic equations with homogeneous Dirichlet boundary data in polygonal domains with corners. To begin our study of this situation, we consider the setting of an angular sector in 2 dimensions : Unlike in the whole-space, on such a sector there exist non-smooth harmonic functions (these depend on the angle of the sector). Here, we construct extended homogenization correctors corresponding to these harmonic functions and prove growth estimates for these which are quasi-optimal, namely optimal up to a logarithmic loss. Our construction of the corner correctors relies on a large-scale regularity theory for a-harmonic functions in the sector, which we also prove and which, as a by-product, yields a Liouville principle. We also propose a nonstandard 2-scale expansion, which is adapted to the sectoral domain and incorporates the corner correctors. Our final result is a quasi-optimal error estimate for this adapted 2-scale expansion.
