多边形和多面体中线性椭圆系统的解析正则性
Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra
AI总结：
针对多面体中的二阶椭圆边值问题，利用二进划分和嵌套开集技术，证明了加权各向异性解析估计，建立了全局解析正则性。
摘要：
我们证明了多面体中二阶椭圆边值问题解的加权各向异性解析估计。所使用的加权解析类与Guo在1993年为建立多面体中hp有限元方法的指数收敛性而引入的相同。我们首先给出多边形中已知加权解析正则性的一个简单证明，依赖于光滑域中具有解析导数控制的椭圆先验估计的新表述。该技术基于角点附近的二进划分。该技术可以成功扩展到多面体，提供各向同性解析正则性。这不是最优的，因为它没有充分利用沿边缘的完全正则性。我们将其与嵌套开集技术相结合，以获得所需的三维各向异性解析正则性结果。我们的证明是全局的，不需要分析奇异函数。
英文摘要：
We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.
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