论文标题
Hölder规律性的椭圆形算子分数域的域的规律性具有混合边界条件
Hölder Regularity for Domains of Fractional Powers of Elliptic Operators with Mixed Boundary Conditions
论文作者
论文摘要
这项工作是关于全球Hölder规律性，用于解决椭圆形偏微分方程的解决方案，但在不规则域上受到混合边界条件。有两个主要结果。首先，我们表明，如果在负sobolev空间中实现了椭圆差分运算符的域，并嵌入了hölder连续函数的空间中，那么$ q> d $就会嵌入到Hölder连续函数中，那么该操作员的合适分数幂的域也是如此。然后，第二个主要结果表明，第一个的前提确实得到了满足。证明是沿着本地化，转换和反思的经典技术进行的，从而可以归结为Ladyzhenskaya或Kinderlehrer的经典结果。我们方法的主要特征之一是，我们不需要Dirichlet边界部分的Lipschitz图表，而是Dirichlet和Neumann边界部件的界面上的有趣的度量/测量理论条件。 Ter Elst和Rehberg在2015年在相关工作中提出了类似的条件，但是如果仅限于4个空间尺寸，那么目前的证明就更为简单。
This work is about global Hölder regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the realization of an elliptic differential operator in a negative Sobolev space with integrability $q > d$ embeds into a space of Hölder continuous functions, then so do the domains of suitable fractional powers of this operator. The second main result then establishes that the premise of the first is indeed satisfied. The proof goes along the classical techniques of localization, transformation and reflection which allows to fall back to the classical results of Ladyzhenskaya or Kinderlehrer. One of the main features of our approach is that we do not require Lipschitz charts for the Dirichlet boundary part, but only an intriguing metric/measure-theoretic condition on the interface of Dirichlet- and Neumann boundary parts. A similar condition was posed in a related work by ter Elst and Rehberg in 2015, but the present proof is much simpler, if only restricted to space dimension up to 4.
