Liouville Principles and a Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space
Liouville Principles and a Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space
复制标题
半空间上随机椭圆算子的刘维尔原理和大规模正则理论
DOI：
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发表时间：
2016
影响因子：
2
通讯作者：
C. Raithel
中科院分区：
文献类型：
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作者：
J. Fischer;C. Raithel
研究了具有随机系数场的二阶线性椭圆型方程解的大尺度正则性。与以往随机椭圆算子正则性理论的工作不同，我们的兴趣在于边界处的正则性：我们考虑齐次Dirichlet边界条件的半空间上的问题，并以均匀化自适应倾斜过剩的相应衰减估计的形式导出相关的$C^{1，alpha}$型大尺度正则性理论。这个正则性理论需要一个相关的刘维尔型定理。结果是基于适应半空间设置，我们构造的均匀化校正器的存在-通过一个完全确定性的参数-作为整个空间上的均匀化校正器的修改。这种适应过程是在较大尺度上进行归纳的，关键是依赖于在较小尺度上已经建立的规律性理论。
We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the regularity at the boundary: We consider problems posed on the half-space with homogeneous Dirichlet boundary conditions and derive an associated $C^{1,alpha}$-type large-scale regularity theory in the form of a corresponding decay estimate for the homogenization-adapted tilt-excess. This regularity theory entails an associated Liouville-type theorem. The results are based on the existence of homogenization correctors adapted to the half-space setting, which we construct - by an entirely deterministic argument - as a modification of the homogenization corrector on the whole space. This adaption procedure is carried out inductively on larger scales, crucially relying on the regularity theory already established on smaller scales.