论文标题
随机穿孔域上的随机均质化
Stochastic homogenization on randomly perforated domains
论文作者
论文摘要
我们研究了$ w^{1，p} $的均匀界限扩展和跟踪操作员的存在，在随机穿孔的域上函数，其中几何被认为是固定的ergodic。这种延伸和痕量算子对于随机均质化的紧凑性很重要。与以前的方法和结果相反，我们对几何形状使用非常弱的假设，我们称之为局部$（δ，m）$ - 规律性，各向同性锥混合和有界平均连接性。第一个概念测量了域的局部Lipschitz的规律性，而第二个概念测量了空间的介观分布。第三个是最棘手的部分，并测量几何学的“介质”连接性。与以前的方法相反，我们不需要夹杂物之间的距离最小，并且允许全球无限的Lipschitz常数和渗透孔。我们将通过基于泊松点过程和Delaunay管道过程将其应用于布尔模型来说明我们的方法。我们最终在$ \ mathbb {r}^d $和$ω$上引入了合适的Sobolev空间，以构建随机的二级收敛方法，并将结果理论应用于随机完善的域上的$ p $ - laplace问题的均质化。
We study the existence of uniformly bounded extension and trace operators for $W^{1,p}$-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions on the geometry which we call local $(δ,M)$-regularity, isotropic cone mixing and bounded average connectivity. The first concept measures local Lipschitz regularity of the domain while the second measures the mesoscopic distribution of void space. The third is the most tricky part and measures the "mesoscopic" connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process. We finally introduce suitable Sobolev spaces on $\mathbb{R}^d$ and $Ω$ in order to construct a stochastic two-scale convergence method and apply the resulting theory to the homogenization of a $p$-Laplace problem on a randomly perforated domain.
