Filters

2020 - 20224

More filters

20224
Reset filters

Settings

End date: 2023 × Start date: 2020 × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: Stochastic homogenization ×

Search Results

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    Stochastic two-scale convergence and Young measures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, Martin; Neukamm, Stefan; Varga, Mario
    In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.
  • Thumbnail Image
    Item
    Stochastic homogenization on perforated domains I: Extension operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, Martin
    This preprint is part of a major rewriting and substantial improvement of WIAS Preprint 2742. In this first part of a series of 3 papers, we set up a framework to study the existence of uniformly bounded extension and trace operators for W1,p-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimaly smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from W1,p to W1,r with the strict inequality r
  • Thumbnail Image
    Item
    Stochastic homogenization on randomly perforated domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Heida, Martin
    We study the existence of uniformly bounded extension and trace operators for W1,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 Rd 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.
  • Thumbnail Image
    Item
    Stochastic homogenization on perforated domains III -- General estimates for stationary ergodic random connected Lipschitz domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Heida, Martin
    This is Part III of a series on the existence of uniformly bounded extension operators on randomly perforated domains in the context of homogenization theory. Recalling that randomly perforated domains are typically not John and hence extension is possible only from W 1,p to W 1,r, r < p, we will show that the existence of such extension operators can be guarantied if the weighted expectations of four geometric characterizing parameters are bounded: The local Lipschitz constant M, the local Lipschitz radius Δ , the mesoscopic Voronoi diameter ∂ and the local connectivity radius R.