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- ItemPositivity and time behavior of a general linear evolution system, non-local in space and time(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Khrabustovskyi, Andrii; Stephan, HolgerWe consider a general linear reaction-diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle, and positivity of the solution, and investigate its asymptotic behavior. Moreover, we give an explicite expression of the limit of the solution for large times. In order to obtain these results we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution of a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction-diffusion system what allows us to investigate its properties.
- ItemHomogenization and Orowan's law for anisotropic fractional operators of any order(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Patrizi, Stefania; Valdinoci, EnricoWe consider an anisotropic Lévy operator Is of any order s 2 (0, 1) and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain is different according to the cases s < 1/2 and s > 1/2. In the isotropic onedimensional case, we also prove a statement related to the so-called Orowans law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior.
- ItemHomogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Flegel, Franziska; Heida, Martin; Slowik, MartinWe study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for the normalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian’s Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence.
- ItemGeneralized Prandtl-Ishlinskii operators arising from homogenization and dimension reduction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Mielke, AlexanderWe consider rate-independent evolutionary systems over a physically domain Ω that are governed by simple hysteresis operators at each material point. For multiscale systems where ε denotes the ratio between the microscopic and the macroscopic length scale, we show that in the limit ε → 0 we are led to systems where the hysteresis operators at each macroscopic point is a generalized Prandtl-Ishlinskii operator.
- ItemDeriving effective models for multiscale systems via evolutionary Gamma-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mielke, AlexanderWe discuss possible extensions of the recently established theory of evolutionary Gamma-convergence for gradient systems to nonlinear dynamical systems obtained by perturbation of a gradient systems. Thus, it is possible to derive effective equations for pattern forming systems with multiple scales. Our applications include homogenization of reaction-diffusion systems, the justification of amplitude equations for Turing instabilities, and the limit from pure diffusion to reaction-diffusion. This is achieved by generalizing the Gamma-limit approaches based on the energy-dissipation principle or the volutionary variational estimate.
- ItemEigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Biskup, Marek; Fukushima, Ryoki; König, WolfgangWe consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors recent work where similar conclusions have been obtained for bounded random potentials.
- ItemA gradient system with a wiggly energy and relaxed EDP-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dondl, Patrick; Frenzel, Thomas; Mielke, AlexanderIf gradient systems depend on a microstructure, we want to derive a macroscopic gradient structure describing the effective behavior of the microscopic system. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gammaconvergence. We call this notion relaxed EDP-convergence since the special structure of the dissipation functional may not be preserved under Gamma-convergence. However, by investigating the kinetic relation we derive the macroscopic dissipation potential.
- ItemAn existence result and evolutionary [Gamma]-convergence for perturbed gradient systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bacho, Aras; Emmrich, Etienne; Mielke, AlexanderWe consider the initial-value problem for the perturbed gradient flows, where a differential inclusion is formulated in terms of a subdifferential of an energy functional, a subdifferential of a dissipation potential and a more general perturbation, which is assumed to be continuous and to satisfy a suitable growth condition. Under additional assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semi-implicit discretization scheme with a variational approximation technique.