Homogenization and Correctors for Stochastic Hyperbolic Equations in Domains with Periodically Distributed Holes

The goal of this paper is to present new results on homogenization and correctors for stochastic linear hyperbolic equations in periodically perforated domains with homogeneous Neumann conditions on the holes. The main tools are the periodic unfolding method, energy estimates, probabilistic and deterministic compactness results. The findings of this paper are stochastic counterparts of the celebrated work [D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63 (2006) 467–496]. The convergence of the solution of the original problem to a homogenized problem with Dirichlet condition has been shown in suitable topologies. Homogenization and convergence of the associated energies results recover the work in [M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptot. Anal. 97 (2016) 301–327]. In addition to that, we obtain corrector results.

Homogenization and correctors of Robin problem for linear stochastic equations in periodically perforated domains

In this paper, we present homogenization and corrector results for stochastic linear parabolic equations in periodically perforated domains with non-homogeneous Robin conditions on the holes. We use the periodic unfolding method and probabilistic compactness results. Homogenization results presented in this paper are stochastic counterparts of some fundamental work given in [Cioranescu, Donato and Zaki in Port. Math. (N.S.) 63 (2006), 467–496]. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a parabolic stochastic equation in fixed domain with Dirichlet condition on the boundary. In contrast to the two scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems.

Homogenization of Perforated Elastic Structures

The paper is dedicated to the asymptotic behavior of $\varepsilon$ ε -periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as $\varepsilon \to 0$ ε → 0 . In case of plate-like or beam-like structures the asymptotic reduction of dimension from $3D$ 3 D to $2D$ 2 D or $1D$ 1 D respectively takes place. An example of the structure under consideration can be obtained by a periodic repetition of an elementary “flattened” ball or cylinder for plate-like or beam-like structures in such a way that the contact surface between two neighboring balls/cylinders has a non-zero measure. Since the domain occupied by the structure might have a non-Lipschitz boundary, the classical homogenization approach based on the extension cannot be used. Therefore, for obtaining Korn’s inequalities, which are used for the derivation of a priori estimates, we use the approach based on interpolation. In case of plate-like and beam-like structures the proof of Korn’s inequalities is based on the displacement decomposition for a plate or a beam, respectively. In order to pass to the limit as $\varepsilon \to 0$ ε → 0 we use the periodic unfolding method.

Homogenization and correctors for linear stochastic equations via the periodic unfolding methods

In this paper, we use the periodic unfolding method and Prokhorov’s and Skorokhod’s probabilistic compactness results to obtain homogenization and corrector results for stochastic partial differential equations (PDEs) with periodically oscillating coefficients. We show the convergence of the solutions of the original problems to the solutions of the homogenized problems. In contrast to the two-scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems

Μερικές διαφορικές εξισώσεις και προβλήματα της επιστήμης των υλικών

The main objective of this thesis is the homogenization of partial dierentialequations (mainly Maxwell'As equations) describing electromagneticphenomena in complex media. In particular, we study the homogenization ofMaxwell'As equations focusing on the periodic unfolding method in complexmedia under Drude-Born-Fedorov type, local in time, constitutive relations.Firstly, we formulate Maxwell'A s problem as an evolution initial value(Cauchy) problem in a Hilbert space supplemented with the constitutiverelations of a bianisotropic medium (the most general linear medium in electromagnetics).Further, we analyze the notion of homogenization and weapply it as examples to equations of elliptic type in divergence form and toMaxwell'As system in bianisotropic media.We present also the method of periodic unfolding in the case of an ellipticpartial dierential equation and in the main part of this work we considerthe problem of the well-posedness of the time-dependent Maxwell'As equationsin a Drude-Born-Fedorov type environment considering the elds to beelements of an appropriate Hilbert space. In order to prove the existence anduniqueness we apply the Faedo-Galerkin method and for the continuous dependencefrom the initial data we use semigroup theory for operators. Therest of the main part of the thesis deals with the homogenization of theconsidered problem, using the periodic unfolding method.In the last chapter, we examine the time-harmonic Maxwell problem ina bianisotropic cavity, which we study by transforming it to an eigenvalueproblem.