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Networks & Heterogeneous Media

2008 , Volume 3 , Issue 3

Special Issue on Homogenization Theory and Related Topics

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Preface
Leonid Berlyand and  V. V. Zhikov
2008, 3(3): i-ii doi: 10.3934/nhm.2008.3.3i +[Abstract](33) +[PDF](40.9KB)
Abstract:
This special issue contains selected papers on Homogenization Theory and related topics. It is dedicated to Eugene Khruslov on the occasion of his seventieth birthday. Professor Khruslov made pioneering contributions into this field.
Homogenization problems were first studied in the late nineteenth century (Poisson, Maxwell, Rayleigh) and early twentieth century (Einstein). These studies were based on deep physical intuition allowing these outstanding physicists to solve several specific important problems such as calculating the effective conductivity of a two-phase conductor and the effective viscosity of suspensions. It was not until the early 1960s that homogenization began to gain a rigorous mathematical footing which enabled it to be applied to a wide variety of problem in physics and mechanics. A number of mathematical tools such as the asymptotic analysis of PDEs, variational bounds, heterogeneous multiscale method, and the probabilistic techniques of averaging were developed. Although this theory is a well-established area of mathematics, many fascinating problems remain open. Interesting examples of such problems can be found in the papers of this issue.

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Homogenization of spectral problems in bounded domains with doubly high contrasts
Natalia O. Babych , Ilia V. Kamotski and  Valery P. Smyshlyaev
2008, 3(3): 413-436 doi: 10.3934/nhm.2008.3.413 +[Abstract](39) +[PDF](351.9KB)
Abstract:
Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed. Two-scale limit equations are derived and relate to certain non-standard self-adjoint operators. In particular they explicitly display the first two terms in the asymptotic expansion for the eigenvalues, with a surprising bound for the error of order $\varepsilon^{5/4}$ proved.
Distribution of minimum values of stochastic functionals
Victor Berdichevsky
2008, 3(3): 437-460 doi: 10.3934/nhm.2008.3.437 +[Abstract](45) +[PDF](252.5KB)
Abstract:
Some mathematical problems of mechanics and physics have a form of the following variational problem. There is a functional, $I$, which is a sum of some quadratic positive functional and a linear functional. The quadratic functional is deterministic. The linear functional is a sum of a large number, $N$, of statistically independent linear functionals. The minimum value of the functional, $I$, is random. One needs to know the probability distribution of the minimum values for large $N$. The probability distribution was found in [2] in terms of solution of some deterministic variational problem. It was clear from the derivation that the class of quadratic and linear functionals for which this probability distribution can be used is not empty. It was not clear though how wide this class is. This paper aims to give some sufficient conditions for validity of the results of [2].
Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain
Leonid Berlyand and  Petru Mironescu
2008, 3(3): 461-487 doi: 10.3934/nhm.2008.3.461 +[Abstract](50) +[PDF](312.5KB)
Abstract:
Let $A$ be an annular type domain in $\mathbb{R}^2$. Let $A_\delta$ be a perforated domain obtained by punching periodic holes of size $\delta$ in $A$; here, $\delta$ is sufficiently small. Suppose that $\J$ is the class of complex-valued maps in $A_\delta$, of modulus $1$ on $\partial A_\delta$ and of degrees $1$ on the components of $\partial A$, respectively $0$ on the boundaries of the holes.

We consider the existence of a minimizer of the Ginzburg-Landau energy

$E_\lambda(u)=\frac 1\2_[\int_{A_\delta}](|\nabla u|^2+\frac\lambda 2(1-|u|^2)^2)$
among all maps in $u\in\J$.

It turns out that, under appropriate assumptions on $\lambda=\lambda(\delta)$, existence is governed by the asymptotic behavior of the $H^1$-capacity of $A_\delta$. When the limit of the capacities is $>\pi$, we show that minimizers exist and that they are, when $\delta\to 0$, equivalent to minimizers of the same problem in the subclass of $\J$ formed by the $\mathbb{S}^1$-valued maps. This result parallels the one obtained, for a fixed domain, in [3], and reduces homogenization of the Ginzburg-Landau functional to the one of harmonic maps, already known from [2].

When the limit is $<\pi$, we prove that, for small $\delta$, the minimum is not attained, and that minimizing sequences develop vortices. In the case of a fixed domain, this was proved in [1].
Non convex homogenization problems for singular structures
Andrea Braides and  Valeria Chiadò Piat
2008, 3(3): 489-508 doi: 10.3934/nhm.2008.3.489 +[Abstract](30) +[PDF](316.6KB)
Abstract:
We prove a homogenization theorem for non-convex functionals depending on vector-valued functions, defined on Sobolev spaces with respect to oscillating measures. The proof combines the use of the localization methods of $\Gamma$-convergence with a 'discretization' argument, which allows to link the oscillating energies to functionals defined on a single Lebesgue space, and to state the hypothesis of $p$-connectedness of the underlying periodic measure in a handy way.
Duality results in the homogenization of two-dimensional high-contrast conductivities
Marc Briane and  David Manceau
2008, 3(3): 509-522 doi: 10.3934/nhm.2008.3.509 +[Abstract](39) +[PDF](214.7KB)
Abstract:
The paper deals with some extensions of the Keller-Dykhneduality relations arising in the classical homogenization of two-dimensional uniformly bounded conductivities, to the case of high-contrast conductivities. Only assuming a $L^1$-bound on the conductivity we prove that the conductivity and its dual converge respectively, in a suitable sense, to the homogenized conductivity and its dual. In the periodic case a similar duality result is obtained under a less restrictive assumption.
Random homogenization of fractional obstacle problems
Luis Caffarelli and  Antoine Mellet
2008, 3(3): 523-554 doi: 10.3934/nhm.2008.3.523 +[Abstract](50) +[PDF](315.5KB)
Abstract:
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.
Direct integral decomposition for periodic function spaces and application to Bloch waves
Carlos Conca , Luis Friz and  Jaime H. Ortega
2008, 3(3): 555-566 doi: 10.3934/nhm.2008.3.555 +[Abstract](47) +[PDF](193.3KB)
Abstract:
In this paper, we study a direct integral decomposition for the spaces $L^2(O)$ and $H^1(O)$ based on $(\xi,Y^*)-$periodic functions. Using this decomposition we can write the Green's operator (associated to the classical Stokes system in fluid mechanics) in terms of a family of self-adjoint compact operators which depend on the parameter $\xi$. As a consequence, we obtain the so-called Bloch waves associated to the Stokes system in the case of a periodic perforated domain.
Globally stable quasistatic evolution in plasticity with softening
G. Dal Maso , Antonio DeSimone , M. G. Mora and  M. Morini
2008, 3(3): 567-614 doi: 10.3934/nhm.2008.3.567 +[Abstract](69) +[PDF](529.5KB)
Abstract:
We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.
Evolution in off-critical diblock copolymer melts
Michael Helmers , Barbara Niethammer and  Xiaofeng Ren
2008, 3(3): 615-632 doi: 10.3934/nhm.2008.3.615 +[Abstract](47) +[PDF](240.8KB)
Abstract:
We study the evolution of diblock copolymer melts in which one component has small volume fraction. In this case one observes phase morphologies which consist of small spheres of the minority component embedded in the other component. Based on the Ohta-Kawasaki free energy one can set up an evolution equation which has the interpretation of a gradient flow. We restrict this gradient flow to morphologies in which the minority phase consists of spheres and derive monopole approximations for different parameter regimes. We use these approximations for simulations of large particle systems.
On the conjugate of periodic piecewise harmonic functions
Dag Lukkassen , Annette Meidell and  Peter Wall
2008, 3(3): 633-646 doi: 10.3934/nhm.2008.3.633 +[Abstract](49) +[PDF](259.2KB)
Abstract:
The paper considers the conjugate of periodic functions which are piecewise harmonic. In particular, we consider the harmonic conjugate of the solution of the problem of stationary heat conduction through a periodic network of fibres and matrix of arbitrary shape. A numerical example is also presented.
E.Ya.Khruslov. On the occasion of his 70th birthday
Vladimir V. Marchenko , Klavdii V. Maslov , Dmitry Shepelsky and  V. V. Zhikov
2008, 3(3): 647-650 doi: 10.3934/nhm.2008.3.647 +[Abstract](45) +[PDF](61.5KB)
Abstract:
On January 7, 2007 Evgueni Yakovlevich Khruslov, a prominent mathematician, Academician of the National Academy of Science of Ukraine, celebrated his 70th birthday.
Evgueni Khruslov was born in Kharkov, Ukraine. In 1954 he graduated from a high school at a city’s suburb. When studying at the school, he preferred exact sciences, like physics and mathematics. However, at the time of graduation, he could not imagine mathematics to be his future professional occupation, and thus he chose a technical college, the Kharkov Polytechnic Institute, to continue the education.

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Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall
Grigory Panasenko and  Ruxandra Stavre
2008, 3(3): 651-673 doi: 10.3934/nhm.2008.3.651 +[Abstract](34) +[PDF](257.5KB)
Abstract:
In this paper we continue the study of a fluid-structure interaction problem with the non periodic case. We consider the non stationary flow of a viscous fluid in a thin rectangle with an elastic membrane as the upper part of the boundary. The physical problem which corresponds to non homogeneous boundary conditions is stated. By using a boundary layer method, an asymptotic solution is proposed. The properties of the boundary layer functions are established and an error estimate is obtained.
Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain
T. A. Shaposhnikova and  M. N. Zubova
2008, 3(3): 675-689 doi: 10.3934/nhm.2008.3.675 +[Abstract](52) +[PDF](179.3KB)
Abstract:
This paper is aimed at a homogenization problem for a parabolic variational inequality with unilateral constraints. The constraints on solutions are imposed on disk-shaped subsets belonging to the boundary of the domain and forming a periodic structure, so that one has a problem with rapidly oscillating boundary conditions on a part of the boundary. Under certain conditions on the relation between the period of the structure and the radius of the disks, the homogenized problem is obtained. With the help of special auxiliary functions, the solutions of the original variational inequalities are shown to converge to the solution of the homogenized problem in Sobolev space as the period of the structure tends to zero.

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