Asymptotic analysis of a boundary optimal control problem on a general branched structure

We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar‐type (non‐smooth) while the literature on smooth oscillating boundary is very few. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems, namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch.     

Periodic unfolding and Robin problems in perforated domains

The periodic unfolding method was introduced in 2002 by D. Cioranescu et al. for the study of classical periodic homogenization. In this Note, we extend this method to perforated domains introducing also a boundary unfolding operator. As an application, we study the homogenization of some elliptic problems with Robin condition on the boundary of the holes. To cite this article: D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Low-cost control problems on perforated and non-perforated domains

We study the homogenization of a class of optimal control problems whose state equations are given by second order elliptic boundary value problems with oscillating coefficients posed on perforated and non-perforated domains. We attempt to describe the limit problem when the cost of the control is also of the same order as that describing the oscillations of the coefficients. We study the situations where the control and the state are both defined over the entire domain or when both are defined on the boundary.     

Periodic unfolding and homogenizationÉclatement périodique et homogénéisation

A novel approach to periodic homogenization is proposed, based on an unfolding method, which leads to a fixed domain problem (without singularly oscillating coefficients). This method is elementary in nature and applies to cases of periodic multi-scale problems in domains with or without holes (including truss-like structures). To cite this article: D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 99–104.     

Asymptotic analysis of Neumann periodic optimal boundary control problem

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.     

Bingham flow in porous media with obstacles of different size

By using the unfolding operators for periodic homogenization, we give a general compactness result for a class of functions defined on bounded domains presenting perforations of two different size. Then we apply this result to the homogenization of the flow of a Bingham fluid in a porous medium with solid obstacles of different size. Next, we give the interpretation of the limit problem in terms of a nonlinear Darcy law. Copyright © 2017 John Wiley & Sons, Ltd.     

振荡Robin混合边值齐次化问题

该文研究了振荡Robin混合边值齐次化问题解的收敛率.该工作的困难之处在于Robin边值上出现的振荡因子以及边界交叉项的处理.该文利用对偶方法巧妙得对振荡积分进行了估计.文中建立了解的H1和L2收敛率,所得结果明显地依赖于维数.该文可以视为将对偶方法和光滑算子,延拓到处理振荡Robin混合边值问题的情形.     

Biharmonic equation in a highly oscillating domain and homogenization of an associated control problem

We consider a Dirichlet boundary control problem posed in an oscillating boundary domain governed by a biharmonic equation. Homogenization of a PDE with a non-homogeneous Dirichlet boundary condition on the oscillating boundary is one of the hardest problems. Here, we study the homogenization of the problem by converting it into an equivalent interior control problem. The convergence of the optimal solution is studied using periodic unfolding operator.     

Σ-convergence of nonlinear monotone operators in perforated domains with holes of small size

This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in ℝ ^N with isolated holes of size ɛ² (ɛ > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent theory of homogenization structures.     

穿孔区域中双曲-抛物方程均匀化问题的一个注记

In this paper, we study a class of hyperbolic-parabolic problems in periodically perforated domains with a homogeneous Neumann condition on the boundary of holes. We focus on the homogenization of these equations, which generalizes those achieved by BensoussanLions-Papanicolau and Migorski. The proof is based on the periodic unfolding method in perforated domains.     

带不完美界面的双曲问题均匀化的一个注记

本文研究了一类二分区域上的具有非周期系数的双曲问题.利用周期Unfolding方法,得到了均匀化及其矫正结果,推广了Donato,Faella和Monsurrò的工作.     

The periodic unfolding method for a class of imperfect transmission problems

The periodic unfolding method was introduced by D. Cioranescu, A. Damlamian and G. Griso for studying the classical periodic homogenization in fixed domains and more recently extended to periodically perforated domains by D. Cioranescu, A. Damlamian, P. Donato, G. Griso, and R. Zaki. Here, the method is adapted to two-component domains which are separated by a periodic interface. The unfolding method is then applied to an elliptic problem with a jump of the solution on the interface, which is proportional to the flux and depends on a real parameter. We prove some homogenization and corrector results, which recover and complete those previously obtained by the first author and S. Monsurr`o. Bibliography: 32 titles. Illustrations: 2 figures.     

A two-scale model for the periodic homogenization of the wave equation

We present a new mathematical object designed to analyze the oscillations occurring on both microscopic and macroscopic scales in a wave equation with oscillating coefficients and data. Through a Bloch wave homogenization method, our study addresses typical problems of two-scale convergence in the interior of the domain, and sheds some light on the behavior near the boundary. A decoupled system of (systems of) transport equations is derived in each energy band, and the total energy field is approximated. We also recover previously known results in homogenization as a restricted part of our model.     

Error estimates on homogenization of free boundary velocities in periodic media

In this paper we consider a free boundary problem which describes contact angle dynamics on inhomogeneous surface. We obtain an estimate on convergence rate of the free boundaries to the homogenization limit in periodic media. The method presented here also applies to more general class of free boundary problems with oscillating boundary velocities.     

Viscosity method for homogenization of parabolic nonlinear equations in perforated domains

In this paper, we develop a viscosity method for homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of the capacity of columns or the capacity of punctured ball is too high or too small, the limit of u? will converge to trivial solutions. The critical decay rates of having nontrivial solution are obtained with the construction of barriers. We also show the limit of u? satisfies a homogenized equation with a term showing the effect of the highly oscillating obstacles or perforated domain in viscosity sense.     

On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

In this paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel periodically oscillating boundary. This domain consists of the body, a large number of thin cylinders joining to the body through the thin transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary of the transmission zone. We prove the homogenization theorems and derive the estimates for the convergence of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

Deterministic homogenization of non-linear non-monotone degenerate elliptic operators

Deterministic homogenization has been till now applied to the study of monotone operators, the determination of the limiting problem being systematically based on the monotonicity of the operator under consideration. Here we mean to show that deterministic homogenization also tackle non-monotone operators. More precisely, under an abstract general hypothesis, we study the homogenization of non-linear non-monotone degenerate elliptic operators. We obtain some general homogenization result, which result is applied to the resolution of several concrete homogenization problems such as the periodic homogenization and the almost periodic homogenization problems. Our main tool is the theory of homogenization structures.     

Homogenization of Boundary Value Problems in Plane Domains with Frequently Alternating Type of Nonlinear Boundary Conditions: Critical Case

In the present paper we consider a boundary homogenization problem for the Poisson’s equation in a bounded domain and with a part of the boundary conditions of highly oscillating type (alternating between homogeneous Neumman condition and a nonlinear Robin type condition involving a small parameter). Our main goal in this paper is to investigate the asymptotic behavior as ε → 0 of the solution to such a problem in the case when the length of the boundary part, on which the Robin condition is specified, and the coefficient, contained in this condition, take so-called critical values. We show that in this case the character of the nonlinearity changes in the limit problem. The boundary homogenization problems were investigate for example in [1, 2, 4]. For the first time the effect of the nonlinearity character change via homogenization was noted for the first time in [5]. In that paper an effective model was constructed for the boundary value problem for the Poisson’s equation in the bounded domain that is perforated by the balls of critical radius, when the space dimension equals to 3. In the last decade a lot of works appeared, e.g., [6–10], in which this effect was studied for different geometries of perforated domains and for different differential operators. We note that in [6–10] only perforations by balls were considered. In papers [11, 12] the case of domains perforated by an arbitrary shape sets in the critical case was studied.     

Homogenization of low-cost control problems on perforated domains

The aim of this paper is to study the asymptotic behaviour of some low-cost control problems in periodically perforated domains with Neumann condition on the boundary of the holes. The optimal control problems considered here are governed by a second order elliptic boundary value problem with oscillating coefficients. It is assumed that the cost of the control is of the same order as that describing the oscillations of the coefficients. The asymptotic analysis of small cost problem is more delicate and need the H-convergence result for weak data. In this connection, an H-convergence result for weak data under some hypotheses is also proved.     

Asymptotic behaviour of linear Dirichlet parabolic problems with variable operators depending on time in varying domains

We study the asymptotic behaviour of the solutions of linear parabolic Dirichlet problems when the coefficients and the domains where the problems are posed vary simultaneously. In the limit problem it appear the H-limit of the operators, and as it is usual in the homogenization of Dirichlet problems, a new term of order zero. We also obtain a corrector result.