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In this paper, the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain is studied. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary whereas the Dirichlet condition is considered on the smooth separate part. Using the unfolding method, under natural hypothesis on the regularity of the domain, we prove the weak $L^2$-convergence of the zero-extended solutions of the nonlinear problem and their flows to the solutions of a limit distributional problem., 16 pages, 2 figures

In this paper we study the asymptotic behaviour of an exact controllability problem for a second order linear evolution equation defined in a two-component composite with e-periodic disconnected inclusions of size e. On the interface we prescribe a jump of the solution that varies according to a real parameter γ. In particular, we suppose that − 1 γ ≤ 1 . The case γ = 1 is the most interesting and delicate one, since the homogenized problem is represented by a coupled system of a P.D.E. and an O.D.E., giving rise to a memory effect. Our approach to exact controllability consists in applying the Hilbert Uniqueness Method, introduced by J.-L. Lions, which leads us to the construction of the exact control as the solution of a transposed problem. Our main result proves that the exact control and the corresponding solution of the e-problem converge to the exact control of the homogenized problem and to the corresponding solution respectively.

The first aim of this paper is to study, by means of the periodic unfolding method, the homogenization of elliptic problems with source terms converging in a space of functions less regular than the usual \begin{document}$ L^2 $\end{document} , in an \begin{document}$ \varepsilon $\end{document} -periodic two component composite with an imperfect transmission condition on the interface. Then we exploit this result to describe the asymptotic behaviour of the exact controls and the corresponding states of hyperbolic problems set in composites with the same structure and presenting the same condition on the interface. The exact controllability is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads us to the construction of the exact controls as solutions of suitable transposed problem.

In this paper, we study the asymptotic behavior of minimizing solutions of a Ginzburg&ndash, Landau type functional with a positive weight and with convex potential near 0 and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis&ndash, Merle&ndash, Rivi&egrave, re.

In this paper we study the asymptotic behavior of the solutions of time dependent micromagnetism problem in a multi-domain consisting of a thin-wire in junction with a thin film. We assume that the volumes of the two parts composing each multi-structure vanish with same rate. We obtain a 1D limit problem on the thin-wire and a 2D limit problem on the thin film, and the two limit problems are uncoupled. The limit problem remains non-convex, but now it becomes completely local.

We study an optimal control problem for certain evolution equations in two component composites with \begin{document}$\varepsilon$\end{document} -periodic disconnected inclusions of size \begin{document}$\varepsilon$\end{document} in presence of a jump of the solution on the interface that varies according to a parameter \begin{document}$γ$\end{document} . In particular the case \begin{document}$γ=1$\end{document} is examinated.

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter $\epsilon$ denotes the thickness of the domain and the roughness periodicity of the boundary. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when $\epsilon$ tends to zero. We obtain the homogenized limit problem for the velocity and the pressure, which preserves the nonlinear character of the flow, and study the effects of the microstructure in the corresponding effective equations. Finally, we give the interpretation of the limit problem in terms of a non linear Darcy law., Comment: 23 pages, 2 figures

The macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction–diffusion processes in periodic porous media is analyzed. Our mathematical model can be used for studying several metabolic processes taking place in living cells, in which biochemical species can diffuse in the cytosol and react both in the cytosol and also on the organellar membranes. The coupling of the concentrations of the biochemical species is realized via various properly scaled nonlinear reaction terms. These nonlinearities, which model, at the microscopic scale, various volume or surface reaction processes, give rise in the macroscopic model to different effects, such as the appearance of additional source or sink terms or of a non-standard diffusion matrix.

The aim of this paper is to describe the asymptotic behavior, as $\varepsilon\to 0$, of an elliptic problem with rapidly oscillating coefficients in an $\varepsilon$-periodic two component composite with an interfacial contact resistance on the interface, in the case of weakly converging data.

In the paper, an equilibrium problem for an elastic body with a thin elastic and a volume rigid inclusion is analyzed. It is assumed that the thin inclusion conjugates with the rigid inclusion at a given point. Moreover, a delamination of the thin inclusion is assumed. Inequality type boundary conditions are considered at the crack faces to prevent a mutual penetration between the faces. A passage to the limit is justified as the rigidity parameter of the thin inclusion goes to infinity. The main goal of the paper is to analyze an optimal control problem with a cost functional characterizing a deviation of the displacement field from a given function. A rigidity parameter of the thin inclusion serves as a control function. An existence theorem to this problem is proved.

A high resolution coccolithophore study was carried out in order to improve the understanding of the paleoceanographic evolution and changes in paleoproductivity occurring off the Iberian Margin (IM) between 445 and 360 ka, i.e. during late Marine Isotope Stage 12 to 11. Coccolithophore assemblages allowed reconstructing surface water changes characterized by millennial-scale oscillations (~ 1.5 kyr cycles) involving Portugal or Iberian Poleward Currents (PC and IPC) prevalence. Changes in paleoproductivity, possibly related to the upwelling of Eastern North Atlantic Central Waters (ENACW) – of sub-tropical (ENACWst) or sub-polar origin (ENACWsp) – were also recognized. This study also permitted detecting abrupt events (stadial/interstadial-type oscillations) and revealed that changes in paleoproductivity are related to opposite dynamics during glacial and interglacial stages, with the reversed setting being established during the deglaciation. Furthermore, a possible control of half and fourth precessional cycle components, on the occurrence of abrupt changes within the assemblages' structure, during the deglaciation, is proposed on the basis of wavelet analysis results applied to selected taxa.

A high-resolution coccolithophore study allowed reconstructing sea surface conditions during the last 24 ka BP off the Iberian Margin. Variations in nannoplankton accumulation rate, cold-water species, Florisphaera profunda, Umbilicosphaera sibogae and Coccolithus pelagicus ssp. azorinus trends suggest the occurrence of substantial changes in surface water dynamics. Palaeoproductivity shows changes during the glacial and the interglacial transition from intense glacial upwelling to persistent interglacial stratification. A drastic productivity decline occurred between � 18 and 13 ka BP related to the arrival of subpolar waters indicated by the presence of the subarctic subspecies C. pelagicus pelagicus and increased percentages of tetra-unsaturated alkenones. Abrupt variability at core MD03-2699 during the last 24 ka BP was observed in palaeoproductivity and sea surface temperatures proxies. Increases of reworked species fluxes were observed during the Last Glacial Maximum and during the transition to the Holocene, likely connected to turbidity currents and/or contourites formed by Mediterranean outflow intensification. In addition, C. pelagicus ssp. pelagicus presence, Gephyrocapsa muellerae and tetra-alkenone percentages reveal rapid coolings coeval with the Holocene Bond cycles and rapid climate coolings recognized for the Northern Hemisphere between 11.02 and 1.7 ka BP and associated with subpolar water arrival at the site. Copyright # 2013 John Wiley & Sons, Ltd.

We study the homogenization of elliptic equations stated in L2-space with degenerate weight. Both coefficients of the differential operator and the weight are ε-periodic and highly oscillating as ε tends to zero. Under minimal hypotheses on the coefficients and the weight we prove estimates of order ε and ε2 for L2-norm of the difference between the exact solution and its appropriate approximations by L2-norm of the right-side function. The spectral method based on Bloch decomposition is used. In the case of nonunique solution provided that the weight is not regular we consider estimates for any of so-called variational solutions.

Let G be a smooth bounded domain in R 2 . Consider the functional E e ( u ) = 1 2 ∫ G ( p 0 + t | x | k | u | l ) | ∇ u | 2 + 1 4 e 2 ∫ G ( 1 − | u | 2 ) 2 on the set H g 1 ( G , C ) = { u ∈ H 1 ( G , C ) ; u = g on ∂ G } where g is a given boundary data with degree d ≥ 0 . In this paper we will study the behavior of minimizers u e of E e and we will estimate the energy E e ( u e ) .

This paper is concerned with the study of homogenization of an optimal control problem governed by a second-order linear evolution equation with a homogeneous Neumann boundary condition in a domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities, with a fixed height, whose size depends on a small parameter . We identify the limit problem and we remark that both limit state equation and limit cost are different from those ones at ε−level.

In this paper, we study the exact controllability of a second order linear evolution equation in a domain with highly oscillating boundary with homogeneous Neumann boundary condition on the oscillating part of boundary. Our aim is to obtain the exact controllability for the homogenized equation. The limit problem with Neumann condition on the oscillating boundary is different and hence we need to study the exact controllability of this new type of problem. In the process of homogenization, we also study the asymptotic analysis of evolution equation in two setups, namely solution by standard weak formulation and solution by transposition method.

In the paper we consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωε, which is the union of a domain Ω₀ and a large number 2N of thin rods with variable thickness of order ε = O(N⁻¹). The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. We investigate the asymptotic behaviour of the solution as ε → 0 under the Robin conditions on the boundaries of the thin rods. By using some special extension operators, the convergence theorem is proved. Розглядається мiшана крайова задача для рiвняння Пуассона у плоскому дворiвневому з’єднаннi Ωε, яке є об’єднанням деякої областi Ω₀ та великої кiлькостi 2N тонких стержнiв iз змiнною товщиною порядку ε = O(N⁻¹) Тонкi стержнi роздiлено на два рiвнi в залежностi вiд їх довжини. Крiм того, тонкi стержнi з кожного рiвня ε-перiодично чергуються. Вивчено асимптотичну поведiнку розв’язку, коли ε → 0, при крайових умовах Робiна на межах тонких стержнiв. Iз використанням спецiальних операторiв продовження доведено теорему збiжностi.

In this paper, using Pontryagin’s maximum principle, we study the asymptotic behaviour of a parabolic optimal control problem in a domain $$\Omega _{\varepsilon }\subset \mathbf {R}^{n},$$ whose boundary $$\partial \Omega _{\varepsilon }$$ contains a highly oscillating part. On this part we consider a homogeneous Neumann boundary condition. We identify the limit problem, which is an optimal control problem for the limit equation. Moreover, we explicitly remark that both limit state equation and limit cost are different from those ones at $$\varepsilon $$ -level.

In this paper we study the asymptotic behavior of a Ginzburg-Landau problem in a e-periodically perforated domain of with mixed Dirichlet-Neumann conditions. The holes can verify two different situations. In the first one they have size e and a homogeneous Dirichlet condition is posed on a flat portion of each hole, whose size is an order smaller than e, the Neumann condition being posed on the remaining part. In the second situation, we consider two kinds of e-periodic holes, one of size of order smaller than e, where a homogeneous Dirichlet condition is prescribed and the other one of size e, on which a non-homogeneous Neumann condition is given. Moreover, in this case as e goes to zero, the two families of holes approach each other. In both situations a homogeneous Dirichlet condition is also prescribed on the whole exterior boundary of the domain. MSC: 35J20, 35J25, 35B25, 35J55, 35B40.

International audience; We are interested in the homogenization of a stationary Bingham flow in a porous medium. The model and the formal expansion of this problem are introduced in Lions and Sanchez-Palencia (J. Math. Pures Appl. 60:341–360, 1981) and a rigorous justification of the convergence of the homogenization process is given in Bourgeat and Mikelic (J. Math. Pures Appl. 72:405–414, 1993), by using monotonicity methods coupled with the two-scale convergence method. In order to get the homogenized problem, we apply here the unfolding method in homogeniza-tion, method introduced in Cioranescu et al. (SIAM J. Math. Anal.40:1585–1620, 2008).

In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change. (C) 2013 Elsevier Inc. All rights reserved.

It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens., 24 pages, 9 figures

In this paper the asymptotic behaviour of a second-order linear evolution problem is studied in a domain, a part of wich has an oscillating boundary. An homogeneous Neumann condition is given on the whole boundary of the domain. Moreover the behaviour of associated optimal control problem is analyzed.

In this paper we consider the properties of a family of probability (continuous and singular) measures, which will be called Binomial measures because of their relationship with the binomial model in probability. These measures arise in many applications with different notations. Many properties in common with Lebesgue measure hold true for this family, sometimes unexpectedly.