We consider the homogenization process corresponding to some heterogeneous problems that are given by the Poisson equation with a nonlinear Robin-type boundary conditions on the interior boundary of some small perforations (or the boundary of some small particles) in the so-called critical case, giving rise to the appearance of a strange term in the limit semilinear equation. We prove the strong convergence, in the corresponding Sobolev space, of the solutions with a suitable corrector term. In contrast with other previous results in the literature, we do not assume any additional regularity on the solution of the limit equation: we prove that when the spatial dimension is n = 3or n = 2, then the inherent H-2 regularity is enough to get such a strong convergence. As an application we consider an optimal control problem in which the cost functional involves the gradient of the state solutions, being independent of the nonlinear term arising in the Robin boundary conditions. By working with the corresponding adjoint problem, we show that the limit of the optimal controls is givenas a suitable optimal control associated with the new cost functional in which the strange term and other related terms arise in some unexpected way. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

We study the asymptotic behavior of an optimal boundary control vε and a solution uε to the parabolic state problem specified in a domain Ωε ⊂ ℝ^n , n ⩾ 3, ε-periodically perforated by tiny balls along a part of its boundary as ε → 0. On the boundary of inclusion, we impose the dynamic condition with a large growth coefficient at the time derivative. The critical relation between the period of structure, diameter of balls, and the growth coefficient is considered. This relation leads to the emergence of the nonlocal “strange” term given by some ordinary differential equation, in the effective state equation. We establish the weak convergence of the optimal control to the optimal control associated with the limit cost functional containing the new “strange” term.

We pass to the limit in the homogenization of an optimal control problem associated with a parabolic equation with a dynamic boundary condition. New unexpected terms appear due to the critical scale.

We study the asymptotic behavior of the solution to the diffusion equation in a planar domain, perforated by tiny sets of different shapes with a constant perimeter and a uniformly bounded diameter, when the diameter of a basic cell, ε , goes to 0. This makes the structure of the heterogeneous domain aperiodical. On the boundary of the removed sets (or the exterior to a set of particles, as it arises in chemical engineering), we consider the dynamic unilateral Signorini boundary condition containing a large-growth parameter β (ε ) . We derive and justify the homogenized model when the problem’s parameters take the “critical values”. In that case, the homogenized problem is universal (in the sense that it does not depend on the shape of the perforations or particles) and contains a “strange term” given by a non-linear, non-local in time, monotone operator H that is defined as the solution to an obstacle problem for an ODE operator. The solution of the limit problem can take negative values even if, for any ε , in the original problem, the solution is non-negative on the boundary of the perforations or particles.

We study the homogenization of a nonlinear problem given by the Poisson equation, in a domain with arbitrarily shaped perforations (or particles) and with a dynamic unilateral boundary condition (of Signorini type), with a large coefficient, on the boundary of these perforations (or particles). This problem arises in the study of chemical reactions of zero order. The consideration of a possible asymmetry in the perforations (or particles) is fundamental for considering some applications in nanotechnology, where symmetry conditions are too restrictive. It is important also to consider perforations (or particles) constituted by small different parts and then with several connected components. We are specially concerned with the so-called critical case in which the relation between the coefficient in the boundary condition, the period of the basic structure, and the size of the holes (or particles) leads to the appearance of an unexpected new term in the effective homogenized equation. Because of the dynamic nature of the boundary condition this ``strange term'' becomes now a non-local in time and non-linear operator. We prove a convergence theorem and find several properties of the ``strange operator'' showing that there is a kind of regularization through the homogenization process. For more information see https://ejde.math.txstate.edu/Volumes/2024/03/abstr.html

We obtain the homogenized problem associated with the Poisson equation in a domain perforated by "tiny" balls (or in a domain defined as the exterior to a periodic set of very small particles) of radius a_ε=C_0ε ^γ with γ = n/n-2, C_0>0 (the so-called, "critical case"). On the boundary of these balls, we assume a dynamic unilateral boundary condition (the so-called "Signorini boundary condition"). We prove that the homogenized problem consists of an elliptic equation coupled with an ordinary differential unilateral problem: in contrast with the case of "big perforations" (or "big particles") for which the equation is a unilateral parabolic problem. In particular, we prove that the solution to the homogenized problem may become regionally negative (at least in the interior of some subset of Q^T=Ω× (0,T) on which f ( x , t ) is negative). Nothing similar may happen in the case of big particles since the corresponding homogenized problem imply that the the solution is always non-negative, even for very negative data f ( x , t )

The paper studies the asymptotic behavior of the optimal control for the Poisson type boundary value problem in a domain perforated by holes of an arbitrary shape with Robin-type boundary conditions on the internal boundaries. The cost functional is assumed to be dependent on the gradient of the state and on the usual L2-norm of the control. We consider the so-called “critical” relation between the problem parameters and the period of the structure $$\varepsilon \to 0$$ . Two “strange” terms arise in the limit. The paper extends, by first time in the literature, previous papers devoted to the homogenization of the control problem which always assumed the symmetry of the periodic holes.

We characterize the homogenization limit of the solution of a Poisson equation in a bounded domain, either periodically perforated or containing a set of asymmetric periodical small particles and on the boundaries of these particles a nonlinear dynamic boundary condition holds involving a Holder nonlinear \(\sigma(u)\). We consider the case in which the diameter of the perforations (or the diameter of particles) is critical in terms of the period of the structure. As in many other cases concerning critical size, a "strange" nonlinear term arises in the homogenized equation. For this case of asymmetric critical particles we prove that the effective equation is a semilinear elliptic equation in which the time arises as a parameter and the nonlinear expression is given in terms of a nonlocal operator H which is monotone and Lipschitz continuous on \(L^2(0,T)\), independently of the regularity of \(\sigma\).

The paper studies the homogenization of the optimal control problem for the Dirichlet cost functional with the Poisson state equation specified in a bounded domain $$\Omega $$ . On part of the boundary $$\partial \Omega $$ , denoted by $$(\partial \Omega )^0$$ , rapidly alternating boundary conditions are considered. It is assumed that on subsets of $$(\partial \Omega )^0$$ , distributed with the period $${\varepsilon }$$ , of diameter of order $$O({\varepsilon }^{\frac{n-1}{n-2}})$$ a Robin type boundary condition is specified involving the large coefficient $${\varepsilon }^{-\frac{n-1}{n-2}}$$ . On the rest part of $$(\partial \Omega )^0$$ , we set the Neumann boundary condition. We suppose that parameters take the so-called critical values. The critical case is characterized by the fact that the effective problem contains the “strange” term.

We consider the convergence of solutions and cost functional in some optimal control problems arising in the study of the adsorption chemical phenomenon in which some microscopic reactant particles are placed over an internal manifold γ of the chemical reactor Ω. The chemical reaction is given by some Robin-type boundary condition on the boundary of the periodic set of particles. We consider the special case in which there is a critical relation between the coefficient of the reaction, the size of the particles and the dimension of the space. This gives rise to a “strange term”, which is not occurring for other scales, and thus the limit cost functional must be suitably defined. In a last section, we use this type of technique to prove a similar “energy convergence” result (improving the H01(Ω)-weak convergence) for the problem without control for the critical scale case.

We consider the homogenization of an optimal control problem in which the control is placed on a part of the boundary and the spatial domain contains a thin layer of "small particles", very close to the controlling boundary, and a Robin boundary condition is assumed on the boundary of those "small particles". This problem can be associated with the climatization modeling of Bioclimatic Double Skin Fa\c{c}ades which was developed in modern architecture as a tool for energy optimization. We assume that the size of the particles and the parameters involved in the Robin boundary condition are critical (and so they justify the occurrence of some "strange terms" in the homogenized problem). The cost functional is given by a weighted balance of the distance (in a H^1-type metric) to a prescribed target internal temperature u_{T} and the proper cost of the control (given by its L^2 norm). We prove the (weak) convergence of states u_{{\epsilon}} and of the controls v_{{\epsilon}} to some functions which are completely identified: u_0 satisfies an artificial boundary condition on the control boundary and v_0 is the optimal control associated to a limit cost functional J_0 in which the "boundary strange term" on the control boundary arises. This information on the limit problem makes much more manageable the study of the optimal climatization of such double skin structures.

The behavior of materials at the nanoscale is a key aspect of modern nanoscience and nanotechnology. This book presents rigorous mathematical techniques showing that some very useful phenomenological properties which can be observed at the nanoscale in many nonlinear reaction-diffusion processes can be simulated and justified mathematically by means of homogenization processes when a certain critical scale is used in the corresponding framework.

Асташова И.В., Барабанов Е.А., Боровских А.В., Бутузов В.Ф., Быков В.В., Ветохин А.Н., Глызин С.Д., Горицкий А.Ю., Денисова Н.В., Изобов Н.А., Ильин А.В., Ильяшенко Ю.С., Капустина Т.О., Кигурадзе И.Т., Козлов В.В., Колесов А.Ю., Коньков А.А., Ломов И.С., Моисеев Е.И., Палин В.В. и др.//Дифференциальные уравнения, 2021

We construct and justify an efficient model arising when homogenizing the Poisson equation in an ε -periodically perforated domain along an ( n − 1)-dimensional manifold by sets of an arbitrary shape and critical size on the boundary of which we impose a nonlinear dynamic condition containing an absorption coefficient of the form ε − k , where k takes the critical value ( n − 1)/( n − 2), n ≥ 3. We show that the transmission conditions on the manifold contain a nonlocal monotone operator.

One of the main goals of this paper is to extend some of the mathematical techniques of some previous papers by the authors showing that some very useful phenomenological properties which can be observed at the nano-scale can be simulated and justified mathematically by means of some homogenization processes when a certain critical scale is used in the corresponding framework. Here the motivating problem in consideration is formulated in the context of the reverse osmosis. We consider, on a part of the boundary of a domain Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}}, a set of very small periodically distributed semipermeable membranes having an ideal infinite permeability coefficient (which leads to Signorini-type boundary conditions) on a part Γ1{\Gamma_{1}} of the boundary. We also assume that a possible chemical reaction may take place on the membranes. We obtain the rigorous convergence of the problems to a homogenized problem in which there is a change in the constitutive nonlinearities. Changes of this type are the reason for the big success of the nanocomposite materials. Our proof is carried out for membranes not necessarily of radially symmetric shape. The definition of the associated critical scale depends on the dimension of the space (and it is quite peculiar for the special case of n=2{n=2}). Roughly speaking, our result proves that the consideration of the critical case of the scale leads to a homogenized formulation which is equivalent to having a global semipermeable membrane, at the whole part of the boundary Γ1{\Gamma_{1}}, with a “finite permeability coefficient of this virtual membrane”, which is the best we can get, even if the original problem involves a set of membranes of any arbitrary finite permeability coefficients.

A homogenized model is constructed (with rigorous justification) for a boundary-value problem for the Poisson equation in a periodically perforated domain with a nonlinear Robin condition on the boundary of the cavities. This condition contains a parameter depending on the period of the structure and a function σ(x, u) responsible for the nonlinearity. The cavities can have an arbitrary shape and the parameters of the problem have “critical values,” which results in a homogenized problem with a different type of nonlinearity.

A strange term arising in the homogenization of elliptic (and parabolic) equations with dynamic boundary conditions given on some boundary parts of critical size is considered. A problem with dynamic boundary conditions given on the union of some boundary subsets of critical size arranged ε-periodically along the boundary and with homogeneous Neumann conditions given on the rest of the boundary is studied. It is proved that the homogenized boundary condition is a Robin-type containing a nonlocal term depending on the trace of the solution u(x, t) on the boundary $$\partial \Omega $$ .