Taras A. Mel’nyk

• Professor
• Professor (Full) at Taras Shevchenko National University of Kyiv

A convergence theorem and asymptotic estimates (as $\varepsilon \to 0$ ) are proved for a solution to a mixed boundary-value problem for the Poisson equation in a junction $\Omega_{\varepsilon}$ of a domain $\Omega_0$ and a large number $N^2$ of $\varepsilon -$periodically situated thin cylinders with thickness of order $ \varepsilon = O(N^{-1}).$...

Convergence theorems and asymptotic estimates (as ε→0) are proved for eigenvalues and eigenfunctions of a boundary value problem for the Laplace operator in a plane thick periodic junction with concentrated masses. This junction consists of the junction’s body and a large number N=O(ε -1 ) of the fine rods. The density of the junction is of order O...

We consider a boundary-value problem for the Poisson equation in a thick junction Ωε, which is the union of a domain Ω0 and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂νuε + εκ(uε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this pro...

We study the asymptotic behavior (as ε→0) of an optimal control problem in a plane thick two-level junction, which is the union of some domain and a large number2N of thin rods with variable thickness of order e = O(N-1).\varepsilon =\mathcal{O}(N^{-1}). The thin rods are divided into two levels depending on the geometrical characteristics and on...

We investigate the asymptotic behavior, as " tends to 0+, of the transverse dis- placement of a Kirchhofi-Love plate composed of two domains ›+ " ( ›¡ " , contained in the (x1;x2)-coordinate plane and depending on " in the following way. The flrst domain ›¡ " is a thin strip with vanishing height h" (in direction x2), as " tends to 0+. The second o...

The article examines a boundary-value problem in a domain consisting of perforated and imperforate regions, with Neumann conditions prescribed at the boundaries of the perforations. Assuming the porous medium has symmetric, periodic structure with a small period $\varepsilon,$ we analyse the limit behavior of the problem as $\varepsilon \to 0.$ A c...

A reduced-dimensional asymptotic modelling approach is presented for the analysis of two-phase flow in a thin cylinder with aperture of order $\mathcal{O}(\varepsilon),$ where $\varepsilon$ is a small positive parameter. We consider a nonlinear Muskat-Leverett two-phase flow model expressed in terms of a fractional flow formulation and Darcy's law...

The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection–diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The lim...

The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection-diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The lim...

We consider for a small parameter [Formula: see text] a parabolic convection–diffusion problem with Péclet number of order [Formula: see text] in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order [Formula: see text] connected through a domain (node) of diameter [Formula: see text] Inhomogeneous Rob...

The article examines spectral problems in a domain $\Omega_{\varepsilon}$, which is the union of a domain $\Omega_{0}$ and a lot of thin trees that are $\varepsilon$-periodically situated along a manifold on the boundary of $\Omega_{0}$. The trees possess a finite number of branching levels. At the boundaries of branches from each branching level,...

This article completes the study of the influence of the intensity parameter α in the boundary condition ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O ( ε ). Inside of the thin networ...

This concise textbook provides a thorough introduction to the function theory of one complex variable. It presents the fundamental concepts with clarity and rigor, offering concise proofs that avoid lengthy and tedious arguments commonly found in mathematics textbooks. The English version is a substantial extension of the Ukrainian one. Some impor...

The main goal of this chapter is to show that analytic functions can be represented as infinite power series. The key to proving this theorem is the Cauchy integral formula established in the previous section. Here we generalize this formula for derivatives and prove the surprising fact that derivatives of analytic functions can be calculated by in...

In this chapter, we continue the study of power series, but already their generalizations, namely power series containing terms \((z -z_0)^n\) with a negative integer n. These series were introduced by the French mathematician Pierre Laurent (1813–1854) in 1843. Laurent series are a valuable tool for studying the behavior of analytic functions near...

Just as a person’s character is manifested in extreme situations, so the properties of analytic functions are determined by their behavior in isolated singularities. In this chapter, we will illustrate this claim with examples of integral calculations. It turns out that in order to calculate the integral of an analytic function along a curve, it is...

Inspired by the properties of analytic functions proved in the previous sections, in the last section we are ready to explore new, no less amazing properties of such functions. In Sect. 9.1 we show that analyticity is sufficient for a nonconstant function being an open map. This property indicates that the modulus of a non-constant analytic functio...

In this chapter we recall some concepts from basic courses in mathematical analysis of real-valued functions of one and several variables as well as from a course in linear algebra, namely complex numbers and their various forms, arithmetic operations on them (Sect. 1.1), and basic topological notions in the vector space \(\mathbb R^2\) (Sect. 1.5)...

Conformal mappings are of immense importance in various branches of mathematics and in many applications. To solve many problems, one needs to be able to construct a bijective conformal mapping from one domain onto another in the complex plane. In this chapter we study how to construct such bijective conformal mappings. We will consider various ele...

In the previous two chapters, it was shown that analytic complex-valued functions enjoy excellent differentiability properties that their real counterparts do not share. It is well known that differentiation and integration are mutually inverse operations and they are the main concerns of calculus. To continue on, the next logical step is to consid...

Analytic functions have surprised us in previous chapters with their properties. In this chapter we will learn about another interesting property of analytic functions, namely the possibility of continuing an analytic function from the domain in which it is defined to a larger one. An important property of the analytic continuation procedure is tha...

In this chapter and onwards, we study properties of complex-valued functions of a complex variable. It turns out that every complex-valued function is determined by the corresponding vector function from \(\mathbb R^2\) into \(\mathbb R^2\). This fact enables us to obtain some properties of complex-valued functions from the first section. Fundament...

This article completes the study of the influence of the intensity parameter $\alpha$ in the boundary condition $\varepsilon \partial_{\boldsymbol{\nu}_\varepsilon} u_\varepsilon - u_\varepsilon \, \overrightarrow{V_\varepsilon}\boldsymbol{\cdot}\boldsymbol{\nu}_\varepsilon = \varepsilon^{\alpha} \varphi_\varepsilon $ given on the boundary of a thi...

We consider time-dependent convection-diffusion problems with high P\'eclet number of order $\mathcal{O}(\varepsilon^{-1})$ in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order $\mathcal{O}(\varepsilon).$ On the lateral surfaces of the thin cylinders and the b...

We consider time-dependent convection-diffusion problems with high P\'eclet number of order $\mathcal{O}(\varepsilon^{-1})$ in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order $\mathcal{O}(\varepsilon).$ On the lateral surfaces of the thin cylinders and the b...

A spectral problem is considered in a domain Ωε$$ {\Omega}_{\varepsilon } $$ that is the union of a domain Ω0$$ {\Omega}_0 $$ and a lot of thin trees situated ε$$ \varepsilon $$‐periodically along some manifold on the boundary of Ω0.$$ {\Omega}_0. $$ The trees have finite number of branching levels. The perturbed Robin boundary condition ∂νuε+εαiki...

We consider for a small parameter $\varepsilon >0$ a parabolic convection-diffusion problem with P\'eclet number of order $\mathcal{O}(\varepsilon^{-1})$ in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order $\mathcal{O}(\varepsilon)$ connected through a domain (node) of diameter $\mathcal{O}(\varep...

A spectral problem is considered in a domain $\Omega_{\varepsilon}$ that is the union of a domain $\Omega_{0}$ and a lot of thin trees situated $\varepsilon$-periodically along some manifold on the boundary of $\Omega_{0}.$ The trees have finite number of branching levels. The perturbed Robin boundary condition $\partial_{\nu}u^{\varepsilon} + \var...

A steady-state convection-diffusion problem with a small diffusion of order O ( ε ) is considered in a thin three-dimensional graph-like junction consisting of thin cylinders connected through a domain (node) of diameter O ( ε ), where ε is a small parameter. Using multiscale analysis, the asymptotic expansion for the solution is constructed and ju...

A steady-state convection-diffusion problem with a small diffusion of order $\mathcal{O}(\varepsilon)$ is considered in a thin three-dimensional graph-like junction consisting of thin cylinders connected through a domain (node) of diameter $\mathcal{O}(\varepsilon),$ where $\varepsilon$ is a small parameter. Using multiscale analysis, the asymptoti...

A spectral problem is considered in a thin $3D$ graph-like junction that consists of three thin curvilinear cylinders that are joined through a domain (node) of a small diameter. A concentrated mass is located in the node. The asymptotic behaviour of the eigenvalues and eigenfunctions is studied, when the thin junction is shrunk into a graph. Ther...

We analyze the contact Hele-Shaw problem with zero surface tension of a free boundary in a thin domain \(\Omega ^{\varepsilon }(t).\) Under suitable conditions on the given data, the one-valued local classical solvability of the problem for each fixed value of the parameter \(\varepsilon \) is proved. Using the multiscale analysis, we study the asy...

We analyze the contact Hele-Shaw problem with zero surface tension of a free boundary in a thin domain $\Omega^{\varepsilon}(t).$ Under suitable conditions on the given data, the one-valued local classical solvability of the problem for each fixed value of the parameter $\varepsilon$ is proved. Using the multiscale analysis, we study the asymptotic...

The textbook consists of three parts. In the first part the main properties of Sobolev spaces are considered. The second one deals with the solvability of typical boundary-value problems in Sobolev spaces for linear elliptic, hyperbolic, and parabolic second order differential equations. In addition, some constructive methods for solving such probl...

A spectral problem is considered in a thin $3D$ graph-like junction that consists of three thin curvilinear cylinders that are joined through a domain (node) of the diameter $\mathcal{O}(\varepsilon),$ where $\varepsilon$ is a small parameter. A concentrated mass with the density $\varepsilon^{-\alpha}$ $(\alpha \ge 0)$ is located in the node. The...

This paper is devoted to study the asymptotic behavior, as $\varepsilon$ vanishes, of a nonlinear monotone Signorini boundary value problem modelizing chemical activity in an $\varepsilon$-periodic structure of thin cylindrical absorbers, like a comb in 2D a or a brush in 3D. The novelty of this paper is the presence of a perturbed coefficient $\va...

A thin graph-like junction $\Omega_\varepsilon \subset \Bbb R^3$ consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter $\mathcal{O}(\varepsilon).$ Here $\varepsilon$ is a small parameter characterizing the thickness of the thin cylinders and the node. In $\Omega_\varepsilon$ we consider a semilinear para...

We consider the case when the thin discs of a thick multilevel junction can have sharp edges, i.e., their thickness tends to zero polynomially while approaching the edges. Three qualitatively different cases in the asymptotic behavior of the solution to a linear elliptic boundary-value problem are discovered depending on the edge form, namely round...

Approximation techniques are demonstrated for semilinear elliptic and parabolic problems with various alternating Robin boundary conditions. With the help of special junction-layer solutions, whose behavior is determined by the type 3:2:2, and the method of matched asymptotic expansions, approximation functions are constructed for the solutions and...

This book presents asymptotic methods for boundary-value problems (linear and semilinear, elliptic and parabolic) in so-called thick multi-level junctions. These complicated structures appear in a large variety of applications. A concise and readable introduction to the topic, the book provides a full review of the literature as well as a presentat...

The method proposed in Chap. 2 is broadened for semilinear initial-boundary-value problems. Here we show how to apply the Minty–Browder method to homogenize nonlinear Robin conditions that have special intensity factor \(\varepsilon ^\alpha \); where a is a real parameter that significantly impacts the asymptotic behavior of the solutions.

Convergence theorems are proved for solutions to linear elliptic boundary-value problems in a thick multilevel junction of type 3:2:2. In the first problem, various alternating perturbed Robin boundary conditions are considered, and alternating Neumann and Dirichlet boundary conditions on the surfaces of the thin discs from different sets in the se...

A mathematical model, which takes into account new experimental results about diverse roles of macrophages in the atherosclerosis development, is proposed. Using technic of upper and lower solutions, the existence and uniqueness of its positive solution are justified. After the nondimensionalisation, small parameters are found and the multiscale an...

In this paper, we consider a domain Ωε⊂RN, N≥2, with a very rough boundary depending on ε. For instance, if N=3 Ωε has the form of a brush with an ε-periodic distribution of thin cylindrical teeth with fixed height and a small diameter of order ε. In Ωε we consider a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending...

A semi-linear boundary-value problem with nonlinear Robin boundary conditions is considered in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter 𝓞(ε). Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε...

Some existing models of the atherosclerosis development are discussed and a new improved mathematical model, which takes into account new experimental results about diverse roles of macrophages in the atherosclerosis development, is proposed. The existence of its positive solution is obtained. This new model is also analyzed with the help of asympt...

A semilinear parabolic problem is considered in a thin 3‐D star‐shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter The purpose is to study the asymptotic behavior of the solution u ε as ε →0, ie, when the star‐shaped junction is transformed in a graph. In addition, the passage to...

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin $3D$ aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter $\mathcal{O}(\varepsilon).$ A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the parame...

Some existing models of the atherosclerosis development are discussed and a new improved mathematical model, which takes into account new experimental results about diverse roles of macrophages in atherosclerosis, is proposed. Using technic of upper and lower solutions, the existence and uniqueness of its positive solution are justified. After the...

A semi-linear parabolic problem is considered in a thin $3D$ star-shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter $\mathcal{O}(\varepsilon).$ The purpose is to study the asymptotic behavior of the solution $u_\varepsilon$ as $\varepsilon \to 0,$ i.e. when the star-shaped juncti...

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin $3D$ aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter $\mathcal{O}(\varepsilon).$ A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the parame...

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain Omega(epsilon) coinciding with two thin rectangles connected through a joint of diameter O(epsilon). A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the small parameter epsilon -> 0. Energetic and...

We consider a semilinear variation inequality in a thick multi-level junction Ω ε , which is the union of a domain Ω 0 (the junction’s body) and a large number of thin cylinders. The thin cylinders are divided into m classes depending on the geometrical characteristics and the semilinear perturbed boundary conditions of the Signorini type given on...

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied in the paper. Model cascade junction consists of the junction's body and a great number 5N=O(ε−1) of ε-alternating thin rods belonging to two classes. The first class...

We study small-parameter asymptotics of eigenelements of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses. There are five qualitatively different cases in the asymptotic behaviour of eigenvalues and eigenfunctions as the small parameter tends to zero (‘light’, ‘intermediate’, ‘slightly heavy’, ‘...

We consider a semi-linear parabolic problem in a thick junction , which is the union of a domain and a lot of joined thin trees situated ε-periodically along some manifold on the boundary of . The trees have finite number of branching levels. The following nonlinear Robin boundary condition is given on the boundaries of the branches from the i-th b...

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain $\Omega_\varepsilon$ coinciding with two thin rectangles connected through a joint of diameter ${\cal O}(\varepsilon)$. A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the small parameter $\varepsi...

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain $\Omega_\varepsilon$ coinciding with two thin rectangles connected through a joint of diameter ${\cal O}(\varepsilon)$. A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the small parameter $\varepsi...

This contributed volume contains a collection of articles on state-of-the-art developments on the construction of theoretical integral techniques and their application to specific problems in science and engineering. Written by internationally recognized researchers, the chapters in this book are based on talks given at the Thirteenth International...

The text-book is designed for a one-year course in complex analysis as part of the basic curriculum of graduate programs in mathematics and related subjects. The main focus lies on the theory of complex-valued functions of a single complex variable. The text contains basic classical concepts and results of the field that are augmented by numerous i...

We consider a semi-linear parabolic problem in a model plane thick fractal junction Ω ε , which is the union of a domain Ω 0 and a lot of joined thin trees situated ε-periodically along some interval on the boundary of Ω 0 . The trees have finite number of branching levels. The following nonlinear Robin boundary condition ∂ ν v ε + ε αi κ i (v ε) =...

We consider a semi-linear parabolic problem in a model plane thick fractal junction $\Omega_{\varepsilon}$, which is the union of a domain $\Omega_{0}$ and a lot of joined thin trees situated $\varepsilon$-periodically along some interval on the boundary of $\Omega_{0}.$ The trees have finite number of branching levels. The following nonlinear Robi...

We construct and substantiate an asymptotic expansion for the solution of an inhomogeneous Neumann boundary-value problem for the Poisson equation whose right-hand side depends on longitudinal and transverse variables in a thin cascade domain. We find the asymptotic energy and establish uniform pointwise estimates for the difference between the sol...

We study the asymptotic behavior of eigenvalues and eigenfunctions of the Laplacian in a 2D thick cascade junction with heavy concentrated masses. We present two-term asymptotic approximations, as ε→0, for the eigenelements in the case of “slightly heavy”, “moderate heavy”, and “super heavy” concentrated masses. Asymptotics of high-frequency cell-v...

We study the asymptotic behavior (as ε→0) of an optimal control problem in a plane thick two-level junction, which is the union of some domain 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 the geometrical characteristics and on the controls given on their ba...

We consider a boundary-value problem for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ωε that is ε-periodically perforated by small holes. The holes are split into two ε-periodic sets depending on the boundary interaction via their boundary surfaces. Therefore, two different nonlinear boundary co...

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction’s body and a great number 5N = O(\epsilon^{-1}) of \epsilon-alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of...

Asymptotic expansion is constructed and justified for the solution to a nonuniform Neumann boundary-value problem for the Poisson equation with the right-hand side that depends both on longitudinal and transversal variables in a thin cascade domain. Asymptotic energetic and uniform pointwise estimates for the difference between the solution of the...

1. INTRODUCTIONProblems in thick cascade junction domains havebeen addressed relatively recently (see [1, 2]). A newtransmission condition at the interface was obtainedthat is dictated by the behavior of the short rods. It wasfound in [3] that the concentrated masses on the shortrods affect the transmission conditions. For the spectral problem in s...

We consider a linear elliptic boundary value problem in a two-level thick junction of type 3 : 2 : 2 which consists of a cylinder with ε-periodically stringed thin disks. The thin disks are divided into two levels depending on their geometric structure and boundary conditions on their surfaces. The first-level thin disks have variable thickness van...

We combine asymptotic algorithms for solving spectral problems with rapidly oscillating coefficients in thin perforated domains with different limit dimensions. The homogenized theorem is proved. Complete asymptotic expansions for the eigenvalues and eigenfunctions are constructed and justified under certain symmetry conditions for thin perforated...

Boundary value and spectral problems for an elliptic differential equation with rapidly oscillating coefficients in a thin perforated region with rapidly changing thickness are investigated. Descriptions of asymptotic algorithms for solutions of such problems in thin regions with different limiting dimensions are combined. For a mixed inhomogeneous...

The asymptotic behaviour (as \epsilon → 0) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is investigated. This cascade junction consists of the junction's body and great number 5N = (\epsilon^{−1}) of \epsilon-alternating thin rods belonging to two classes...

We investigate the asymptotic behavior of a solution of a quasilinear parabolic boundary-value problem in a two-level thick junction of the type 3:2:2. This junction consists of a cylinder on which thin disks of variable thickness are ε-periodically threaded. The thin disks are divided into two levels, depending on their geometric structure and the...

We consider quasilinear optimal control problems involving a thick two-level junction Ωε which consists of the junction body Ω0 and a large number of thin cylinders with the cross-section of order O(ε 2). The thin cylinders are divided into two levels depending on the geometrical characteristics, the quasilinear boundary conditions and controls giv...

We consider a parabolic Signorini boundary value problem in a thick plane junction Ω ε which is the union of a domain Ω 0 and a large number of ε-periodically situated thin rods. The Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is done as ε→0, i.e., when the number of the thin rods i...

Рассматриваются краевые и спектральные задачи для эллиптического дифференциального уравнения с быстро осциллирующими коэффициентами в тонких перфорированных областях с быстро изменяющейся толщиной. В работе совмещается описание асимптотических алгоритмов для решений таких задач в тонких областях с различными предельными размерностями. Для смешанной...

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 $\varepsilon =\mathcal{O}(N^{-1})$. The thin rods are divided into two levels depending on their length. In addition, the thin rods from each leve...

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a mixed boundary-value problem for the Laplace operator in a plane thick periodic junction with concentrated masses is investigated. This junction consists of the junction's body and a large number N=O(ε⁻¹) of thin rods. The density of the junction is order O(ε-α) on the rods (th...

We consider quasilinear and linear boundary-value problems for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ωε that is ε-periodically perforated by small holes of order 𝒪(ε). The holes are divided into three ε-periodical sets depending on boundary conditions on their surfaces. On the boundaries o...

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ωε that is ε-periodically perforated by small holes of order . The holes are divided into three ε-periodical sets depending on boundary conditions. The homogeneous Dirichlet boundary conditions are imposed for holes of one set, whereas, for holes...

We consider a mixed boundary-value problem for the Poisson equation in a thick junction $\Omega_\epsilon$ which is the union of a domain $\Omega_0$ and a large number of $\epsilon$-periodically situated thin cylinders. The non-uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is...

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences betwe...

In the paper, we deal with the homogenization problem for the Poisson equation in a singularly perturbed three-dimensional junction of a new type. This junction consists of a body and a large number of thin curvilinear cylinders, joining to body through a random transmission zone with rapidly oscillating boundary, periodic in one direction. Inhomog...

We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ωε that is ε-periodically perforated by small holes of size O\mathcal {O}(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σε(u ε) + εκ m (u ε...

Spectral boundary-value problems are considered in a new kind of perturbed domain, namely, thick multi-level junctions. Boundary-value problems in thick one-level junctions (thick junctions) have been intensively investigated recently (see, for instance, [BlGaGr07], [BlGaMe08], [Me08] and, the references there). In [MeNa97]–[Me(3)01], classificatio...

The asymptotic behavior of solutions to boundary value problems for the Poisson equation is studies in a thick two-level junction of type 3:2:2 with alternating boundary conditions. The thick junction consists of a cylinder with ε-periodically stringed thin disks of variable thickness. The disks are divided into two classes depending on their geome...