Homogenization of a Boundary-Value Problem with Varying type of Boundary Conditions in a Thick Two-Level Junction

Mel'nyk, Taras A.; Vashchuk, P. S.

doi:10.1007/s11072-005-0053-3

Cited by 7 publications

(3 citation statements)

References 19 publications

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“…In 90's Gaudiello has studied Neumann problem with non-homogeneous boundary data [25], Kozlov, Maz'ya, and Movchan studied such problems with the help of asymptotic expansion in the name of multi-structures [29], and Nazarov analyzed in the name of singularly degenerating domains [39]. Mel'nyk and his collaborators have contributed many works on this direction using asymptotic expansion method [33,35,38,28,32,22,23,37,27]. All these above works are of pillar type periodic oscillations except a few.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method

Aiyappan¹,

Cardone²,

Perugia³

et al. 2021

Preprint

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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 where as 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.

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Section: Introductionmentioning

confidence: 99%

Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method

Aiyappan¹,

Cardone²,

Perugia³

et al. 2021

Preprint

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“…In , the authors have studied homogenization of PDEs posed on the oscillating boundary domains using Tartar's Oscillating test functions method. In , Mel'nyk and Vashchuk have studied the homogenization of Poisson equation on a thick two level junction with varying boundary conditions. In , he has derived H 1 norm estimates for the homogenized solutions of elliptic and parabolic type PDEs.…”

Section: Introductionmentioning

confidence: 99%

Optimal control problem in a domain with branched structure and homogenization

Aiyappan

Nandakumaran

2016

Math Methods in App Sciences

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We consider a linear parabolic problem in a thick junction domain which is the union of a fixed domain and a collection of periodic branched trees of height of order 1 and small width connected on a part of the boundary. We consider a threebranched structure, but the analysis can be extended to n-branched structures. We use unfolding operator to study the asymptotic behavior of the solution of the problem. In the limit problem, we get a multi-sheeted function in which each sheet is the limit of restriction of the solution to various branches of the domain. Homogenization of an optimal control problem posed on the above setting is also investigated. One of the novelty of the paper is the characterization of the optimal control via the appropriately defined unfolding operators. Finally, we obtain the limit of the optimal control problem. IntroductionIn this article, we consider a parabolic problem in a thick junction domain , > 0, a small parameter, and also the corresponding optimal control problem. Various materials with complex structures including multi-layer thick junctions are widely used in many fields of science. Such structures are usually known as complex structures because of its complexity both in construction and analysis. Other complex structures are perforated domains, composite materials, grid domains, and domains with oscillating boundaries to name a few.As mentioned earlier, constructions with thick junction (also multi-level) are used in many technologies, like microstrip radiator, nano technologies ([1, 2]), biological systems, fractal-type constructions, etc. Studying PDE problems in such complex structures has paramount importance. We refer to the work in [3][4][5][6] and the references therein for the study in multi-branched structures. Although the importance of optimal control may be at the junctions, we consider the controls on the entire oscillating part from which we can also understand the contribution from each branch at each level. One can apply need based controls at the appropriate junctions.The domain under consideration consists of multiple layer thick junctions known as branched structure (Figure 1). Such a domain has a fixed part and lot of thin periodically distributed parts (or handle trees) attached along certain part of the boundary of the domain at different levels. The trees have finite number of branching levels and in this paper, we take three branching levels, but one can consider any finite number of branches. The height of each branch is of O.1/, whereas the thickness is of O. /. We consider the domain in two-dimensional space. Such a domain has already been considered by Mel'nyk ([6]). Indeed the results can be extended to three dimensional problem and higher dimensions as well. Asymptotic analysis for a Robin problem in a thick junction has been investigated in [5]. In fact, our work is motivated from the work of Mel'nyk, where he considers a semi-linear parabolic problem with the source term vanishes on the oscillating interior part. He has studied the problem usin...

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“…In recent years, a rich collection of new results on asymptotic analysis of boundary-value problems in thick multi-structures is appeared (see [1]- [8]). …”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Homogenization of the signorini boundary-value problem in a thick plane junction

Kazmerchuk

Mel'nyk

2009

Nonlinear Oscill

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We consider a mixed boundary-value problem for the Poisson equation in a plane thick junction Ω ε which is the union of a domain Ω 0 and a large number of ε−periodically situated thin rods. The nonuniform Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is made as ε → 0, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove the convergence theorem and show that the nonuniform Signorini conditions are transformed (as ε → 0) in the limiting variational inequalities in the region that is filled up by the thin rods in the limit passage. The existence and uniqueness of the solution to this non-standard limit problem is established. The convergence of the energy integrals is proved as well.

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Homogenization of a Boundary-Value Problem with Varying type of Boundary Conditions in a Thick Two-Level Junction

Cited by 7 publications

References 19 publications

Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method

Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method

Optimal control problem in a domain with branched structure and homogenization

Homogenization of the signorini boundary-value problem in a thick plane junction

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