A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain coinciding with two thin rectangles connected through a joint of diameter . A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the small parameter . Energetic and uniform pointwise estimates for the difference between the solution of the starting problem () and the solution of the corresponding limit problem () are proved, from which the influence of the geometric irregularity of the joint is observed.
N.S.Bakhvalov and G.P.Panasenko, Homogenization: Averaging Processes in Periodic Media, Nauka, Moscow, 1984(in Russian); English translation: Kluwer, Dordrecht/Boston/London, 1989.
2.
D.Blanchard and A.Gaudiello, Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem, ESAIM Control. Optim. Calc. Var.9 (2003), 449–460.
3.
D.Blanchard, A.Gaudiello and T.A.Mel’nyk, Boundary homogenization and reduction of dimension in a Kirchhoff–Love plate, SIAM J. Math. Anal.39 (2008), 1764–1787.
4.
A.O.Borisyuk, Experimental study of wall pressure fluctuations in rigid and elastic pipes behind an axisymmetric narrowing, Journal of Fluids and Structures26 (2010), 658–674.
5.
G.Cardone, A.Corbo-Esposito and G.Panasenko, Asymptotic partial decomposition for diffusion with sorption in thin structures, Nonlinear Analysis65 (2006), 79–106.
6.
G.A.Chechkin, V.V.Jikov, D.Lukkassen and A.L.Piatnitski, On homogenization of networks and junctions, Asymptotic Analysis30 (2002), 61–80.
7.
G.A.Chechkin and T.A.Mel’nyk, Spatial-skin effect for eigenvibrations of a thick cascade junction with “heavy” concentrated masses, Mathematical Methods in Applied Sciences37 (2014), 56–74.
8.
D.Cioranescu, J.Saint and J.Paulin, Homogenization of Reticulated Structures, Appl. Math. Sci., Vol. 139, Springer-Verlag, New York, 1999.
9.
U.De Maio, T.Durante and T.A.Mel’nyk, Asymptotic approximation for the solution to the Robin problem in a thick multi-level junction, Mathematical Models and Methods in Applied Sciences15 (2005), 1897–1921.
10.
A.Gaudiello and A.G.Kolpakov, Influence of non degenerated joint on the global and local behavior of joined rods, International Journal of Engineering Science49 (2010), 295–309.
11.
A.Gaudiello and E.Zappale, Junction in a thin multidomain for a fourth order problem, Mathematical Models and Methods in Applied Sciences16 (2006), 1887–1918.
12.
A.M.Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs, Vol. 102, American Mathematical Society, Providence, RI, 1992.
13.
A.V.Klevtsovskiy and T.A.Mel’nyk, Asymptotic expansions of the solution of an elliptic boundary-value problem for a thin cascade domain, Nonlinear Oscillations16 (2013), 214–237, English translation: J. Math. Sci.198 (2014), 303–327.
14.
T.A.Mel’nyk, Homogenization of the Poisson equation in a thick periodic junction, Zeitschrift für Analysis und ihre Anwendungen18 (1999), 953–975.
15.
T.A.Mel’nyk, Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure, J. Math. Anal. Appl.424 (2015), 1237–1260.
16.
T.A.Mel’nyk and S.A.Nazarov, Asymptotic structure of the spectrum of the Neumann problem in a thin comb-like domain, C. R. Acad. Sci., Paris319 (1994), 1343–1348.
17.
T.A.Mel’nyk and A.V.Popov, Asymptotic analysis of boundary-value and spectral problems in thin perforated regions with rapidly changing thickness and different limiting dimensions, Matem. Sbornik203 (2012), 97–124, English translation: Sbornik: Mathematics203 (2012), 1169–1195.
18.
S.A.Nazarov, Junctions of singularly degenerating domains with different limit dimensions, J. Math. Sci.80 (1996), 1989–2034.
19.
S.A.Nazarov, Asymptotic Theory of Thin Plates and Rods. Dimension Reduction and Integral Bounds, Nauchnaya Kniga, Novosibirsk, 2002(in Russian).
20.
S.A.Nazarov and B.A.Plamenevskii, Asymptotics of the spectrum of the Neumann problem in a singularly degenerate thin domains. I, Algebra i Analiz2 (1990), 85–111.
21.
S.A.Nazarov and A.S.Slutskii, Arbitrary plane systems of anisotropic beams, Tr. Mat. Inst. Steklov.236 (2002), 234–261.
22.
S.A.Nazarov and A.S.Slutskii, Asymptotic analysis of an arbitrary spatial system of thin rods, Proceedings of the St. Petersburg Mathematically Society10 (2004), 59–109.
23.
G.P.Panasenko, Asymptotic analysis of bar systems, Int. Russian Journal of Math. Phys.2 (1994), 325–352.
24.
G.P.Panasenko, Method of asymptotic partial decomposition of domain, Mathematical Models and Methods in Applied Sciences8 (1998), 139–156.
25.
G.P.Panasenko, Multi-Scale Modelling for Structures and Composites, Springer, Dordrecht, 2005.
26.
G.Panasenko, Method of asymptotic partial domain decomposition for non-steady problems: Wave equation on a thin structure, in: Analytic Methods in Interdisciplinary Applications, V.V.Mityushev and M.Ruzhansky, eds, Springer Proceedings in Mathematics & Statistics, Vol. 116, Springer, Cham, 2015.
27.
G.Panasenko and K.Pileckas, Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. I. The case without boundary-layer in time, Nonlinear Analysis122 (2015), 125–168.
28.
G.Panasenko and K.Pileckas, Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. II. General case, Nonlinear Analysis125 (2015), 582–607.
29.
S.E.Pastukhova, Homogenization of non-linear elasticity problems for thin periodic structures, Doklady RAN383 (2002), 596–600.
30.
V.V.Zhikov, Homogenization of elasticity problems on singular structures, Izv. Math.66 (2002), 299–365.
31.
V.V.Zhikov and S.E.Pastukhova, Homogenization of elasticity problems on periodic grids of critical thickness, Matem. Sbornik194 (2003), 61–96.
