Scalar boundary value problems on junctions of thin rods and plates

Bunoiu, R.; Cardone, G.; Nazarov, S. A.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1495-1528.

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative thickness, appears in the differential equation, too, while the asymptotic structures crucially depend on the contrastness ratio. Asymptotic error estimates are derived in anisotropic weighted Sobolev norms.

[1] A.A. Arsen'Ev, The existence of resonance poles and resonances under scattering in the case of boundary conditions of the second and third kind.Ž. Vyčisl. Mat. i Mat. Fiz. 16 (1976) 718-724. | MR | Zbl

[2] J. Beale, Thomas Scattering frequencies of reasonators. Commun. Pure Appl. Math. 26 (1973) 549-563. | MR | Zbl

[3] L. Berlyand, G. Cardone, Y. Gorb and G.P. Panasenko, Asymptotic analysis of an array of closely spaced absolutely conductive inclusions. Netw. Heterog. Media 1 (2006) 353-377. | MR | Zbl

[4] D. Blanchard, A. Gaudiello and G. Griso, Junction of a periodic family of elastic rods with a 3d plate. I. J. Math. Pures Appl. 88 (2007) 1-33. | MR | Zbl

[5] D. Blanchard, A. Gaudiello and G. Griso, Junction of a periodic family of elastic rods with a thin plate. II. J. Math. Pures Appl. 88 (2007) 149-190. | MR | Zbl

[6] D. Blanchard and G. Griso, Microscopic effects in the homogenization of the junction of rods and a thin plate. Asymptot. Anal. 56 (2008) 1-36. | MR | Zbl

[7] D. Blanchard and G. Griso, Asymptotic behavior of a structure made by a plate and a straight rod. Chin. Annal. Math. Ser. B 34 (2013) 399-434. | MR

[8] D. Borisov, R. Bunoiu and G. Cardone, On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition. Annal. Henri Poincaré 11 (2010) 1591-1627. | MR | Zbl

[9] D. Borisov, R. Bunoiu and G. Cardone, Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows. J. Math. Sci. 176 (2011) 774-785. | MR | Zbl

[10] D. Borisov and R. Bunoiu, Cardone G., On a waveguide with an infinite number of small windows. C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011) 53-56. | MR | Zbl

[11] D. Borisov, R. Bunoiu and G. Cardone, Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics. Z. Angew. Math. Phys. 64 (2013) 439-472. | MR | Zbl

[12] D. Borisov and G. Cardone, Homogenization of the planar waveguide with frequently alternating boundary conditions. J. Phys. A: Math. Theor. 42 (2009) 365-205. | MR | Zbl

[13] D. Borisov and G. Cardone, Complete asymptotic expansions for the eigenvalues of the Dirichlet Laplacian in thin three-dimensional rods. ESAIM: COCV 17 (2011) 887-908. | Numdam | MR | Zbl

[14] D. Borisov and G. Cardone, Planar Waveguide with “Twisted” Boundary Conditions: Small Width. J. Math. Phys. 53 (2012) 023-503. | MR | Zbl

[15] D. Borisov, G. Cardone, L. Faella and C. Perugia, Uniform resolvent convergence for strip with fast oscillating boundary. J. Differ. Eqs. 255 (2013) 4378-4402. | MR | Zbl

[16] G. Cardone, A. Corbo Esposito and G.P. Panasenko, Asymptotic partial decomposition for diffusion with sorption in thin structures. Nonlinear Anal. 65 (2006) 79-106. | MR | Zbl

[17] G. Cardone, A. Corbo Esposito and S.E. Pastukhova, Homogenization of a scalar problem for a combined structure with singular or thin reinforcement. Z. Anal. Anwend. 26 (2007) 277-301. | MR | Zbl

[18] G. Cardone, R. Fares and G.P. Panasenko, Asymptotic expansion of the solution of the steady Stokes equation with variable viscosity in a two-dimensional tube structure. J. Math. Phys. 53 (2012) 103-702. | MR

[19] G. Cardone, G.P. Panasenko and Y. Sirakov, Asymptotic analysis and numerical modeling of mass transport in tubular structures. Math. Models Methods Appl. Sci. 20 (2010) 397-421. | MR | Zbl

[20] G. Cardone, S.A. Nazarov and A.L. Piatnitski, On the rate of convergence for perforated plates with a small interior Dirichlet zone. Z. Angew. Math. Phys. 62 (2011) 439-468. | MR | Zbl

[21] P.G. Ciarlet, Mathematical elasticity. Vol. II. Theory of plates. Studies Math. Appl. 27 (1997). | MR | Zbl

[22] D. Cioranescu, O.A. Oleĭnik and G. Tronel, Korn's inequalities for frame type structures and junctions with sharp estimates for the constants. Asymptot. Anal. 8 (1994) 1-14. | MR | Zbl

[23] D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Appl. Math. Sci. 136 (1999). | MR | Zbl

[24] R.R. Gadyl'Shin, On the eigenvalues of a dumbbell with a thin handle. Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005) 45-110; Izv. Math. 69 (2005) 265-329. | MR | Zbl

[25] A. Gaudiello, R. Monneau, J. Mossino, F. Murat and A. Sili, Junction of elastic plates and beams. ESAIM: COCV 13 (2007) 419-457. | Numdam | MR | Zbl

[26] A. Gaudiello and A. Sili, Asymptotic analysis of the eigenvalues of a Laplacian problem in a thin multidomain. Indiana Univ. Math. J. 56 (2007) 1675-1710. | MR | Zbl

[27] A. Gaudiello and A. Sili, Asymptotic analysis of the eigenvalues of an elliptic problem in an anisotropic thin multidomain. Proc. Roy. Soc. Edinburgh Sect. A 141 (2011) 739-754. | MR | Zbl

[28] I. Gruais, Modélisation de la jonction entre une plaque et une poutre en élasticité linéarisée. RAIRO Modél. Math. Anal. Numér. 27 (1993) 77-105. | Numdam | MR | Zbl

[29] I. Gruais, Modeling of the junction between a plate and a rod in nonlinear elasticity. Asymptot. Anal. 7 (1993) 179-194. | MR | Zbl

[30] A.M. Il'In, A boundary value problem for an elliptic equation of second order in a domain with a narrow slit. I. The two-dimensional case. Mat. Sb. 99 (1976) 514-537. | MR | Zbl

[31] Il'In A.M., Matching of asymptotic expansions of solutions of boundary value problems. Moscow, Nauka (1989); Translations: Math. Monogr., vol. 102. AMS, Providence (1992). | MR | Zbl

[32] P. Joly and S. Tordeux. Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates. ESAIM: M2AN 42 (2008) 193-221. | Numdam | MR | Zbl

[33] V.A. Kondratiev, Boundary problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obshch. 16 (1967) 209−292; Trans. Moscow Math. Soc. 16 (1967) 227−313. | MR | Zbl

[34] V. Kozlov, V. Maz'Ya and A. Movchan, Asymptotic analysis of fields in multi-structures. Oxford Math. Monogr. Oxford University Press (1999). | MR | Zbl

[35] V.A. Kozlov, V.G. Maz'Ya and A.B. Movchan, Asymptotic analysis of a mixed boundary value problem in a multi-structure. Asymptot. Anal. 8 (1994) 105-143. | MR | Zbl

[36] V.A. Kozlov, V.G. Maz'Ya and A.B. Movchan, Asymptotic representation of elastic fields in a multi-structure. Asymptot. Anal. 11 (1995) 343-415. | MR | Zbl

[37] V.A. Kozlov, V.G. Maz'Ya and A.B. Movchan, Fields in non-degenerate 1D-3D elastic multi-structures. Quart. J. Mech. Appl. Math. 54 (2001) 177-212. | MR | Zbl

[38] O.A. Ladyzhenskaya, The boundary value problems of mathematical physics. Moscow, Nauka (1973); Appl. Math. Sci., vol. 49. Springer-Verlag, New York (1985). | MR

[39] N.S. Landkof, Foundations of modern potential theory. Die Grundlehren der mathematischen Wissenschaften, vol. 180. Springer-Verlag, New York-Heidelberg (1972). | MR | Zbl

[40] H. Le Dret, Problèmes variationnels dans le multi-domaines: modélisation des jonctions et applications. Res. Appl. Math., vol. 19. Masson, Paris (1991). | MR | Zbl

[41] D. Leguillon and E. Sanchez-Palencia, Approximation of a two-dimensional problem of junction. Comput. Mech. 6 (1990) 435-455. | Zbl

[42] J.L. Lions, Magenes E., Non-homogeneous boundary value problems and applications. Springer-Verlag, New York-Heidelberg (1972). | Zbl

[43] J.-L. Lions, Some more remarks on boundary value problems and junctions. Proc. of Asymptotic methods for elastic structures, Lisbon 1993. De Gruyter, Berlin (1995) 103-118. | MR | Zbl

[44] V.G. Maz'Ya, S.A. Nazarov and B.A. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Tbilisi Univ. 1981; Operator Theory. Adv. Appl., vol. 112. Birkhäuser, Basel (2000). | Zbl

[45] S.A. Nazarov, Asymptotic Theory of Thin Plates and Rods. Dimension Reduction and Integral Estimates, vol. 1. Nauchnaya Kniga, Novosibirsk (2001).

[46] S.A. Nazarov, Selfadjoint extensions of the operator of the Dirichlet problem in weighted function spaces. Mat. Sb. 137 (1988) 224-241; Math. USSR-Sb. 65 (1990) 229-247. | MR | Zbl

[47] S.A. Nazarov, Asymptotic behavior of the solution of a boundary value problem in a thin cylinder with a nonsmooth lateral surface. Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993) 202-239; Russian Acad. Sci. Izv. Math. 42 (1994) 183-217. | MR | Zbl

[48] S.A. Nazarov, Junctions of singularly degenerating domains with different limit dimensions. I. Tr. Semin. im. I. G. Petrovskogo 18 (1995) 3-78; J. Math. Sci. 80 (1996) 1989-2034. | MR | Zbl

[49] S.A. Nazarov, Korn's inequalities for junctions of bodies and thin rods. Math. Meth. Appl. Sci. 20 (1997) 219-243. | MR | Zbl

[50] S.A. Nazarov, Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions. Proc. St. Petersburg Math. Society, V, 77-125; Amer. Math. Soc. Transl. Ser. 2, 193, Amer. Math. Soc., Providence (1999). | MR | Zbl

[51] S.A. Nazarov, Asymptotic expansions at infinity of solutions of a problem in the theory of elasticity in a layer. Tr. Mosk. Mat. Obs. 60 (1999) 3-97; Trans. Moscow Math. Soc. (1999) 1-85. | MR | Zbl

[52] S.A. Nazarov, Junctions of singularly degenerating domains with different limit dimensions. II. Tr. Semin. im. I. G. Petrovskogo 20 (2000) 155-195; 312-313; J. Math. Sci. 97 (1999) 155-195. | MR | Zbl

[53] S.A. Nazarov, Asymptotic analysis and modeling of the junction of a massive body and thin rods. Tr. Semin. im. I. G. Petrovskogo 24 (2004) 95-214, 342-343; J. Math. Sci. 127 (2005) 2192-2262. | MR | Zbl

[54] S.A. Nazarov, Estimates for the accuracy of modeling boundary value problems on the junction of domains with different limit dimensions. Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004) 119-156; Izv. Math. 68 (2004) 1179-1215. | MR | Zbl

[55] S.A. Nazarov, Elliptic boundary value problems on hybrid domains. Funktsional. Anal. i Prilozhen 38 (2004) 55-72; Funct. Anal. Appl. 38 (2004) 283-297. | MR | Zbl

[56] S.A. Nazarov, Korn's inequalities for elastic joints of massive bodies, thin plates, and rods. Uspekhi Mat. Nauk 63 (2008) 379, 37-110; Russian Math. Surveys 63 (2008) 35-107. | MR | Zbl

[57] S.A. Nazarov, Asymptotic behavior of the solutions of the spectral problem of the theory of elasticity for a three-dimensional body with a thin coupler. Sibirsk. Mat. Zh. 53 (2012) 345-364; Sib. Math. J. 53 (2012) 274-290. | MR

[58] S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries. Moscow: Nauka. (1991); de Gruyter Expositions Math., vol. 13. Walter de Gruyter & Co., Berlin (1994). | MR | Zbl

[59] G.P. Panasenko, Multi-scale Modeling for Structures and Composites. Springer, Dordrecht (2005). | MR

[60] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annal. Math. Studies, vol. 27, Princeton University Press, Princeton (1951). | Zbl

[61] J. Sanchez-Hubert, Sanchez-Palencia E., Coques élastiques minces. Propriétés asymptotiques. Recherches en Mathématiques Appliquées. Paris, Masson (1997). | Zbl

[62] V.I. Smirnov, A course of higher mathematics. Advanced calculus, vol. II. Sneddon Pergamon Press, London (1964). | Zbl

[63] V.I. Smirnov, A course of higher mathematics. Integral equations and partial differential equations, vol. IV. Sneddon Pergamon Press, London (1964). | Zbl

[64] M. Van Dyke, Perturbation methods in fluid mechanics. Appl. Math. Mech., vol. 8 Academic Press, New York, London (1964). | MR | Zbl

[65] V.S. Vladimirov, Generalized Functions in Mathematical Physics, Mir Moscow (1979). | MR | Zbl