# Scalar boundary value problems on junctions of thin rods and plates

• Volume: 48, Issue: 5, page 1495-1528
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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Bunoiu, R., Cardone, G., and Nazarov, S. A.. "Scalar boundary value problems on junctions of thin rods and plates." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.5 (2014): 1495-1528. <http://eudml.org/doc/273234>.

@article{Bunoiu2014,
abstract = {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.},
author = {Bunoiu, R., Cardone, G., Nazarov, S. A.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {junction of thin plate and rods; asymptotic analysis; dimension reduction; boundary layers; error estimates; anisotropic weighted Sobolev norms},
language = {eng},
number = {5},
pages = {1495-1528},
publisher = {EDP-Sciences},
title = {Scalar boundary value problems on junctions of thin rods and plates},
url = {http://eudml.org/doc/273234},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Bunoiu, R.
AU - Cardone, G.
AU - Nazarov, S. A.
TI - Scalar boundary value problems on junctions of thin rods and plates
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 5
SP - 1495
EP - 1528
AB - 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.
LA - eng
KW - junction of thin plate and rods; asymptotic analysis; dimension reduction; boundary layers; error estimates; anisotropic weighted Sobolev norms
UR - http://eudml.org/doc/273234
ER -

## References

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1. [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. Zbl0329.35020MR418653
2. [2] J. Beale, Thomas Scattering frequencies of reasonators. Commun. Pure Appl. Math.26 (1973) 549–563. Zbl0254.35094MR352730
3. [3] L. Berlyand, G. Cardone, Y. Gorb and G.P. Panasenko, Asymptotic analysis of an array of closely spaced absolutely conductive inclusions. Netw. Heterog. Media1 (2006) 353–377. Zbl1116.35310MR2247782
4. [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. Zbl1116.74038MR2334771
5. [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. Zbl1127.74025MR2348767
6. [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. Zbl1173.35332MR2376221
7. [7] D. Blanchard and G. Griso, Asymptotic behavior of a structure made by a plate and a straight rod. Chin. Annal. Math. Ser. B34 (2013) 399–434. Zbl1330.74127MR3048668
8. [8] D. Borisov, R. Bunoiu and G. Cardone, On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition. Annal. Henri Poincaré11 (2010) 1591–1627. Zbl1210.82077MR2769705
9. [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. Zbl1290.81038MR2838974
10. [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. Zbl1211.35098MR2755696
11. [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. Zbl1282.35033MR3068832
12. [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. Zbl1178.81088MR2534513
13. [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. Zbl1223.35248MR2826984
14. [14] D. Borisov and G. Cardone, Planar Waveguide with “Twisted” Boundary Conditions: Small Width. J. Math. Phys.53 (2012) 023–503. Zbl1274.81108MR2920471
15. [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. Zbl1286.35025MR3105925
16. [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. Zbl1103.35029MR2226260
17. [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. Zbl1156.35312MR2322834
18. [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. Zbl06245381MR3050619
19. [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. Zbl1200.35021MR2647026
20. [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. Zbl1264.74189MR2803480
21. [21] P.G. Ciarlet, Mathematical elasticity. Vol. II. Theory of plates. Studies Math. Appl. 27 (1997). Zbl0953.74004MR1477663
22. [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. Zbl0791.73010MR1265122
23. [23] D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Appl. Math. Sci. 136 (1999). Zbl0929.35002MR1676922
24. [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. Zbl1075.35023MR2136257
25. [25] A. Gaudiello, R. Monneau, J. Mossino, F. Murat and A. Sili, Junction of elastic plates and beams. ESAIM: COCV 13 (2007) 419–457. Zbl1133.35322MR2329170
26. [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. Zbl1226.49044MR2354696
27. [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. A141 (2011) 739–754. Zbl1220.35173MR2819710
28. [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. Zbl0767.73034MR1204630
29. [29] I. Gruais, Modeling of the junction between a plate and a rod in nonlinear elasticity. Asymptot. Anal.7 (1993) 179–194. Zbl0788.73040MR1226973
30. [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. Zbl0381.35028MR407439
31. [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). Zbl0754.34002MR1007834
32. [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. Zbl1132.35348MR2405145
33. [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. Zbl0194.13405MR226187
34. [34] V. Kozlov, V. Maz’ya and A. Movchan, Asymptotic analysis of fields in multi-structures. Oxford Math. Monogr. Oxford University Press (1999). Zbl0951.35004MR1860617
35. [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. Zbl0812.35019MR1288812
36. [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. Zbl0846.73009MR1388837
37. [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. Zbl0988.74014MR1832468
38. [38] O.A. Ladyzhenskaya, The boundary value problems of mathematical physics. Moscow, Nauka (1973); Appl. Math. Sci., vol. 49. Springer-Verlag, New York (1985). MR793735
39. [39] N.S. Landkof, Foundations of modern potential theory. Die Grundlehren der mathematischen Wissenschaften, vol. 180. Springer-Verlag, New York-Heidelberg (1972). Zbl0253.31001MR350027
40. [40] H. Le Dret, Problèmes variationnels dans le multi-domaines: modélisation des jonctions et applications. Res. Appl. Math., vol. 19. Masson, Paris (1991). Zbl0744.73027MR1130395
41. [41] D. Leguillon and E. Sanchez-Palencia, Approximation of a two-dimensional problem of junction. Comput. Mech.6 (1990) 435–455. Zbl0736.73042
42. [42] J.L. Lions, Magenes E., Non-homogeneous boundary value problems and applications. Springer-Verlag, New York-Heidelberg (1972). Zbl0223.35039
43. [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. Zbl0829.73052MR1333206
44. [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). Zbl1127.35301
45. [45] S.A. Nazarov, Asymptotic Theory of Thin Plates and Rods. Dimension Reduction and Integral Estimates, vol. 1. Nauchnaya Kniga, Novosibirsk (2001).
46. [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. Zbl0683.35033MR971695
47. [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. Zbl0807.35031MR1220589
48. [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. Zbl0862.35027MR1425043
49. [49] S.A. Nazarov, Korn’s inequalities for junctions of bodies and thin rods. Math. Meth. Appl. Sci.20 (1997) 219–243. Zbl0880.35040MR1430494
50. [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). Zbl0968.35038MR1736907
51. [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. Zbl0944.74011MR1702684
52. [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. Zbl0976.35013MR1846015
53. [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. Zbl1145.74390MR2360841
54. [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. Zbl1167.35343MR2108526
55. [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. Zbl1127.35328MR2117508
56. [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. Zbl1155.74027MR2406182
57. [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. Zbl06105679MR2975940
58. [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). Zbl0806.35001MR1283387
59. [59] G.P. Panasenko, Multi-scale Modeling for Structures and Composites. Springer, Dordrecht (2005). Zbl1078.74002MR2133084
60. [60] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annal. Math. Studies, vol. 27, Princeton University Press, Princeton (1951). Zbl0044.38301
61. [61] J. Sanchez-Hubert, Sanchez-Palencia E., Coques élastiques minces. Propriétés asymptotiques. Recherches en Mathématiques Appliquées. Paris, Masson (1997). Zbl0881.73001
62. [62] V.I. Smirnov, A course of higher mathematics. Advanced calculus, vol. II. Sneddon Pergamon Press, London (1964). Zbl0121.25904
63. [63] V.I. Smirnov, A course of higher mathematics. Integral equations and partial differential equations, vol. IV. Sneddon Pergamon Press, London (1964). Zbl0121.25904
64. [64] M. Van Dyke, Perturbation methods in fluid mechanics. Appl. Math. Mech., vol. 8 Academic Press, New York, London (1964). Zbl0329.76002MR176702
65. [65] V.S. Vladimirov, Generalized Functions in Mathematical Physics, Mir Moscow (1979). Zbl0515.46034MR564116

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