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Keywords:
membrane; Laplacian; estimation of frequencies
Summary:
The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2-dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to transform a Riemannian manifold with nonempty boundary $(M, \partial M, g)$ to a compact Riemannian manifold $(M\sharp M, \widetilde g)$ without boundary. An easy numerical investigation on a concrete semi-ellipsoidic membrane with clamped boundary tests the sharpness of the method.
References:
[1] Antman, S. S.: Ordinary differential equations of nonlinear elasticity. I: Foundations of the theories of nonlinearly elastic rods and shells. Arch. Ration. Mech. Anal. 61 (1976), 307-351. DOI 10.1007/BF00250722 | MR 0418580 | Zbl 0354.73046
[2] Carroll, M. M., Naghdi, P. M.: The influence of the reference geometry on the response of elastic shells. Arch. Ration. Mech. Anal. 48 (1972), 302-318. DOI 10.1007/BF00250856 | MR 0356661 | Zbl 0283.73034
[3] Cheng, S.-Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143 (1975), 289-297. DOI 10.1007/BF01214381 | MR 0378001 | Zbl 0329.53035
[4] Ericksen, J. L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1 (1958), 295-323. DOI 10.1007/BF00298012 | MR 0099135 | Zbl 0081.39303
[5] Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Universitext, Springer, Berlin (2004). DOI 10.1007/978-3-642-18855-8 | MR 2088027 | Zbl 1068.53001
[6] Jost, J.: Riemannian Geometry and Geometric Analysis. Universitext, Springer, Berlin (2011). DOI 10.1007/978-3-642-21298-7 | MR 2829653 | Zbl 1227.53001
[7] Marsden, J. E., Hughes, T. J. R.: Mathematical Foundations of Elasticity. Dover Publications, New York (1994). MR 1262126 | Zbl 0545.73031
[8] Naghdi, P. M.: The theory of shells and plates. Handbook der Physik, vol. VIa/2 C. Truesdell Springer, Berlin (1972), 425-640. MR 0763159
[9] Sabatini, L.: Estimations of the topological invariants and of the spectrum of the Laplacian for Riemannian manifolds with boundary. Submitted to Result. Math. (2017). MR 1362899
[10] Simo, J. C., Fox, D. D.: On a stress resultant geometrically exact shell model. I: Formulation and optimal parametrization. Comput. Methods Appl. Mech. Eng. 72 (1989), 267-304. DOI 10.1016/0045-7825(89)90002-9 | MR 0989670 | Zbl 0692.73062
[11] Simo, J. C., Fox, D. D., Rifai, M. S.: On a stress resultant geometrically exact shell model. II: The linear theory; computational aspects. Comput. Methods Appl. Mech. Eng. 73 (1989), 53-92. DOI 10.1016/0045-7825(89)90098-4 | MR 0992737 | Zbl 0724.73138
[12] Yang, P. C., Yau, S.-T.: Eigenvalues of the laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 7 (1980), 55-63. MR 0577325 | Zbl 0446.58017
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