Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T02:34:32.570Z Has data issue: false hasContentIssue false
  • English
  • Français

First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof

Published online by Cambridge University Press:  14 November 2011

Shari Moskow  and
Michael Vogelius
Shari Moskow
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A.
Michael Vogelius
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

Let λε be a Dirichlet eigenvalue of the ‘periodically, rapidly oscillating’ elliptic operator –∇·(a(x/ε)∇) and let ∇ be a corresponding (simple) eigenvalue of the homogenised operator –∇·(A∇). We characterise the possible limit points of the ratio (λε–λ)/ε as ε→0. Our characterisation is quite explicit when the underlying domain is a (planar) convex, classical polygon with sides of rational or infinite slopes. In particular, in this case it implies that there is often a continuum of such limit points.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 6 , 1997 , pp. 1263 - 1299
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Avellaneda, M. and Lin, F.-H.. Homogenization of elliptic problems with If boundary data. Appl. Math. Optim. 15 (1987), 93–107.CrossRefGoogle Scholar
2Babuska, I., Kellogg, R. B. and Pitkaranta, J.. Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33 (1979), 447–71.CrossRefGoogle Scholar
3Bensoussan, A., Lions, J. L. and Papanicolaou, G.. Boundary layer analysis in homogenization of diffusion equations with Dirichlet conditions in the half space. In Proceedings of the International Symposium on Stochastic Differential Equations, Kyoto 1976, ed. Ito, K. (Wiley, 1978).Google Scholar
4Bensoussan, A., Lions, J. L. and Papanicolaou, G.. Asymptotic Analysis of Periodic Structures (Amsterdam: North-Holland, 1980).Google Scholar
5Bergh, J. and Lofstrom, J.. Interpolation Spaces (New York: Springer, 1976).CrossRefGoogle Scholar
6Grisvard, P.. Elliptic Problems in Nonsmooth Domains (Boston: Pitman, 1985).Google Scholar
7Kesavan, S.. Homogenization of elliptic eigenvalue problems: Part 1. Appl. Math. Optim. 5 (1979), 153–67.CrossRefGoogle Scholar
8Kesavan, S.. Homogenization of elliptic eigenvalue problems: Part 2. Appl. Math. Optim. 5 (1979), 197–216.CrossRefGoogle Scholar
9Landis, E. M. and Panasenko, G. P.. A theorem on the asymptotics of solutions of elliptic equations with coefficients periodic in all variables except one. Soviet Math. Dokl. 18 (1977), 1140–3.Google Scholar
10Lions, J. L.. Some Methods for the Mathematical Analysis of Systems (Beijing: Science Press and New York: Gordon and Breach, 1981).Google Scholar
11Lions, J. L. and Magenes, E.. Non-Homogeneous Boundary Value Problems and Applications, I (Berlin: Springer, 1972).Google Scholar
12Moskow, S. and Vogelius, M.. First order corrections to the homogenized eigenvalues of a periodic composite medium. The case of Neumann boundary conditions (Preprint, 1996).Google Scholar
13Murat, F. and Tartar, L.. Calcul des variations et homogénéisation. In Les Méthodes D'Homogénéisation: Théorie et Applications en Physique (Colléction de la Direction des Etudes et Recherches d'Electricité de France, Eyrolles, Paris, 1985).Google Scholar
14Oleinik, O. A. and Iosif'jan, G. A.. On the behavior at infinity of solutions of second order elliptic equations in domains with noncompact boundary. Math. SSSR Sb. 40 (1981), 527–48.CrossRefGoogle Scholar
15Oleinik, O. A., Shamaev, A. S. and Yosifian, G. A.. Mathematical Problems in Elasticity and Homogenization (Amsterdam: North-Holland, 1992).Google Scholar
16Osborn, J.. Spectral approximation for compact operators. Math. Comp. 29 (1975), 712–25.CrossRefGoogle Scholar
17Santosa, F. and Vogelius, M.. First-order corrections to the homogenized eigenvalues of a periodic composite medium. SIAM J. Appl. Math. 53 (1993), 1636–68.CrossRefGoogle Scholar
18Vogelius, M.. A note on Spectral Approximation, Technical Report TR77-64. (College Park: University of Maryland, 1977).Google Scholar
19Vogelius, M. and Papanicolaou, G.. A projection method applied to diffusion in a periodic structure. SIAM J. Appl. Math. 42 (1982), 1302–22.CrossRefGoogle Scholar