Homogenization of the spectral problem on the Riemannian manifold consisting of two domains connected by many tubes
2013 ◽
Vol 143
(6)
◽
pp. 1255-1289
◽
Keyword(s):
The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limitsare positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.
2000 ◽
Vol 130
(1)
◽
pp. 35-51
1987 ◽
Vol 101
(2)
◽
pp. 349-362
Keyword(s):
1997 ◽
Vol 20
(2)
◽
pp. 397-402
◽
1994 ◽
Vol 36
(1)
◽
pp. 77-80
◽
1970 ◽
Vol 11
(1)
◽
pp. 84-84
◽
2009 ◽
Vol 19
(11)
◽
pp. 2065-2100
◽
1998 ◽
Vol 174
(1)
◽
pp. 13-72
◽
2006 ◽
Vol 136
(6)
◽
pp. 1131-1155
◽