Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function

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• 作者：I. Pankratova^a ; ^b ; ^{pankratova@cmap.polytechnique.fr} ; A. Piatnitski^a ; ^c ; ^{andrey@sci.lebedev.ru}
• 关键词：Spectral problem ; Sign-changing density ; Homogenization ; Thin cylinder
• 刊名：Journal of Differential Equations
• 出版年：2011
• 出版时间：1 April 2011
• 年：2011
• 卷：250
• 期：7
• 页码：3088-3134
• 全文大小：426 K

文摘

The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodic in the axial direction of the cylinder, and that the spectral density function changes sign. We show that the behavior of the spectrum depends essentially on whether the average of the density function is zero or not. In both cases we construct the effective 1-dimensional spectral problem and prove the convergence of spectra.