Uniform spectral asymptotics for singularly perturbed locally periodic operators.(English)Zbl 1026.35012

The authors study the spectral asymptotics of a singular perturbed second order elliptic operator with locally periodic oscillating coefficients of the form ${\mathcal A}^{\varepsilon}=-\varepsilon^2 \frac{\partial}{\partial x_i}\left(a^{ij} \left(x,\frac{x}{\varepsilon}\right) \frac{\partial}{\partial x_j}\right)+c \left(x,\frac{x} {\varepsilon}\right),$ defined in a bounded open set $$G$$ of $$\mathbb{R}^n.$$ The coefficients $$a^{ij}(x,z)$$, $$c(x,z)$$ are assumed to be real and sufficiently smooth and can be seen as periodic functions with respect to $$z$$ with period $$1$$ in all the coordinate directions and the matrix $$\{a^{ij}(x,z)\}$$ is symmetric, uniform positive definite. They consider the following eigenvalue problem: ${\mathcal A}^{\varepsilon}p^{\varepsilon}= \lambda^{\varepsilon} p^{\varepsilon} \text{ in } G,\quad p^{\varepsilon}=0 \text{ on } \partial G.\tag{1}$ They introduce an auxilliary eigenvalue problem to $${\mathcal A}^{\varepsilon}$$ $-\frac{\partial}{\partial z_i}\left(a^{ij}(x,z) \frac{\partial p}{\partial z_j}\right)+c(x,z)p= \lambda p \text{ for } z \in \text{ the unit torus}.\tag{2}$ First they mention the following theorem obtained previously in this field by them and some other authors:
Theorem: Assume that $$a^{ij}(x,z)=a^{ij}(z)$$ and $$c(x,z)=c(z)$$. The $$k$$th eigenpair $$(\lambda^{\varepsilon}_k$$, $$p^{\varepsilon}_k)$$ of (1) satisfies $$p^{\varepsilon}_k=u^{\varepsilon}_k p_1(\frac{x}{2})$$ and $$\lambda^{\varepsilon}_k=\lambda_1+\varepsilon^2 \nu_k +o(\varepsilon^2)$$, where $$(\lambda_1,p_1(z))$$ is the first eigenpair of the cell eigenproblem (2) and, up to a subsequence, the sequence $$u^{\varepsilon}_ k$$ converges weakly in $$H_0^1(G)$$ to $$u_k$$ such that $$(\nu_k,u_k)$$ is a $$k$$th eigenpair for the homogenized problem $-\frac{\partial}{\partial x_i}\left(a^{ij}_{\text{eff}}\frac{\partial u}{\partial x_j} \right)= \nu u \text{ in } G, \quad u=0 \text{ on }\partial G,$ where $$a^{ij}_{\text{eff}}$$ are homogenized coefficients of $$a^{ij}.$$
The first main theorem, which the authors obtained is as follows: Let $$p_1(x,z)$$ and $$\lambda_1(x)$$ be the first eigenvector and eigenvalue of (2) normalized by $$\|p_1(x,\cdot)\|_{L^2(T^n)}=1$$. Under some assumptions, for $$k \geq 1,$$ let $$\lambda^{\varepsilon}_k$$ and $$p^{\varepsilon}_k$$ be the $$k$$th normalized eigenvalue and eigenvector of (1). Then $p^{\varepsilon}_k(x)=u^{\varepsilon}_k \left(\frac{x}{\sqrt{\varepsilon}}\right) p_1 \left(x,\frac{x}{\varepsilon}\right),\quad \lambda^{\varepsilon}_k=\lambda_ 1(0)+\varepsilon \mu_k + o(\varepsilon),$ where up to a subsequence, the sequence $$u^{\varepsilon}_k(y)/\|u^{\varepsilon} _k \|_{L^2_{(\mathbb{R}^n)}}$$ converges weakly in $$H^1(\mathbb{R}^n)$$, and strongly in $$L^ 2(\mathbb{R}^n),$$ to $$u_k(y)$$, and $$(\mu_k,u_k)$$ is the $$k$$th eigenvalue and eigenvector of the homogenized problem: $- \frac{\partial}{\partial y_i}\left(a^{ij}_{\text{eff}} \frac{\partial u}{\partial y_j}\right)+ (c_{\text{eff}}+D_{ij}y_iy_j)u= \mu u \text{ for }y_i \text{ in }\mathbb{R}^n,\quad u\in L^2(\mathbb{R}^n),$ where $$D=\left\{D_{ij}\right\}$$ is the Hessian matrix $$(1/2)\nabla_x \nabla_x \lambda_1(0)$$ and $$a^{ij}_{\text{eff}}$$, $$c_{\text{eff}}$$ are homogenized coefficients.
The second main theorem is on an error estimate for the ground state asymptotics of a form: $|\lambda_k^{\varepsilon} - \lambda_1(0) - \varepsilon \mu_1|\leq C \varepsilon ^ {3/2}, \Biggl\|p_1^{\varepsilon}-\frac{q_1^{\varepsilon}}{\|q_1^{\varepsilon}\|}\Biggr\|_{L^2(\mathbb{R}^ n)}\leq C \varepsilon ^{1/2} .$

MSC:

 35B25 Singular perturbations in context of PDEs 35P05 General topics in linear spectral theory for PDEs 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
Full Text:

References:

 [1] Allaire G., SIAM J. Math. Anal. 23 (6) pp 1482– (1992) · Zbl 0770.35005 [2] Allaire G., Comput. Methods Appl. Mech. Engrg. 187 pp 91– (2000) · Zbl 1126.82346 [3] Allaire G., Ann. Mat. Pura Appl. [4] Allaire G., J. Math. Pures et Appli. 77 pp 153– (1998) · Zbl 0901.35005 [5] Allaire G., C. R. Acad. Sci. Paris Série I 324 pp 939– (1997) · Zbl 0879.35153 [6] Anselone P., Collectively Compact Operator Approximation Theory and Applications to Integral Equations (1971) · Zbl 0228.47001 [7] Bensoussan A., Asymptotic Analysis for Periodic Structures (1978) · Zbl 0404.35001 [8] Capdeboscq Y., C. R. Acad. Sci. Paris Série I 327 pp 807– (1998) · Zbl 0918.35135 [9] Capdeboscq Y., Proc. Roy. Soc. Edinburgh [10] Castro C., SIAM J. Appl. Math. 60 (4) pp 1205– (2000) · Zbl 0967.34074 [11] Piatnitski A., Commun. Math. Phys. 197 pp 527– (1998) · Zbl 0937.58023 [12] Kozlov S., Transc. Moscow Math. Soc. 2 pp 101– (1984) [13] Glimm J., Quantum Physics. A Functional Integral Point of View (1981) · Zbl 0461.46051 [14] Jikov V.V., Homogenization of Differential Operators and Integral Functionals (1994) [15] Vanninathan M., Proc. Indian Acad. Sci. Math. Sci. 90 pp 239– (1981) · Zbl 0486.35063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.