## 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
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