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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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