Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function (Q630584)

Let \(Q \subset\mathbb R^{d-1}\) be a smoothly bounded domain. The authors consider the following spectral problem: \[ \begin{alignedat}{2} {\mathcal A}^\varepsilon u^\varepsilon(x): &=- \text{div} \big( a^\varepsilon(x)\nabla u^\varepsilon(x)\big)= \lambda^\varepsilon \rho^\varepsilon(x) u^\varepsilon(x), \quad && x\in G_\varepsilon,\\ {\mathcal B}^\varepsilon u^\varepsilon(x): &= \big( a^\varepsilon(x)\nabla u^\varepsilon, n\big)=0, && x\in \Sigma_\varepsilon,\\ u^\varepsilon(-1,x') &= u^\varepsilon(1,x')=0, && x\in \partial(\varepsilon Q), \end{alignedat} \] where \(G_\varepsilon:= [-1,1] \times\varepsilon Q\) with boundary \(\Sigma_\varepsilon= (-1,1) \times \partial(\varepsilon Q)\) and \[ a^\varepsilon(x):= a\bigg(x_1,\frac{y}{\varepsilon}\bigg), \quad \rho^\varepsilon(x):= \rho\bigg(x_1,\frac{y}{\varepsilon}\bigg) \quad \text{with} \quad x=:(x_1,y) \in\mathbb R\times\mathbb R^{d-1}. \] The coefficient \(a(x_1,y)\) is a symmetric, uniform elliptic \(d \times d\)-matrix. Both \(a(\cdot,y)\) and the spectral density function \(\rho(\cdot,y)\) are assumed to be 1-periodic. The authors study the asymptotic behavior of the spectrum and the corresponding eigenfunctions with respect to \(\varepsilon\). An estimate for the rate of convergence is also obtained. The asymptotic behavior depends essentially on whether the average of the spectral density function over the period is equal to zero or not. For the case of a positive average the limit spectral problem is for a one-dimensional Sturm-Liouville problem while for a zero average it is for a quadratic operator pencil. The authors also consider the case when the local average of the spectral density function changes sign.