Tartar's method of compensated compactness in averaging the spectrum of a mixed problem for an elliptic equation in a perforated domain with third boundary condition

We study the problem described in the title of this paper in the domain $\Omega_\varepsilon$ obtained from a domain $\Omega\in\mathbb R^d$ by periodic perforation with period $\varepsilon Q$, where $Q$ is the unit cube in $\mathbb R^d$. For this problem we use the method of compensated compactness to obtain the first two terms of the asymptotics of the $k$-th eigenvalue in powers of $\varepsilon$ as $\varepsilon\to0$: $\lambda_{\varepsilon,k}=\varepsilon^{-1}\Lambda+\lambda_k+\dotsb$, where $\Lambda$ is a constant independent of $k$ and $\lambda_k$ is the $k$-th eigenvalue of the averaged problem (which turns out to be the Dirichlet problem in the domain $\Omega$) for $k\in\mathbb N$.