Bulk-surface spectral homogenization with indefinite mass in periodic perforated domains

We study the homogenization of an elliptic eigenvalue problem in a periodically perforated domain, where the spectral parameter appears both in the equation, through a bulk density $q$, and in a Steklov condition on the boundaries of the holes, through a surface density $\rho$, and where the resulting bulk–surface spectral measure $qdy+\rho d{S}_y$ is indefinite. We show that the asymptotic behavior of the spectrum depends essentially on whether the combined mean $m_{\mathrm{e}\mathrm{f}\mathrm{f}}={\int }_{Y^{\ast }}qdy+{\int }_{\mathrm{\Gamma }}\rho d{S}_y$ is positive, negative, or zero. In all three cases, we prove convergence of eigenvalue clusters with multiplicity and of eigenspaces and, under additional regularity, $O(\epsilon )$ estimates for the shifted or rescaled eigenvalues, spectral projectors, and aligned eigenspaces.