A mixed eigenvalue problem on domains tending to infinity in several directions

The aim of this paper is to analyze the asymptotic behavior of the eigenvalues of elliptic operators in divergence form with mixed boundary type conditions for domains that become unbounded in several directions, while they stay bounded in some directions (cylindrical domains). The limiting behavior of such eigenvalues is shown to depend on an ensemble of eigenvalue problems defined on a domain that is unbounded only in one direction. The asymptotic behavior of the eigenfunctions is also discussed. This work is a continuation of the work done in [M. Chipot, P. Roy and I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity, Asymptot. Anal. 85 (2013) 199–227].