Localization of eigenfunctions in a thin cylinder with a locally periodic oscillating boundary

We study a Dirichlet spectral problem for second-order elliptic operator with locally periodic coefficients in thin cylinder. The lateral boundary of the cylinder is assumed to be periodic. When thickness ε tends zero, eigenvalues are order ε−2 and described terms first eigenvalue μ(x1) an auxiliary cell parametrized by x1, while eigenfunctions localize rate ε.