  We consider the Laplacian in a domain squeezed between two parallel
hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet
boundary conditions on one of the hypersurfaces and Neumann boundary conditions
on the other. We derive two-term asymptotics for eigenvalues in the limit when
the distance between the hypersurfaces tends to zero. The asymptotics are
uniform and local in the sense that the coefficients depend only on the
extremal points where the ratio of the area of the Neumann boundary to the
Dirichlet one is locally the biggest.
