  We consider the Laplacian in a domain squeezed between two parallel curves in
the plane, subject to Dirichlet boundary conditions on one of the curves and
Neumann boundary conditions on the other. We derive two-term asymptotics for
eigenvalues in the limit when the distance between the curves 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 curvature radii of the
Neumann boundary to the Dirichlet one is the biggest. We also show that the
asymptotics can be obtained from a form of norm-resolvent convergence which
takes into account the width-dependence of the domain of definition of the
operators involved.
