Laplacian eigenvalues with a large negative Robin parameter on a part of the boundary

We consider the Laplacian eigenvalues for smooth planar domains with strongly attractive Robin conditions imposed on a part of the boundary and Neumann condition on the remaining boundary. The asymptotics of individual eigenvalues is described in terms of an effective operator on an interval with boundary conditions at the endpoints. For several typical geometries a more precise asymptotics in terms of the boundary curvature is obtained.