The Laplace operator with Neumann boundary conditions under perturbations of the domain (I and II)

We will consider the Laplace operator with Neumann boundary condition and will try to understand its behavior when the domain undergoes a perturbation.
We will provide conditions on the perturbations under which the continuity of the resolvent and the spectrum of the operator is obtained. 
We will also analyze in detail the case of a dumbbell domain perturbation which, roughly speaking, consists of two fixed domains joined by a thin channel.  The thin channel is of thickness epsilon>0 and it degenerates to a line segment as epsilon goes to 0.   We will show that, in general, there is a net contribution to the spectral behavior of the limiting problem coming from the channel and will analyze how the shape of the thin channel influences the spectral behavior of the limiting problem. We will also obtain the first term in the asymptotic expansion of the eigenvalues as the thickness of the thin channel goes to 0.  If time allows, we will show some extensions of these results to other situations like: other boundary conditions (like Robin),  higher order operators (biharmonic),  nonlinear elliptic problems and reaction diffusion equations in dumbbell domains.