  We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a
Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected
by a thin tube with vanishing width. All the eigenvalues are collapsing to
zero, the speed being driven by some power of the width which multiplies the
eigenvalues of a one dimensional problem. In two dimensions of the space, the
behaviour is fundamentally different from the third or higher dimensions and
the limit problems are of different nature. This phenomenon is due to the fact
that only in dimension two the boundary of the tube has not vanishing surface
measure.
