  We consider a thin multidomain of $R^N$, N>1, consisting of two vertical
cylinders, one placed upon the other: the first one with given height and small
cross section, the second one with small thickness and given cross section. In
this multidomain we study the asymptotic behavior, when the volumes of the two
cylinders vanish, of a Laplacian eigenvalue problem and of a $L^2$-Hilbert
orthonormal basis of eigenvectors. We derive the limit eigenvalue problem
(which is well posed in the union of the limit domains, with respective
dimension 1 and N-1) and the limit basis. We discuss the limit models and we
precise how these limits depend on the dimension N and on limit of the ratio
between the volumes of the two cylinders.
