Asymptotic Analysis of the Eigenvalues of a Laplacian Problem in a Thin Multidomain

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.