Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains

Taking advantage from the so-called Lemma on small eigenvalues by Colin de Verdière, we study ramification for multiple of Dirichlet Laplacian in bounded perforated domains. The asymptotic behavior turns out to depend expansion suitable associated eigenfunctions. We treat case planar domains details, thanks a generalization u-capacity which compute dimension 2. In this are proved split essentially different rates convergence perturbed or coefficients front their if rate two eigenbranches be same.