| Abstract: |
| We address the problem of splitting of eigenvalues of the Neumann Laplacian under singular domain perturbations. We consider a domain perturbed by the excision of a small spherical hole shrinking to an interior point. We establish that the splitting of multiple eigenvalues is a generic property: if the center of the hole is located outside a set of Hausdorff dimension N-1 and the radius is sufficiently small, multiple eigenvalues split into branches of lower multiplicity. The proof relies on the validity of an asymptotic expansion for the perturbed eigenvalues in terms of the scaling parameter. |
|