Explicit terms in the small volume expansion of the shift of Neumann Laplacian eigenvalues due to a grounded inclusion in two dimensions

The first terms in the small volume asymptotic expansion of the shift of Neumann Laplacian eigenvalues caused by a grounded inclusion of area ε2 are derived. A novel explicit formula to compute them from the capacity, the eigenvalues and the eigenfunctions of the unperturbed domain, the size and the position of the inclusion, is given. The key step in the derivation is the filtering of the spectral decomposition of the Neumann function with the residue theorem. As a consequence of the formula, when a bifurcation of a double eigenvalue occurs (as for example in the case of a generic inclusion inside a disk) one eigenvalue decays like O(1/log⁡ε), the other like O(ε2).