Small volume expansion of the splitting of multiple Neumann Laplacian eigenvalues due to a grounded inclusion in two dimensions

The first terms of the small volume asymptotic expansion for the splitting of Neumann boundary condition Laplacian eigenvalues due to a grounded inclusion of size { \epsilon} are derived. An explicit formula to compute the first term from the eigenvalues and eigenfunctions of the unperturbed domain, the inclusion size and position is given. As a consequence, when an eigenvalue of double multiplicity splits in two distinct eigenvalues, one decays like O(1/log({ \epsilon})), the other like O({ \epsilon}^2).

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