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The aim of the present paper is to investigate the behavior of the spectrumof the Neumann Laplacian in domains with little holes excised from theinterior. More precisely, we consider the eigenvalues of the Laplacian withhomogeneous Neumann boundary conditions on a bounded, Lipschitz domain. Then,we singularly perturb the domain by removing Lipschitz sets which are "small"in a suitable sense and satisfy a uniform extension property. In this context,we provide an asymptotic expansion for all the eigenvalues of the perturbedproblem which are converging to simple eigenvalues of the limit one. Theeigenvalue variation turns out to depend on a geometric quantity resembling thenotion of (boundary) torsional rigidity: understanding this fact is one of themain contributions of the present paper. In the particular case of a holeshrinking to a point, through a fine blow-up analysis, we identify the exactvanishing order of such a quantity and we establish some connections betweenthe location of the hole and the sign of the eigenvalue variation.

Quantitative spectral stability for the Neumann Laplacian in domains with small holes