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In this paper we establish new quantitative stability estimates with respectto domain perturbations for all the eigenvalues of both the Neumann and theDirichlet Laplacian. Our main results follow from an abstract lemma statingthat it is actually sufficient to provide an estimate on suitable projectionoperators. Whereas this lemma could be applied under different regularityassumptions on the domain, here we use it to estimate the spectrum in Lipschitzand in so-called Reifenberg-flat domains. Our argument also relies on suitableextension techniques and on an estimate on the decay of the eigenfunctions atthe boundary which could be interpreted as a boundary regularity result.

Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains