# Stability of the spectrum for perturbations of the boundary

Consider the Laplace operator on a smooth bounded open set with Dirichlet boundary conditions. I need some result of the following type: if one perturbs the boundary in a suitable sense to be determined, say depending on a small parameter $\epsilon$, then it is possible to order eigenvalues and eigenfunctions so that they are smooth functions of $\epsilon$ and spatial variables. At first this seemed a very natural result, but the total scarcity of references suggests otherwise, and I am starting to think that this kind of result is rather difficult, if one wants to achieve some generality. Has anyone encountered any result in this direction?

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Assume your domains are diffeomorphic to a ball. Then instead of looking at the Laplace operator on the domain, you can look at the Laplace-Beltrami on the ball with a strange metric. This reduces the problem to comparing the Laplace operator for two different metrics. If I remember correctly, the main terms will be the same, so all that is left is a small perturbation in an appropriate operator theoretical sense. There might be quite a few details here that are annoying to work out, in particular if the closeness of metrics corresponds to the closeness you want. –  Helge Jul 21 '12 at 0:00
Indeed, but I guess one can obtain at most the asymptotics of eigenvalues and some quantities connected with the eigenfunctions, while I need more or less a sort of Fourier expansion depending on the parameter (probably too much) –  Piero D'Ancona Jul 21 '12 at 17:25

This is true, as long as your domain depends smoothly upon one real parameter. Say that you are insterested in the $n$ first eigenvalues. Using a Lyapunov-Schmidt procedure, you may reduce to the situation of an $n\times n$ symmetric matrix $S(\epsilon)$. Then look at Kato's book in the Grundlehren series.

If instead your domain depends on two or more parameters, the matrix $S$ will depend on several variables, and the eigenvalues will not be smooth functions, unless they remain simple.

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Thank you Denis. Actually the domains I have in mind are sections of a higher dimensional domain, and I am looking for sensible assumptions on the global domain such that the sections have (locally) similar spectral properties. Clearly in the greatest generality this seems hopeless, but maybe for domains with a special structure... –  Piero D'Ancona Jun 4 '12 at 19:09
By the way, when I say smooth I mean $C^2$, but I could probably go a long way with Lipschitz only –  Piero D'Ancona Jun 4 '12 at 19:11
I remember that F. Murat wrote a paper on the dependency of elliptic BVPs upon the domain. –  Denis Serre Jun 5 '12 at 5:05
Look at Perturbation of the boundary in boundary-value problems in partial differential equations'' by Dan Henry, London Math Society Lecture Notes #318