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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?

flag	asked Jun 4 2012 at 17:24 Piero D'Ancona 3,667●1●6●18

	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 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 at 17:25

marrocos · Answer 1 · 2012-07-20 23:46:35Z UTC

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Hello, My name is Marcus Morocco. My doctoral thesis was on exactly these issues. I calculated the expressions for the first and second derivatives of eigenvalues and eigenvectors of the laplace opelador ccom Neumann boundary condition. If interest can send the file.

answered Jul 20 at 23:46

marrocos
11●1

Thank you for the info. Did you publish your results? – Piero D'Ancona Jul 21 at 17:23

Rafe Mazzeo · Answer 2 · 2012-06-04 18:08:17Z UTC

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Look at ``Perturbation of the boundary in boundary-value problems in partial differential equations'' by Dan Henry, London Math Society Lecture Notes #318

answered Jun 4 2012 at 18:08

Rafe Mazzeo
1,304●2●4

	Thank you, I'll take a look – Piero D'Ancona Jun 4 2012 at 19:09
	Ah, now I recall I already checked that book, but it gave information on the eigenvalues mostly. I would rather need some kind of Fourier expansion depending on the parameter(s). It is easy to conceive some special cases (e.g. a ball with varying radius) and I was hoping some more general class of examples could be known. – Piero D'Ancona Jun 4 2012 at 19:21

Denis Serre · Answer 3 · 2012-06-04 17:59:32Z UTC

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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.

answered Jun 4 2012 at 17:59

Denis Serre
18.8k●1●31●94

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 2012 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 2012 at 19:11

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I remember that F. Murat wrote a paper on the dependency of elliptic BVPs upon the domain. – Denis Serre Jun 5 2012 at 5:05

Stability of the spectrum for perturbations of the boundary

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