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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asked Jun 4, 2012 at 17:24
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$\begingroup$ 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. $\endgroup$– HelgeJul 21, 2012 at 0:00
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$\begingroup$ 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) $\endgroup$– Piero D'AnconaJul 21, 2012 at 17:25
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3 Answers
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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
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$\begingroup$ 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... $\endgroup$ Jun 4, 2012 at 19:09
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$\begingroup$ By the way, when I say smooth I mean $C^2$, but I could probably go a long way with Lipschitz only $\endgroup$ Jun 4, 2012 at 19:11
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1$\begingroup$ I remember that F. Murat wrote a paper on the dependency of elliptic BVPs upon the domain. $\endgroup$ Jun 5, 2012 at 5:05
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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
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$\begingroup$ 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. $\endgroup$ Jun 4, 2012 at 19:21
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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.
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1$\begingroup$ Thank you for the info. Did you publish your results? $\endgroup$ Jul 21, 2012 at 17:23