Re: dependence of eigenfunction of Laplacian on domain
From: Thomas Mautsch (mautsch_at_math.ethz.ch)
Date: 09/23/04
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Date: 23 Sep 2004 20:04:03 +0200
In news:<cisrat$pf3$1@nntp.itservices.ubc.ca> schrieb Robert Israel:
> In article <86zn3i8jlz.fsf@eeebch.et.tu-dresden.de>,
> Tobias Naehring <naehring@iee.et.tu-dresden.de> wrote:
[ ... ]
>>In the following, I call an eigenfunction of the Laplacian $\Delta$ as
>>{\it main eigenfunction} if it corresponds to the eigenvalue with smallest
>>absolute value.
^^^^^^^^???
>
> A mathematical physicist might call this a "ground state".
>
>>-- The question in rough words:
>
>>Is there a main eigenfunction of the two-dimensional
>>Laplacian which is continuously dependent on the domain?
>
>>-- The more precise question:
>
>>Let $G$ be some bounded domain of $\IR^2$ with smooth boundary and let
>>$\Phi_p:G\rightarrow G_p:=\Phi_p(G)$ be a family of diffeomorphisms
>>smoothly parameterised by $p\in(0,1)$. Thereby, each $G_p$ shall be
>>bounded.
>
>>Is there for each $p\in(0,1)$ a main eigenfunction $\phi_p$ of the
>>Laplacian $\Delta$ on $G_p$ such that the family
>>$(\phi_p)_{p\in(0,1)}$ is continuously parameterised by $p\in(0,1)$???
>
> You neglected to mention the boundary condition. I assume it's
> Dirichlet, i.e. f = 0 on the boundary.
>
> f -> f \circ (\Phi_p)^(-1) maps eigenfunctions of the Laplacian on
> G_p to eigenfunctions of some other elliptic second-order linear operator
> Delta_p on G. If everything is nice enough, I think we should be able to
> consider Delta_{p+h} as a small perturbation of Delta_p, and the ground
> state eigenfunction should vary continuously. I suspect that the
^^^^^^^^^^^^
> necessary technical details could be found in Kato's book:
>
> Perturbation theory for linear operators, T. Kato, Springer 1976,
> ISBN 3540075585 (Berlin), 0387075585 (New York)
I think, the normalized ground state will even
vary *smoothly* with the domain, because there is no problem
with eigenvalue-crossing. -
The lowest eigenvalue to the Dirichlet problem is always of multiplicity 1
(because the corresponding eigenfunction has to be positive).
BTW: Perelman uses (without mentioning it) in his paper
"The entropy formula for the Ricci flow and its geometric applications"
http://arxiv.org/abs/math.DG/0211159
a similar fact that the normalized ground state
of a Laplace operator with a metric-dependend potential
(on a closed manifold) depends smoothly on the metric,
so maybe you can find the information you seek in the
notes om Perelman's papers:
http://www.math.lsa.umich.edu/research/ricciflow/perelman.html
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