Ngô Quốc Anh

August 8, 2010

The nodal set of eigenfunctions

Filed under: PDEs — Ngô Quốc Anh @ 17:21

An eigenfunction \varphi is meant to be a solution of Dirichlet’s problem

\begin{cases}-\Delta \varphi = \lambda\varphi,&\text{ in } \Omega,\\\varphi=0,&\text{ on } \partial\Omega,\end{cases}

where

\displaystyle\Delta = \sum\limits_{k = 1}^n {\frac{{{\partial ^2}}}{{\partial x_k^2}}}

is the Laplacian, \Omega is a bounded smooth domain in \mathbb R^n, and \lambda is a constant (i.e. the corresponding eigenvalue).

It is well known that the first eigenfunction is positive in \Omega, and all higher eigenfunctions must change sign.

Definition. The nodal set of an eigenfunction \varphi is defined to be

\displaystyle N = \overline {\left\{ {x \in \Omega :\varphi (x) = 0} \right\}} .

To gain a better understanding of nodal sets let us consider a few examples.

Example 1. Let us consider the case when \Omega=(0,l) where the first four eigenfunctions had been shown.

Red points are nodal sets. In this case, we simply call nodal nodes. This comes from the fact that all eigenvalues \lambda_k and eigenfunctions \varphi_k are already known

\displaystyle{\varphi _k}(x) = \sqrt {\frac{2}{l}} \sin \left( {\frac{{n\pi }}{l}x} \right), \quad k \in \mathbb{N}.

(more…)

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