# 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

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