M721: Index Theory

Local differential operators

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In this post we define a differential operator on Euclidean space and give some familiar examples.

For $\alpha = (\alpha_1, \ldots, \alpha_n)$, the differential operator $D^\alpha=\frac{\partial_1^{\alpha}}{\partial x_{\alpha_1}}\cdots \frac{\partial_n^{\alpha}}{\partial x_{\alpha_n}}$ takes a smooth function on $\mathbb{R}^n$ to a smooth function on $\mathbb{R}^n$. We call $|\alpha|=\alpha_1+\ldots + \alpha_n$ the order of this differential operator.

A linear map $D:C^\infty(\mathbb{R}^n,\mathbb{R}^N)\to C^\infty(\mathbb{R}^n,\mathbb{R}^M)$ is a (linear) differential operator of order $m$ if it is of the form $\displaystyle D = \sum_{\alpha: \,|\alpha|=m} A^{\alpha}D^\alpha$, where $A^\alpha$ is an $M\times N$ matrix over $C^\infty(\mathbb{R}^n,\mathbb{R})$. Given $\xi\in \mathbb{R}^n$, the symbol $\sigma_D(\xi)$ is an $M\times N$ matrix over $\mathbb{R}$ such that $\sigma_D(\xi) = \sum_{\alpha: |\alpha|=m} A^\alpha \xi^\alpha$, where $\xi^\alpha = \xi_1^{\alpha_1}\cdots \xi_n^{\alpha_n}$.

We say that $D$ is elliptic if $M=N$ and for all $\xi\in \mathbb{R}^n-\{0\}$, $\sigma_D(\xi)$ is an isomorphism of $\mathbb{R}^M$ to itself.

Examples

$\bullet$ Grad: $C^\infty(\mathbb{R}^3,\mathbb{R})\to C^\infty(\mathbb{R}^3,\mathbb{R}^3)$,

$Grad = \left(\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right) \frac{\partial }{\partial x_1} + \left(\begin{array}{ccc} 0 \\ 1 \\ 0 \end{array}\right) \frac{\partial }{\partial x_2}+ \left(\begin{array}{ccc} 0 \\ 0 \\ 1 \end{array}\right)\frac{\partial }{\partial x_3}$

We have
$Grad\, f = \left(\begin{array}{c} \frac{\partial f}{\partial x_1}\\ \frac{\partial f}{\partial x_2}\\ \frac{\partial f}{\partial x_3} \end{array}\right) = \left(\begin{array}{c} \frac{\partial }{\partial x_1} \\ \frac{\partial }{\partial x_2} \\ \frac{\partial }{\partial x_3} \end{array}\right) f$

$\leadsto \sigma_\text{grad}(\xi) = \left(\begin{array}{c} \xi_1\\ \xi_2\\ \xi_3 \end{array}\right)$

$\bullet$ Div: $C^\infty(\mathbb{R}^3,\mathbb{R})\to C^\infty(\mathbb{R}^3,\mathbb{R})$,

$Div\, = \frac{\partial }{\partial x_1} + \frac{\partial }{\partial x_2} + \frac{\partial }{\partial x_3}$

We have
$Div f = \frac{\partial f}{\partial x_1} + \frac{\partial f}{\partial x_2} + \frac{\partial f}{\partial x_3} = \left(\frac{\partial}{\partial x_1} + \frac{\partial}{\partial x_2} + \frac{\partial}{\partial x_3}\right) f$

$\leadsto \sigma_{Div}(\xi) = \xi_1+ \xi_2+ \xi_3$.

$\bullet$ Curl: $C^\infty(\mathbb{R}^3,\mathbb{R}^3)\to C^\infty(\mathbb{R}^3,\mathbb{R}^3)$,

$Curl\, = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array} \right) \frac{\partial}{\partial x_1} + \left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{array} \right) \frac{\partial}{\partial x_2} +\left(\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \frac{\partial}{\partial x_3}$

We have
$Curl \left(\begin{array}{c} f_1 \\ f_2 \\ f_3 \end{array}\right) = \left(\begin{array}{c} \frac{\partial f_3}{\partial x_2} - \frac{\partial f_2}{\partial x_3} \\ \frac{\partial f_1}{\partial x_3} - \frac{\partial f_3}{\partial x_1} \\ \frac{\partial f_2}{\partial x_1} - \frac{\partial f_1}{\partial x_2} \end{array}\right) = \left(\begin{array}{ccc}0&-\frac{\partial}{\partial x_3}&\frac{\partial}{\partial x_2}\\ \frac{\partial}{\partial x_3}&0&-\frac{\partial}{\partial x_1}\\-\frac{\partial}{\partial x_2}&\frac{\partial}{\partial x_1}&0\end{array}\right)\left(\begin{array}{c} f_1\\f_2\\ f_3\end{array}\right)$

$\leadsto \sigma_{\text{Curl}}(\xi) = \left(\begin{array}{ccc}0&-\xi_3&\xi_2\\ \xi_3&0&-\xi_1\\-\xi_2&\xi_1&0\end{array}\right)$.

$\bullet$ Exterior derivative $d: C^\infty(\Lambda^{1}T^\ast \mathbb{R}^3)\to C^\infty(\Lambda^{2}T^\ast \mathbb{R}^3)$,

We have
$d(f_1 dx_1 + f_2 dx_2 + f_3 dx_3) = df_1\wedge dx_1+df_2\wedge dx_2+df_3\wedge dx_3 \\ = \left(\frac{\partial f_1}{\partial x_2} dx_2\wedge dx_1+\frac{\partial f_1}{\partial x_3} dx_3\wedge dx_1\right) + \left(\frac{\partial f_2}{\partial x_1} dx_1\wedge dx_2 + \frac{\partial f_2}{\partial x_3} dx_3\wedge dx_2\right) + \left(\frac{\partial f_3}{\partial x_1} dx_1\wedge dx_3 + \frac{\partial f_3}{\partial x_2} dx_2\wedge dx_3\right) \\ = \left(\frac{\partial f_3}{\partial x_2}-\frac{\partial f_2}{\partial x_3} \right) dx_2\wedge dx_3 + \left(\frac{\partial f_1}{\partial x_3} - \frac{\partial f_3}{\partial x_1}\right) dx_3\wedge dx_1 + \left(\frac{\partial f_2}{\partial x_1}-\frac{\partial f_1}{\partial x_2}\right) dx_1\wedge dx_2.$

Now, with respect to the ordered bases $\{dx_1, dx_2, dx_3\}$ and $\{dx_2\wedge dx_3, dx_3\wedge dx_1, dx_1\wedge dx_2\}$ for $C^\infty(\Lambda^{1}T^\ast \mathbb{R}^3)$ and $C^\infty(\Lambda^{2}T^\ast \mathbb{R}^3)$, respectively, we may write this as

$d\left(\begin{array}{c} f_1 \\ f_2 \\ f_3 \end{array}\right) = \left(\begin{array}{c} \frac{\partial f_3}{\partial x_2} - \frac{\partial f_2}{\partial x_3} \\ \frac{\partial f_1}{\partial x_3} - \frac{\partial f_3}{\partial x_1} \\ \frac{\partial f_2}{\partial x_1} - \frac{\partial f_1}{\partial x_2} \end{array}\right)$,

so by the previous example we see that $\sigma_d = \sigma_{\text{Curl}}$.

$\bullet$ Laplacian $\Delta$: $C^\infty(\mathbb{R}^3,\mathbb{R})\to C^\infty(\mathbb{R}^3,\mathbb{R})$,

$\Delta = \frac{\partial^2 }{\partial x_1^2}+ \frac{\partial^2 }{\partial x_2^2}+ \frac{\partial^2 }{\partial x_3^2}$

We have
$\Delta f = \frac{\partial^2 f}{\partial x_1^2}+ \frac{\partial^2 f}{\partial x_2^2}+ \frac{\partial^2 f}{\partial x_3^2} = \left( \frac{\partial^2 }{\partial x_1^2}+ \frac{\partial^2 }{\partial x_2^2}+ \frac{\partial^2 }{\partial x_3^2} \right) f$

$\leadsto \sigma_\Delta(\xi) = \xi_1^2+ \xi_2^2+ \xi_3^2$.

All of the examples above are order 1 except for the Laplacian, which is order 2.

Which of the above are elliptic?

For dimensional reasons, the only candidates are $\sigma_{\text{Div}}$, $\sigma_{d}$ and $\sigma_\Delta$. The first two are not elliptic since, for example, $\sigma_{\text{Div}}((1,-1,0))=0$ and $\sigma_d((1,0,0)) = \left(\begin{array}{ccc}0&0&0\\ 0&0&-1\\ 0&1&0\end{array}\right)$ has determinant $0$. $\sigma_\Delta$ is elliptic, on the other hand, since $\sigma_\Delta(\xi)=0 \Leftrightarrow \xi=0$.

Written by aclightf

August 22, 2012 at 1:57 pm

Posted in Uncategorized

One Response

1. Hi Ash,

$\bullet$ Grad: $C^\infty(\mathbb{R}^3,\mathbb{R})\to C^\infty(\mathbb{R}^3,\mathbb{R}^3)$,

$Grad = \left(\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right) \frac{\partial }{\partial x_1} + \left(\begin{array}{ccc} 0 \\ 1 \\ 0 \end{array}\right) \frac{\partial }{\partial x_2}+ \left(\begin{array}{ccc} 0 \\ 0 \\ 1 \end{array}\right)\frac{\partial }{\partial x_3}$
Thus grad $f = \left(\begin{array}{c} \frac{\partial f}{\partial x_1}\\ \frac{\partial f}{\partial x_2}\\ \frac{\partial f}{\partial x_3} \end{array}\right)$
$\leadsto \sigma_\text{grad}(\xi) = \left(\begin{array}{c} \xi_1\\ \xi_2\\ \xi_3 \end{array}\right)$.