M721: Index Theory

Ruminations of a Graduate Class

Local differential operators

with one comment

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.

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Written by aclightf

August 22, 2012 at 1:57 pm

Posted in Uncategorized

One Response

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  1. Hi Ash,

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

    Your examples should be rewritten, in order to fit your notation from the first paragraph. For example, in your grad example, you could write instead:

    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) .

    jfdavis

    November 13, 2012 at 1:17 am


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