Namespaces
Variants
Actions

Difference between revisions of "Principal part of a differential operator"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
The homogeneous differential operator formed from the given operator by discarding all the terms not containing derivatives of maximal order. The principal part of the differential operator
 
The homogeneous differential operator formed from the given operator by discarding all the terms not containing derivatives of maximal order. The principal part of the differential operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747601.png" /></td> </tr></table>
+
$$L=\sum_{|\alpha|\leq m}a_\alpha D^\alpha$$
  
is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747602.png" />. The principal part of a differential operator is sometimes defined by the introduction of supplementary weights assigned to the differentiations with respect to the various arguments. For instance, the principal part of the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747603.png" /> is sometimes defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747604.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747605.png" /> is given weight 2 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747606.png" /> weight 1).
+
is $\sum_{|\alpha|=m}a_\alpha D^\alpha$. The principal part of a differential operator is sometimes defined by the introduction of supplementary weights assigned to the differentiations with respect to the various arguments. For instance, the principal part of the differential operator $D_1-D_2^2+\alpha D_2$ is sometimes defined as $D_1-D_2^2$ (if $D_1$ is given weight 2 and $D_2$ weight 1).
  
  
Line 10: Line 11:
 
The principal part is also called the principal symbol (cf. also [[Symbol of an operator|Symbol of an operator]]).
 
The principal part is also called the principal symbol (cf. also [[Symbol of an operator|Symbol of an operator]]).
  
The zero sets of the principal symbol are called the characteristics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747607.png" /> (cf. also [[Characteristic|Characteristic]]).
+
The zero sets of the principal symbol are called the characteristics of $L$ (cf. also [[Characteristic|Characteristic]]).
  
Further, a constant-coefficient differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747608.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p0747609.png" /> is said to be of real principal type if the principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476010.png" /> is real and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476012.png" /> (cf. also [[Principal type, partial differential operator of|Principal type, partial differential operator of]]).
+
Further, a constant-coefficient differential operator $L$ in $\mathbf R^n$ is said to be of real principal type if the principal symbol $l$ is real and if $l'(\xi)\neq0$ for $\xi\in\mathbf R^n\setminus 0$ (cf. also [[Principal type, partial differential operator of|Principal type, partial differential operator of]]).
  
For a differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476013.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476014.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476015.png" /> coefficients in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476016.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476017.png" /> the principal symbol can be regarded as an invariantly-defined function on the cotangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074760/p07476018.png" />.
+
For a differential operator $L$ of order $m$ with $C^\infty$ coefficients in a $C^\infty$-manifold $X$ the principal symbol can be regarded as an invariantly-defined function on the cotangent bundle of $X$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>

Latest revision as of 14:26, 8 August 2014

The homogeneous differential operator formed from the given operator by discarding all the terms not containing derivatives of maximal order. The principal part of the differential operator

$$L=\sum_{|\alpha|\leq m}a_\alpha D^\alpha$$

is $\sum_{|\alpha|=m}a_\alpha D^\alpha$. The principal part of a differential operator is sometimes defined by the introduction of supplementary weights assigned to the differentiations with respect to the various arguments. For instance, the principal part of the differential operator $D_1-D_2^2+\alpha D_2$ is sometimes defined as $D_1-D_2^2$ (if $D_1$ is given weight 2 and $D_2$ weight 1).


Comments

The principal part is also called the principal symbol (cf. also Symbol of an operator).

The zero sets of the principal symbol are called the characteristics of $L$ (cf. also Characteristic).

Further, a constant-coefficient differential operator $L$ in $\mathbf R^n$ is said to be of real principal type if the principal symbol $l$ is real and if $l'(\xi)\neq0$ for $\xi\in\mathbf R^n\setminus 0$ (cf. also Principal type, partial differential operator of).

For a differential operator $L$ of order $m$ with $C^\infty$ coefficients in a $C^\infty$-manifold $X$ the principal symbol can be regarded as an invariantly-defined function on the cotangent bundle of $X$.

References

[a1] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983)
How to Cite This Entry:
Principal part of a differential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_part_of_a_differential_operator&oldid=32767
This article was adapted from an original article by A.A. Dezin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article