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A locally summable generalized derivative of a locally summable function (see [[Generalized function|Generalized function]]).
 
A locally summable generalized derivative of a locally summable function (see [[Generalized function|Generalized function]]).
  
More explicitly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s0859701.png" /> is an open set in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s0859702.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s0859703.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s0859704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s0859705.png" /> are locally summable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s0859706.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s0859707.png" /> is the Sobolev generalized partial derivative with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s0859708.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s0859709.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597010.png" />:
+
More explicitly, if $  \Omega $
 +
is an open set in an $  n $-
 +
dimensional space $  \mathbf R  ^ {n} $
 +
and if $  F $
 +
and $  f $
 +
are locally summable functions on $  \Omega $,  
 +
then $  f $
 +
is the Sobolev generalized partial derivative with respect to $  x _ {j} $
 +
of $  F $
 +
on $  \Omega $:
  
<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/s/s085/s085970/s08597011.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  F }{\partial  x _ {j} }
 +
( x)  = \
 +
f ( x) ,\  x \in \Omega ,\ \
 +
j = 1 \dots n ,
 +
$$
  
 
if the following equation holds:
 
if the following equation holds:
  
<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/s/s085/s085970/s08597012.png" /></td> </tr></table>
+
$$
 +
\int\limits _  \Omega  F ( x)
 +
\frac{\partial  \phi }{\partial  x _ {j} }
 +
\
 +
d x  = - \int\limits _  \Omega  f ( x) \phi ( x)  d x
 +
$$
 +
 
 +
for all infinitely-differentiable functions  $  \phi $
 +
on  $  \Omega $
 +
with compact support. The Sobolev generalized derivative is only defined almost-everywhere on  $  \Omega $.
 +
 
 +
An equivalent definition is as follows: Suppose that a locally summable function  $  F $
 +
on  $  \Omega $
 +
can be modified in such a way that, on a set of  $  n $-
 +
dimensional measure zero, it will be locally absolutely continuous with respect to  $  x _ {j} $
 +
for almost-all points  $  ( x _ {1} \dots x _ {j-} 1 , x _ {j+} 1 \dots x _ {n} ) $,
 +
in the sense of the  $  ( n - 1 ) $-
 +
dimensional measure. Then  $  F $
 +
has an ordinary partial derivative with respect to  $  x _ {j} $
 +
for almost-all  $  x \in \Omega $.  
 +
If the latter is locally summable, then it is called a Sobolev generalized derivative.
  
for all infinitely-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597014.png" /> with compact support. The Sobolev generalized derivative is only defined almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597015.png" />.
+
A third equivalent definition is as follows: Given two functions  $  F $
 +
and  $  f $,
 +
suppose there is a sequence  $  \{ F _ {k} \} $
 +
of continuously-differentiable functions on $  \Omega $
 +
such that for any domain  $  \omega $
 +
whose closure lies in  $  \Omega $,
  
An equivalent definition is as follows: Suppose that a locally summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597017.png" /> can be modified in such a way that, on a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597018.png" />-dimensional measure zero, it will be locally absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597019.png" /> for almost-all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597020.png" />, in the sense of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597021.png" />-dimensional measure. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597022.png" /> has an ordinary partial derivative with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597023.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597024.png" />. If the latter is locally summable, then it is called a Sobolev generalized derivative.
+
$$
 +
\int\limits _  \omega  | F _ {k} ( x) - F ( x) |  dx  \rightarrow  0 ,
 +
$$
  
A third equivalent definition is as follows: Given two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597026.png" />, suppose there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597027.png" /> of continuously-differentiable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597028.png" /> such that for any domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597029.png" /> whose closure lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597030.png" />,
+
$$
 +
\int\limits _  \omega  \left |
 +
\frac{\partial  F _ {k} ( x) }{\partial  x _ {j} }
 +
- f ( x) \right |  d x  \rightarrow  0 ,\  k \rightarrow \infty .
 +
$$
  
<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/s/s085/s085970/s08597031.png" /></td> </tr></table>
+
Then  $  f $
 +
is the Sobolev generalized derivative of  $  F $
 +
on  $  \Omega $.
  
<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/s/s085/s085970/s08597032.png" /></td> </tr></table>
+
Sobolev generalized derivatives of  $  F $
 +
on  $  \Omega $
 +
of higher orders (if they exist) are defined inductively:
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597033.png" /> is the Sobolev generalized derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597034.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597035.png" />.
+
$$
  
Sobolev generalized derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597037.png" /> of higher orders (if they exist) are defined inductively:
+
\frac{\partial  ^ {2} F }{\partial  x _ {i} \partial  x _ {j} }
 +
,\
  
<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/s/s085/s085970/s08597038.png" /></td> </tr></table>
+
\frac{\partial  ^ {3} F }{\partial  x _ {i} \partial  x _ {j} \partial  x _ {k} }
 +
,\dots.
 +
$$
  
 
They do not depend on the order of differentiation; e.g.,
 
They do not depend on the order of differentiation; e.g.,
  
<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/s/s085/s085970/s08597039.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  ^ {2} F }{\partial  x _ {i} \partial  x _ {j} }
 +
  = \
  
almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597040.png" />.
+
\frac{\partial  ^ {2} F }{\partial  x _ {j} \partial  x _ {i} }
 +
 
 +
$$
 +
 
 +
almost-everywhere on $  \Omega $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.L. Sobolev,  "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)  {{MR|1125990}} {{MR|0986735}} {{MR|0052039}} {{ZBL|0732.46001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)  {{MR|}} {{ZBL|0397.00003}} {{ZBL|0384.00004}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.L. Sobolev,  "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)  {{MR|1125990}} {{MR|0986735}} {{MR|0052039}} {{ZBL|0732.46001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)  {{MR|}} {{ZBL|0397.00003}} {{ZBL|0384.00004}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:14, 6 June 2020


A locally summable generalized derivative of a locally summable function (see Generalized function).

More explicitly, if $ \Omega $ is an open set in an $ n $- dimensional space $ \mathbf R ^ {n} $ and if $ F $ and $ f $ are locally summable functions on $ \Omega $, then $ f $ is the Sobolev generalized partial derivative with respect to $ x _ {j} $ of $ F $ on $ \Omega $:

$$ \frac{\partial F }{\partial x _ {j} } ( x) = \ f ( x) ,\ x \in \Omega ,\ \ j = 1 \dots n , $$

if the following equation holds:

$$ \int\limits _ \Omega F ( x) \frac{\partial \phi }{\partial x _ {j} } \ d x = - \int\limits _ \Omega f ( x) \phi ( x) d x $$

for all infinitely-differentiable functions $ \phi $ on $ \Omega $ with compact support. The Sobolev generalized derivative is only defined almost-everywhere on $ \Omega $.

An equivalent definition is as follows: Suppose that a locally summable function $ F $ on $ \Omega $ can be modified in such a way that, on a set of $ n $- dimensional measure zero, it will be locally absolutely continuous with respect to $ x _ {j} $ for almost-all points $ ( x _ {1} \dots x _ {j-} 1 , x _ {j+} 1 \dots x _ {n} ) $, in the sense of the $ ( n - 1 ) $- dimensional measure. Then $ F $ has an ordinary partial derivative with respect to $ x _ {j} $ for almost-all $ x \in \Omega $. If the latter is locally summable, then it is called a Sobolev generalized derivative.

A third equivalent definition is as follows: Given two functions $ F $ and $ f $, suppose there is a sequence $ \{ F _ {k} \} $ of continuously-differentiable functions on $ \Omega $ such that for any domain $ \omega $ whose closure lies in $ \Omega $,

$$ \int\limits _ \omega | F _ {k} ( x) - F ( x) | dx \rightarrow 0 , $$

$$ \int\limits _ \omega \left | \frac{\partial F _ {k} ( x) }{\partial x _ {j} } - f ( x) \right | d x \rightarrow 0 ,\ k \rightarrow \infty . $$

Then $ f $ is the Sobolev generalized derivative of $ F $ on $ \Omega $.

Sobolev generalized derivatives of $ F $ on $ \Omega $ of higher orders (if they exist) are defined inductively:

$$ \frac{\partial ^ {2} F }{\partial x _ {i} \partial x _ {j} } ,\ \frac{\partial ^ {3} F }{\partial x _ {i} \partial x _ {j} \partial x _ {k} } ,\dots. $$

They do not depend on the order of differentiation; e.g.,

$$ \frac{\partial ^ {2} F }{\partial x _ {i} \partial x _ {j} } = \ \frac{\partial ^ {2} F }{\partial x _ {j} \partial x _ {i} } $$

almost-everywhere on $ \Omega $.

References

[1] S.L. Sobolev, "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) MR1125990 MR0986735 MR0052039 Zbl 0732.46001
[2] S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian) Zbl 0397.00003 Zbl 0384.00004

Comments

In the Western literature the Sobolev generalized derivative is called the weak or distributional derivative.

References

[a1] L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1973) MR2067351 MR0209834 MR0117544 MR0107812 MR0041345 MR0035918 MR0032815 MR0031106 MR0025615 Zbl 0962.46025 Zbl 0653.46037 Zbl 0399.46028 Zbl 0149.09501 Zbl 0085.09703 Zbl 0089.09801 Zbl 0089.09601 Zbl 0078.11003 Zbl 0042.11405 Zbl 0037.07301 Zbl 0039.33201 Zbl 0030.12601
[a2] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 MR0617913 Zbl 0435.46002
How to Cite This Entry:
Sobolev generalized derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sobolev_generalized_derivative&oldid=48743
This article was adapted from an original article by S.M. Nikol'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article