View source for Sobolev generalized derivative

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" />:

<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>

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>

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" />.

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.

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" />,

<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>

<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>

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:

<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>

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>

almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085970/s08597040.png" />.

====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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''1–2''' , Hermann  (1973)  {{MR|2067351}} {{MR|0209834}} {{MR|0117544}} {{MR|0107812}} {{MR|0041345}} {{MR|0035918}} {{MR|0032815}} {{MR|0031106}} {{MR|0025615}} {{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}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5  {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR></table>