Difference between revisions of "User:Jjtorrens/BancoDePruebas"
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+ | 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> | |
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− | + | if the following equation holds: | |
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− | + | <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. | |
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− | + | 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" />, | |
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− | + | <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> | |
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− | + | <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> | |
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− | + | 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" />. | |
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− | + | 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: | |
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− | + | <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., | |
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− | + | <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" />. | |
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1 | + | <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)</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)</TD></TR></table> |
====Comments==== | ====Comments==== | ||
− | + | In the Western literature the Sobolev generalized derivative is called the weak or distributional derivative. | |
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Schwartz, "Théorie des distributions" , '''1–2''' , Hermann (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5</TD></TR></table> |
Latest revision as of 00:23, 5 May 2012
2020 Mathematics Subject Classification: Primary: 46E35 [MSN][ZBL]
$\newcommand{\abs}[1]{\lvert #1\rvert}
\newcommand{\norm}[1]{\lVert #1\rVert}
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A locally summable generalized derivative of a locally summable function (see Generalized function).
More explicitly, if is an open set in an
-dimensional space
and if
and
are locally summable functions on
, then
is the Sobolev generalized partial derivative with respect to
of
on
:
![]() |
if the following equation holds:
![]() |
for all infinitely-differentiable functions on
with compact support. The Sobolev generalized derivative is only defined almost-everywhere on
.
An equivalent definition is as follows: Suppose that a locally summable function on
can be modified in such a way that, on a set of
-dimensional measure zero, it will be locally absolutely continuous with respect to
for almost-all points
, in the sense of the
-dimensional measure. Then
has an ordinary partial derivative with respect to
for almost-all
. If the latter is locally summable, then it is called a Sobolev generalized derivative.
A third equivalent definition is as follows: Given two functions and
, suppose there is a sequence
of continuously-differentiable functions on
such that for any domain
whose closure lies in
,
![]() |
![]() |
Then is the Sobolev generalized derivative of
on
.
Sobolev generalized derivatives of on
of higher orders (if they exist) are defined inductively:
![]() |
They do not depend on the order of differentiation; e.g.,
![]() |
almost-everywhere on .
References
[1] | S.L. Sobolev, "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) |
[2] | S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian) |
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) |
[a2] | K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 |
Jjtorrens/BancoDePruebas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jjtorrens/BancoDePruebas&oldid=25993