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''subcaloric function''
 
''subcaloric function''
  
 
The analogue of a [[Subharmonic function|subharmonic function]] for the heat equation
 
The analogue of a [[Subharmonic function|subharmonic function]] for the heat equation
  
<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/s091/s091000/s0910001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
 +
 
 +
\frac{\partial  u }{\partial  t }
 +
- \Delta  ^ {2} u  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s0910002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s0910003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s0910004.png" /> is the [[Laplace operator|Laplace operator]]. For example, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s0910005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s0910006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s0910007.png" />, of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s0910008.png" /> will be a subparabolic function in the rectangle
+
where $  u= u( x, t) $,  
 +
$  x = ( x _ {1} \dots x _ {n} ) \in \mathbf R  ^ {n} $
 +
and $  \Delta  ^ {2} u = \sum _ {j=} 1  ^ {n} \partial  ^ {2} u / \partial  x _ {j}  ^ {2} $
 +
is the [[Laplace operator|Laplace operator]]. For example, a function $  v = v( x, t) $,
 +
$  x \in \mathbf R $,  
 +
$  t > 0 $,  
 +
of class $  C  ^ {2} $
 +
will be a subparabolic function in the rectangle
  
<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/s091/s091000/s0910009.png" /></td> </tr></table>
+
$$
 +
= \{ {( x, t) \in \mathbf R \times \mathbf R _ {+} } : {a < x < b, 0 < t < h } \}
 +
$$
  
 
if
 
if
  
<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/s091/s091000/s09100010.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  v }{\partial  t }
 +
-  
 +
\frac{\partial  ^ {2} v }{\partial  x  ^ {2} }
 +
  \leq  0
 +
$$
  
everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100011.png" />. In a more general case, let the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100012.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100013.png" /> be a sufficiently small equilateral triangle with base parallel to the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100014.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100015.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100016.png" /> that is continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100017.png" /> is said to be subparabolic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100018.png" /> if its value at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100019.png" /> is not greater than the value at this point of that solution of (*) in any sufficiently small triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100021.png" />, that has the same values on the sides of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100022.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091000/s09100023.png" />.
+
everywhere in $  D $.  
 +
In a more general case, let the point $  ( x _ {0} , t _ {0} ) \in D $,  
 +
let $  \Delta $
 +
be a sufficiently small equilateral triangle with base parallel to the axis $  t= 0 $
 +
and let $  ( x _ {0} , t _ {0} ) \in \Delta \subset  D $.  
 +
A function $  v = v( x, t) $
 +
that is continuous in the closed domain $  \overline{D}\; $
 +
is said to be subparabolic in $  D $
 +
if its value at any point $  ( x _ {0} , t _ {0} ) \in D $
 +
is not greater than the value at this point of that solution of (*) in any sufficiently small triangle $  \Delta $,
 +
$  ( x _ {0} , t _ {0} ) \in \Delta $,  
 +
that has the same values on the sides of $  \Delta $
 +
as $  v( x, t) $.
  
 
Many properties of subharmonic functions, including the maximum principle, are also valid for subparabolic functions.
 
Many properties of subharmonic functions, including the maximum principle, are also valid for subparabolic functions.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''2''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.G. Petrovskii,  "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Petrovskii,  "Zur ersten Randwertaufgabe der Wärmeleitungsgleichung"  ''Compos. Math.'' , '''1'''  (1935)  pp. 383–419</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''2''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.G. Petrovskii,  "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Petrovskii,  "Zur ersten Randwertaufgabe der Wärmeleitungsgleichung"  ''Compos. Math.'' , '''1'''  (1935)  pp. 383–419</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:24, 6 June 2020


subcaloric function

The analogue of a subharmonic function for the heat equation

$$ \tag{* } \frac{\partial u }{\partial t } - \Delta ^ {2} u = 0, $$

where $ u= u( x, t) $, $ x = ( x _ {1} \dots x _ {n} ) \in \mathbf R ^ {n} $ and $ \Delta ^ {2} u = \sum _ {j=} 1 ^ {n} \partial ^ {2} u / \partial x _ {j} ^ {2} $ is the Laplace operator. For example, a function $ v = v( x, t) $, $ x \in \mathbf R $, $ t > 0 $, of class $ C ^ {2} $ will be a subparabolic function in the rectangle

$$ D = \{ {( x, t) \in \mathbf R \times \mathbf R _ {+} } : {a < x < b, 0 < t < h } \} $$

if

$$ \frac{\partial v }{\partial t } - \frac{\partial ^ {2} v }{\partial x ^ {2} } \leq 0 $$

everywhere in $ D $. In a more general case, let the point $ ( x _ {0} , t _ {0} ) \in D $, let $ \Delta $ be a sufficiently small equilateral triangle with base parallel to the axis $ t= 0 $ and let $ ( x _ {0} , t _ {0} ) \in \Delta \subset D $. A function $ v = v( x, t) $ that is continuous in the closed domain $ \overline{D}\; $ is said to be subparabolic in $ D $ if its value at any point $ ( x _ {0} , t _ {0} ) \in D $ is not greater than the value at this point of that solution of (*) in any sufficiently small triangle $ \Delta $, $ ( x _ {0} , t _ {0} ) \in \Delta $, that has the same values on the sides of $ \Delta $ as $ v( x, t) $.

Many properties of subharmonic functions, including the maximum principle, are also valid for subparabolic functions.

References

[1] V.I. Smirnov, "A course of higher mathematics" , 2 , Addison-Wesley (1964) (Translated from Russian)
[2] I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian)
[3] I.G. Petrovskii, "Zur ersten Randwertaufgabe der Wärmeleitungsgleichung" Compos. Math. , 1 (1935) pp. 383–419

Comments

See [a1] for an account of subparabolic functions from a potential-theoretic point of view.

References

[a1] J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1983)
[a2] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
How to Cite This Entry:
Subparabolic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subparabolic_function&oldid=12693
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article