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''hyper-harmonic function, meta-harmonic function, of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p0735102.png" />''
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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p0735103.png" /> of real variables defined in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p0735104.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p0735105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p0735106.png" />, having continuous partial derivatives up to and including the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p0735107.png" /> and satisfying the poly-harmonic equation everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p0735108.png" />:
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{{TEX|auto}}
 +
{{TEX|done}}
  
<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/p073/p073510/p0735109.png" /></td> </tr></table>
+
''hyper-harmonic function, meta-harmonic function, of order  $  m $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351010.png" /> is the [[Laplace operator|Laplace operator]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351011.png" /> one obtains harmonic functions (cf. [[Harmonic function|Harmonic function]]), while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351012.png" /> one obtains biharmonic functions (cf. [[Biharmonic function|Biharmonic function]]). Each poly-harmonic function is an analytic function of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351013.png" />. Some other properties of harmonic functions also carry over, with corresponding changes, to poly-harmonic functions.
+
A function  $  u( x) = u( x _ {1} \dots x _ {n} ) $
 +
of real variables defined in a region  $  D $
 +
of a Euclidean space  $  \mathbf R  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
having continuous partial derivatives up to and including the order  $  2m $
 +
and satisfying the poly-harmonic equation everywhere in  $  D $:
  
For poly-harmonic functions of any order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351014.png" />, representations using harmonic functions are generalized to get results known for biharmonic functions [[#References|[1]]]–[[#References|[5]]]. For example, for a poly-harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351015.png" /> of two variables there is the representation
+
$$
 +
\Delta  ^ {m} u  \equiv  \Delta ( \Delta \dots ( \Delta u))  = 0,\  m \geq  1,
 +
$$
  
<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/p073/p073510/p07351016.png" /></td> </tr></table>
+
where  $  \Delta $
 +
is the [[Laplace operator|Laplace operator]]. For  $  m = 1 $
 +
one obtains harmonic functions (cf. [[Harmonic function|Harmonic function]]), while for  $  m= 2 $
 +
one obtains biharmonic functions (cf. [[Biharmonic function|Biharmonic function]]). Each poly-harmonic function is an analytic function of the coordinates  $  x _ {j} $.  
 +
Some other properties of harmonic functions also carry over, with corresponding changes, to poly-harmonic functions.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351018.png" />, are harmonic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351019.png" />. For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351020.png" /> of two variables to be a poly-harmonic function, it is necessary and sufficient that it be the real (or imaginary) part of a [[Poly-analytic function|poly-analytic function]].
+
For poly-harmonic functions of any order  $  m > 1 $,  
 +
representations using harmonic functions are generalized to get results known for biharmonic functions [[#References|[1]]]–[[#References|[5]]]. For example, for a poly-harmonic function $  u $
 +
of two variables there is the representation
  
The basic boundary value problem for a poly-harmonic function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351021.png" /> is as follows: Find a poly-harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351022.png" /> in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351023.png" /> that is continuous along with its derivatives up to and including the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351024.png" /> in the closed region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351025.png" /> and which satisfies the following conditions on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351026.png" />:
+
$$
 +
u( x _ {1} , x _ {2} )  = \sum_{k=0}^ { m-1} r  ^ {2k} \omega _ {k} ( x _ {1} , x _ {2} ),\ \
 +
r  ^ {2} = x _ {1}  ^ {2} + x _ {2}  ^ {2} ,
 +
$$
  
<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/p073/p073510/p07351027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
where  $  \omega _ {k} $,
 +
$  k = 0 \dots m- 1 $,
 +
are harmonic functions in  $  D $.  
 +
For a function  $  u( x _ {1} , x _ {2} ) $
 +
of two variables to be a poly-harmonic function, it is necessary and sufficient that it be the real (or imaginary) part of a [[Poly-analytic function|poly-analytic function]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351028.png" /> is the derivative along the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351030.png" /> are given sufficiently smooth functions on the sufficiently smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351031.png" />. Many studies deal with solving problem (*) in the ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351032.png" /> [[#References|[1]]], [[#References|[6]]]. To solve the problem (*) in the case of an arbitrary region, one uses methods of integral equations, as well as variational methods [[#References|[1]]], [[#References|[6]]].
+
The basic boundary value problem for a poly-harmonic function of order  $  m > 1 $
 +
is as follows: Find a poly-harmonic function  $  u = u( x) $
 +
in a region  $  D $
 +
that is continuous along with its derivatives up to and including the order  $  m- 1 $
 +
in the closed region  $  \overline{D}\; = D \cup C $
 +
and which satisfies the following conditions on the boundary  $  C $:
 +
 
 +
$$ \tag{* }
 +
\left . \begin{array}{c}
 +
u \mid  _ {C}  =  f _ {0} ( y),
 +
\\
 +
 
 +
\left .  
 +
\frac{\partial  u }{\partial  n }
 +
\right | _ {C}  = \
 +
f _ {1} ( y) \dots \left .  
 +
\frac{\partial  ^ {m-1}u }{\partial  n  ^ {m-1} }
 +
\right | _ {C}  = \
 +
f _ {m-1} ( y),\  y \in C
 +
\end{array}
 +
\right \} ,
 +
$$
 +
 
 +
where  $  \partial  u / \partial  n $
 +
is the derivative along the normal to $  C $
 +
and $  f _ {0} ( y) \dots f _ {m-1} ( y) $
 +
are given sufficiently smooth functions on the sufficiently smooth boundary $  C $.  
 +
Many studies deal with solving problem (*) in the ball in $  \mathbf R  ^ {n} $[[#References|[1]]], [[#References|[6]]]. To solve the problem (*) in the case of an arbitrary region, one uses methods of integral equations, as well as variational methods [[#References|[1]]], [[#References|[6]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.N. Vekua,  "New methods for solving elliptic equations" , North-Holland  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. Privalov,  B.M. Pchelin,  "Sur la théorie générale des fonctions polyharmoniques"  ''C.R. Acad. Sci. Paris'' , '''204'''  (1937)  pp. 328–330  ''Mat. Sb.'' , '''2''' :  4  (1937)  pp. 745–758</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Nicolesco,  "Les fonctions poly-harmoniques" , Hermann  (1936)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Nicolesco,  "Nouvelles recherches sur les fonctions polyharmoniques"  ''Disq. Math. Phys.'' , '''1'''  (1940)  pp. 43–56</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C. Tolotti,  "Sulla struttura delle funzioni iperarmoniche in pui variabili independenti"  ''Giorn. Math. Battaglini'' , '''1'''  (1947)  pp. 61–117</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.N. Vekua,  "New methods for solving elliptic equations" , North-Holland  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. Privalov,  B.M. Pchelin,  "Sur la théorie générale des fonctions polyharmoniques"  ''C.R. Acad. Sci. Paris'' , '''204'''  (1937)  pp. 328–330  ''Mat. Sb.'' , '''2''' :  4  (1937)  pp. 745–758</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Nicolesco,  "Les fonctions poly-harmoniques" , Hermann  (1936)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Nicolesco,  "Nouvelles recherches sur les fonctions polyharmoniques"  ''Disq. Math. Phys.'' , '''1'''  (1940)  pp. 43–56</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C. Tolotti,  "Sulla struttura delle funzioni iperarmoniche in pui variabili independenti"  ''Giorn. Math. Battaglini'' , '''1'''  (1947)  pp. 61–117</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
See [[#References|[a1]]] for an updated bibliography and for a slightly more general definition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351033.png" /> is poly-harmonic on the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351034.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351035.png" /> locally uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073510/p07351036.png" />.
+
See [[#References|[a1]]] for an updated bibliography and for a slightly more general definition: $  u $
 +
is poly-harmonic on the domain $  \Omega $
 +
if  $  [ {| \Delta  ^ {n} u | } / {( 2n)! } ]  ^ {n/2} \rightarrow 0 $
 +
locally uniformly on $  \Omega $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Aronszain,  T.M. Creese,  L.J. Lipkin,  "Polyharmonic functions" , Clarendon Press  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Chelsea, reprint  (1986)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Aronszain,  T.M. Creese,  L.J. Lipkin,  "Polyharmonic functions" , Clarendon Press  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,  "Partial differential equations" , Chelsea, reprint  (1986)</TD></TR></table>

Latest revision as of 17:01, 13 January 2024


hyper-harmonic function, meta-harmonic function, of order $ m $

A function $ u( x) = u( x _ {1} \dots x _ {n} ) $ of real variables defined in a region $ D $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, having continuous partial derivatives up to and including the order $ 2m $ and satisfying the poly-harmonic equation everywhere in $ D $:

$$ \Delta ^ {m} u \equiv \Delta ( \Delta \dots ( \Delta u)) = 0,\ m \geq 1, $$

where $ \Delta $ is the Laplace operator. For $ m = 1 $ one obtains harmonic functions (cf. Harmonic function), while for $ m= 2 $ one obtains biharmonic functions (cf. Biharmonic function). Each poly-harmonic function is an analytic function of the coordinates $ x _ {j} $. Some other properties of harmonic functions also carry over, with corresponding changes, to poly-harmonic functions.

For poly-harmonic functions of any order $ m > 1 $, representations using harmonic functions are generalized to get results known for biharmonic functions [1][5]. For example, for a poly-harmonic function $ u $ of two variables there is the representation

$$ u( x _ {1} , x _ {2} ) = \sum_{k=0}^ { m-1} r ^ {2k} \omega _ {k} ( x _ {1} , x _ {2} ),\ \ r ^ {2} = x _ {1} ^ {2} + x _ {2} ^ {2} , $$

where $ \omega _ {k} $, $ k = 0 \dots m- 1 $, are harmonic functions in $ D $. For a function $ u( x _ {1} , x _ {2} ) $ of two variables to be a poly-harmonic function, it is necessary and sufficient that it be the real (or imaginary) part of a poly-analytic function.

The basic boundary value problem for a poly-harmonic function of order $ m > 1 $ is as follows: Find a poly-harmonic function $ u = u( x) $ in a region $ D $ that is continuous along with its derivatives up to and including the order $ m- 1 $ in the closed region $ \overline{D}\; = D \cup C $ and which satisfies the following conditions on the boundary $ C $:

$$ \tag{* } \left . \begin{array}{c} u \mid _ {C} = f _ {0} ( y), \\ \left . \frac{\partial u }{\partial n } \right | _ {C} = \ f _ {1} ( y) \dots \left . \frac{\partial ^ {m-1}u }{\partial n ^ {m-1} } \right | _ {C} = \ f _ {m-1} ( y),\ y \in C \end{array} \right \} , $$

where $ \partial u / \partial n $ is the derivative along the normal to $ C $ and $ f _ {0} ( y) \dots f _ {m-1} ( y) $ are given sufficiently smooth functions on the sufficiently smooth boundary $ C $. Many studies deal with solving problem (*) in the ball in $ \mathbf R ^ {n} $[1], [6]. To solve the problem (*) in the case of an arbitrary region, one uses methods of integral equations, as well as variational methods [1], [6].

References

[1] I.N. Vekua, "New methods for solving elliptic equations" , North-Holland (1967) (Translated from Russian)
[2] I.I. Privalov, B.M. Pchelin, "Sur la théorie générale des fonctions polyharmoniques" C.R. Acad. Sci. Paris , 204 (1937) pp. 328–330 Mat. Sb. , 2 : 4 (1937) pp. 745–758
[3] M. Nicolesco, "Les fonctions poly-harmoniques" , Hermann (1936)
[4] M. Nicolesco, "Nouvelles recherches sur les fonctions polyharmoniques" Disq. Math. Phys. , 1 (1940) pp. 43–56
[5] C. Tolotti, "Sulla struttura delle funzioni iperarmoniche in pui variabili independenti" Giorn. Math. Battaglini , 1 (1947) pp. 61–117
[6] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)

Comments

See [a1] for an updated bibliography and for a slightly more general definition: $ u $ is poly-harmonic on the domain $ \Omega $ if $ [ {| \Delta ^ {n} u | } / {( 2n)! } ] ^ {n/2} \rightarrow 0 $ locally uniformly on $ \Omega $.

References

[a1] N. Aronszain, T.M. Creese, L.J. Lipkin, "Polyharmonic functions" , Clarendon Press (1983)
[a2] P.R. Garabedian, "Partial differential equations" , Chelsea, reprint (1986)
How to Cite This Entry:
Poly-harmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poly-harmonic_function&oldid=12716
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article