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''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|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.

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

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

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|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} $[[#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====
<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====
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====
<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>

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[[Category:Mathematics]]