Poly-harmonic function

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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. 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=48232
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article