Poly-harmonic function

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)