# Poly-harmonic function

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

A function of real variables defined in a region of a Euclidean space , , having continuous partial derivatives up to and including the order and satisfying the poly-harmonic equation everywhere in :

where is the Laplace operator. For one obtains harmonic functions (cf. Harmonic function), while for one obtains biharmonic functions (cf. Biharmonic function). Each poly-harmonic function is an analytic function of the coordinates . Some other properties of harmonic functions also carry over, with corresponding changes, to poly-harmonic functions.

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

where , , are harmonic functions in . For a function 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 is as follows: Find a poly-harmonic function in a region that is continuous along with its derivatives up to and including the order in the closed region and which satisfies the following conditions on the boundary :

 (*)

where is the derivative along the normal to and are given sufficiently smooth functions on the sufficiently smooth boundary . Many studies deal with solving problem (*) in the ball in [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)