# 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 . 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 , . To solve the problem (*) in the case of an arbitrary region, one uses methods of integral equations, as well as variational methods , .