10 - Ellipsoidal biharmonic functions

Summary

Eigensolutions of the ellipsoidal biharmonic equation

A function is called biharmonic if it is annihilated by two successive applications of the Laplacian. That is, if we apply the Laplace operator on a biharmonic function we end up with a harmonic function. The most amazing result concerning biharmonic functions was proved by Almansi in 1899 [4]. Almansi proved that, if U is a biharmonic function, then there exist harmonic functions u₁ and u₂ such that

U (r)=u₁(r)+r²u₂(r)

(10.1)

where r denotes the ordinary Euclidean distance.

The Almansi formula (10.1) provides an algebraic representation of a biharmonic function in terms of harmonic functions, and at a first glance it seems that, with this formula, we can solve boundary value problems for the biharmonic operator in ellipsoidal geometry. Indeed, replacing the functions u₁ and u₂ in (10.1) with ellipsoidal harmonics, we end up with ellipsoidal biharmonic functions. Since the ellipsoidal harmonics form a complete set of harmonic eigenfunctions, the corresponding biharmonics form a complete set of biharmonic eigenfunctions. Nevertheless, the effectiveness of these biharmonic eigenfunctions depends on their orthogonality properties, and these properties are not inherited from the orthogonality of the ellipsoidal harmonics, since the Euclidean distance is a function of ρ, μ, and ν. Note that since r is a spherical coordinate, the Almansi representation is tailor-made for problems in spherical geometry. In order to deal with the orthogonality problem we need a cumbersome algebraic analysis which is based on the exact form of the particular ellipsoidal harmonic. Therefore, only biharmonics of the few first degrees can be calculated in closed form.