Directional dependent Green’s function and Kelvin images

Abstract

The inverse of a linear differential operator is an integral operator with a kernel which is commonly known as the Green’s function of the differential operator. Therefore, the knowledge of the Green’s function of a linear problem leads directly to an integral representation of its solution. Any Green’s function is split into a singular part that carries the localized singularity of the Dirac measure and a regular part that is controlled by the Dirichlet boundary condition. In some relatively simple cases, this regular part can be interpreted as the contribution of imaginary sources which lie in the complement of the fundamental domain. If a problem is associated with the Laplace operator, such as the biharmonic operator or the Papkovitch potentials, which both govern Linear Elastostatics, the construction of such Green’s functions are of extremely large importance. All these are well-behaving procedures as long as we live in the highly symmetric geometry represented by the spherical system. But, if we live in a directional-dependent environment, such as the one imposed by the ellipsoidal geometry, the above procedures become extremely complicated, if not impossible. In the present work, the Green’s functions and their Kelvin image systems are obtained for the interior and the exterior regions of an ellipsoid. It is amazing, although not unjustified, that besides the point image source, that is needed for the isotropic spherical case, in the case of ellipsoidal domains, the necessary image system involves a full two-dimensional distribution of imaginary sources to account for the anisotropic character of the ellipsoidal domains.