Abstract
In the present paper, electrostatic image theory is studied for Green’s function for the Laplace operator in the case where the fundamental domain is either the exterior or the interior of a prolate spheroid. In either case, an image system is developed to consist of a point image inside the complement of the fundamental domain and an additional symmetric continuous surface image over a confocal prolate spheroid outside the fundamental domain, although the process of calculating such an image system is easier for the exterior than for the interior Green’s function. The total charge of the surface image is zero and its centroid is at the origin of the prolate spheroid. In addition, if the source is on the focal axis outside the prolate spheroid, then the image system of the exterior Green’s function consists of a point image on the focal axis and a line image on the line segment between the two focal points.
Highlights
Let Ω be a regular bounded three-dimensional domain and ∂Ω be its boundary
An image system for the Laplace operator in a domain is a system of fictitious sources inside the complement of the fundamental domain that produces a potential satisfying the prescribed boundary conditions, namely, R(r, rs)
The image system constructed consists of a Kelvin-image type point image inside the complement of the fundamental domain and an additional continuous surface image over a confocal prolate spheroid outside the fundamental domain, the mathematical process of calculating such an image system is easier for the exterior than for the interior Green’s function
Summary
Let Ω be a regular bounded three-dimensional domain and ∂Ω be its boundary. the interior Green’s function Gi for the Laplace operator in Ω is the solution of the following interior boundary value problem. (1b) where rs is a given point inside Ω, while the exterior Green’s function Ge for the Laplace operator in Ωc is the solution of the following boundary value problem. An image system for the Laplace operator in a domain is a system of fictitious sources inside the complement of the fundamental domain that produces a potential satisfying the prescribed boundary conditions, namely, R(r, rs). These fictitious sources are commonly called images because they are located not in the real domain of interest for the problem but in its complement.
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