Potential distribution, torsion problem and conformal mapping for a doubly-connected region with inner elliptic contour

This paper deals with three boundary value problems of the Laplace equation: 1. (1) potential distribution of a doubly-connected region with inner circular or elliptic boundary—a problem in electrostatics. 2. (2) Saint-Venant torsion of a bar with same region—a problem in elasticity. 3. (3) conformal mapping of the same region into a ring region—a problem in complex variable function. It is proved here that the first is a Dirichlet problem while the second and third are modified Dirichlet problems of Laplace's equation, and a modified Dirichlet problem can be easily reduced to two Dirichlet problems. In order to solve the Dirichlet problems, an eigenexpansion form is proposed. The eigenexpansion form always satisfies the Laplace equation and a particular condition along the inner boundary. The undetermined coefficients in the eigenexpansion form are determined by the use of the variational principle. Several numerical examples and calculated results are given.