ABSTRACT
As in the case of hyperbolic and parabolic equations, a unique solution of an elliptic equation can be obtained when some additional conditions are imposed. This chapter provides several examples of well-posed elliptic boundary value problems. From physical reasoning it is clear that, for instance, if the temperature distribution on the surface of a body is known, the solution of such a boundary value problem which consists of the Laplace or Poisson equation together with a boundary condition should exist and be unique. The physical sense of each of the boundary conditions is clear. The first boundary value problem when the surface temperature is prescribed is called Dirichlet’s problem. The chapter considers two-dimensional problems which have a symmetry allowing the use of polar coordinates. Boundary value problems for the Laplace equation in a rectangular domain can be solved with the method of separation of variables.
