Potential theory has its origins in gravitational theory and electromagnetic theory. The common element of these two is the inverse square law governing the interaction of two bodies. The concept of potential function arose as a result off the work done in moving a unit charge from one point of space to another in the presence of another charged body. A basic potential function 1/r, the reciprocal distance function, has the important property that it satisfies Laplace's equation except when r = 0. Such a function is called a harmonic function. Since the potential energy at a point due to a distribution of charge can be regarded as the sum of a large number of potential energies due to point charges, the corresponding potential function should also satisfy Laplace's equation. Applications to electromagnetic theory led to the problem of determining a harmonic function on a region having prescribed values on the boundary of the region. This problem came to be known as the Dirichlet problem. A similar problem, connected with steadystate heat distribution, asks for a harmonic function with prescribed flux or normal derivative at each point of the boundary. This problem is known as the Neumann problem. Another problem, Robin's problem, asks for a harmonic function satisfying a condition at points of the boundary which is a linear combination of prescribed values and prescribed flux.
In this chapter, explicit formulas will be developed for solving the Dirichlet problem for a ball in n-space, uniqueness of the solution will be demonstrated, and the solution will be proved to have the right “boundary values.” Not nearly as much is possible for the Neumann problem. Explicit formulas for solving the Neumann problem for a ball are known only for the n = 2 and n = 3 cases.
KeywordsHarmonic Function Dirichlet Problem Neumann Problem Borel Subset Orthogonal Transformation
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