Laplace's Equation on a Circle

Laplace's equation in two dimensions is given by

.

Let a circular membrane have a Dirichlet condition everywhere on the boundary, where the condition is for . The formal solution is

(P. Franklin, An Introduction to Fourier Methods and the Laplace Transformation, New York: Dover, 1958 p. 158). Solutions to Laplace's equation are called harmonic functions. One of the properties of harmonic functions is that they will not attain any local minima or maxima inside the boundary; thus the minima and maxima are on the boundaries, as defined by the Dirichlet conditions. Another property is that the solution at any point has a value that is the average of the values over the area of a circle defined with at its center.