Laplace's Equation on a Circle
Given Dirichlet boundary conditions on the perimeter of a circle, Laplace's equation can be solved for the value of the surface in the interior of the circle. This Demonstration does so for a particularly simple boundary condition given by cycles of a sine wave.
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.