Kakutani's Solution of the Dirichlet Problem
Kakutani's Solution of the Dirichlet Problem
Choose one of the three functions using the setter bar. Move the locator to a point in the unit circle. You will see the value of the selected function at this point displayed below the circle. Now increase the duration of motion. A Brownian path will emerge from the chosen point. Keep increasing the duration until the Brownian path hits the boundary of the circle. The value of the chosen function at the point where the path hits the boundary will appear below the value of the function at the point from which the path emerged. Now increase the number of paths, increasing the time duration, if necessary, to make sure they all hit the boundary. If the selected function is harmonic (the first and second choices), the average of the values of the function at the points where the Brownian paths hit the boundary will converge to the value of the function at the starting point of the path. If the selected function is not harmonic (the third choice), the average will converge to the value of a different (harmonic) function that coincides with the given function on the boundary.