Harmonic Functions

PDEs · Axiom Academy

Exploring the beautiful theory of functions satisfying Laplace's equation 1. Definition: The Laplace Equation The Laplacian operator ∇² represents the sum of all second partial derivatives. In two dimensions: A harmonic function has zero "curvature" when you average all directions around a point. Watch the animation to see how the Laplacian measures this balance. One of the most beautiful properties of harmonic functions is the mean value property : the value at any point equals the average of values on any sphere (or circle in 2D) centered at that point. In 2D, this becomes an integral over a circle. This property completely characterizes harmonic functions! Harmonic functions possess a remarkable property: they are infinitely differentiable (C ∞ ), regardless of how smooth the boundary data is! This is a profound result: solving Laplace's equation automatically "smooths out" any irregularities, as long as the solution is defined in the interior. This smoothness follows from the mean value property through careful analysis. Each derivative can be expressed as an average of nearby values. 4. Connection to Complex Analysis There is a deep connection between harmonic functions in 2D and complex analytic functions. This link is one of the most beautiful bridges in mathematics. Conversely, given any harmonic function u in a simply connected domain, there exists a harmonic conjugate v such that f = u + iv is analytic. Let's examine some fundamental examples of harmonic functions:

This is the written version of the interactive lesson above. See the full PDEs course.