What is a harmonic function?

Let $$\mathcal{H} = \lbrace u: \mathbb{D} \rightarrow [0,1] | u_{xx} + u_{yy} = 0 \rbrace.$$
When I first looked at this, my initial reaction was “that looks like that Laplace condition.” Its turns out there is a direct connection to Laplace equations and Harmonic functions.

A solution of Laplace’s equation is called a “harmonic function” (for reasons explained below). Since the Laplace equation is linear, the sum of two or more individual solutions is also a solution.

Regularity theorem for harmonic functions

Harmonic functions are infinitely differentiable in open sets. In fact, harmonic functions are real analytic.

Maximum principle

Harmonic functions satisfy the following maximum principle: if K is a nonempty compact subset of U, then f restricted to K attains its maximum and minimum on the boundary of K. If U is connected, this means that f cannot have local maxima or minima, other than the exceptional case where f is constant. Similar properties can be shown for subharmonic functions.