Maximum principle, subharmonic and superharmonic functions

The Laplace operator Δ of a function of n variables is defined by

$\Delta f = \sum_{i=1}^n \frac{\partial^2 f}{\partial x^2_i}$

If Δ f = 0 in some region Ω, f is said to be harmonic on Ω. In that case f takes on its maximum and minimum over Ω at locations on the boundary ∂Ω of Ω. Here is an example of a harmonic function over a square which clearly takes on its maximum on two sides of the boundary and its minimum on the other two sides.

Plot of x^2 - y^2 over [-1,1] cross [-1,1].

The theorem above can be split into two theorems and generalized:

If Δ f ≥ 0, then f takes on its maximum on ∂Ω.

If Δ f ≤ 0, then f takes on its minimum on ∂Ω.

These two theorems are called the maximum principle and minimum principle respectively.

Now just as functions with Δf equal to zero are called harmonic, functions with Δf non-negative or non-positive are called subharmonic and superharmonic. Or is it the other way around?

If Δ f ≥ 0 in Ω, then f is called subharmonic in Ω. And if Δ f ≤ 0 then f is called superharmonic. Equivalently, f is superharmonic if –f is subharmonic.

The names subharmonic and superharmonic may seem backward: the theorem with the greater than sign is for subharmonic functions, and the theorem with the less than sign is for superharmonic functions. Shouldn’t the sub-thing be less than something and the super-thing greater?

Indeed they are, but you have to look f and not Δf. That’s the key.

If a function f is subharmonic on Ω, then f is below the harmonic function interpolating f from ∂Ω into the interior of Ω. That is, if g satisfies Laplace’s equation

$\begin{align*} \Delta g &= 0 \mbox{ on }\Omega \\ g &= f \mbox{ on } \partial\Omega \end{align*}$

then f ≤ g on Ω.

For example, let f(x) = ||x|| and let Ω be the unit ball in ℝⁿ. Then Δ f ≥ 0 and so f is subharmonic. (The norm function f has a singularity at the origin, but this example can be made rigorous.) Now f is constantly 1 on the boundary of the ball, and the constant function 1 is the unique solution of Laplace’s equation on the unit ball with such boundary condition, and clearly f is less than 1 on the interior of the ball.

Which one is subharmonic?

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