Ngô Quốc Anh

September 26, 2010

Subharmonic functions

Filed under: Giải tích 7 (MA4247) — Ngô Quốc Anh @ 1:53

In this entry, we shall discuss a geometric meaning of subharmonic functions. This will help us to easily remember the definition of subharmonic functions.

In mathematics, a harmonic function is a twice continuously differentiable function f : U\to \mathbb R (where U is an open subset of \mathbb R^n) which satisfies Laplace’s equation, i.e.

\displaystyle\frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0

everywhere on U. This is usually written as

\textstyle \Delta f = 0.

In 1D, this condition is about to say that f is harmonic if and only if f is linear. Concerning to the case of functions with one-variable, we have the s0-called convexity saying that function f is convex if and only if the function lies below or on the straight line segment connecting two points, for any two points in the interval. Mathematically, a function f is said to be convex if

\textstyle \Delta f \geqslant 0.

In higher dimension, the notion of linearity and convexity become harmonicity and subharmonicity. Precisely, two points mentioned above become a hyper-surface, for e.g. like a curve in 2D and a straight line becomes a graph of harmonic function. In practice, the closed interval connecting those two points will be replaced by a closed ball. Therefore, we have

Definition. A C^2 function that satisfies \Delta f \ge 0 is called subharmonic. More generally, a function is subharmonic if and only if, in the interior of any ball in its domain, its graph lies below that of the harmonic function interpolating its boundary values on the ball.

Let us consider several examples in 2D.

  • \log functions.

It is well-known that in 2D function \log|z|, where z=(x,y), is harmonic. Therefore, every functions lying below the graph of \log|z| turns out to be subharmonic.

  • \sin functions.

Again, one can easily show that e^x \sin y is harmonic.

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