(2021-07-27) Laplace equation & harmonic functions
Stationary solutions of the wave equation.
In three dimensions, the Laplace equation reads:
0
=
¶ 2 U
+
¶ 2 U
+
¶ 2 U
¶ x 2
¶ y 2
¶ z 2
=
DU
[D is the Laplacian operator]
Because of the Cauchy-Riemann equationa,
the real and imaginary parts of any analytic function
are two-dimensional harmonic functions.
Another two-dimensional harmonic function is:
Log ( x2 + y2 )
It can be construed as a special case of the previous kind as (twice) the real part of the
following multivalued function of a nonzero complex variable,
whose ambiguity resides only in the imaginary part (k is any integer):
Log (x+iy) = ½ Log (x2 + y2) +
[ Arg (x+iy) + 2kp ] i
Mean-value property
The value of an harmonic function at the center of a ball equal the mean-value of that
function over the surface of the ball.
Liouville's theorem for harmonic functions
Edward Nelson (1932-2014) observed that this is a simple consequence
of the above mean-value property:
Suppose U is a bounded harmonic function. Let A and B be two disctinct points
and consider two balls of radius R centered on them.
