Properties of harmonic functions
Given is a volume V that is enclosed by the closed surface S.
Suppose f(x,y,z) is a function that satisfies the Laplace equation at
each point of a volume V:
Properties of f:
Possibly existing maxima and minima of f(x,y,z) lie on S.
At any point inside the volume V, f equals the average values of f
at the neighboring points.
Proof:
Expand f(x,y,z) around an arbitrary point A = (x0,y0,z0)
inside V. At a small distance d from A and in the
different space directions, f is then given by:
The
higher order terms can be neglected for sufficiently small d.
Summation
of these 6 values gives:
The
latter "="-sign is valid because f satisfies the Laplace equation in
point A.
We thus see that
Note
that this result is valid for an arbitrary choice of the Cartesian coordinate
system.
In
other words:
f(A) is equal to the average of the 6 neighboring points.
f is therefore not a maximum and not a minimum. As A is an
arbitrary point inside V, this means that possibly existing maxima and minima of f
can only lie on the surface S of V.
