12c. Harmonic functions

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As another application of the second derivatives, we can now talk about harmonic functions in general, extending what we know from chapter 8. We first have:

Definition

The Laplace operator in [math]N[/math] dimensions is:

[[math]] \Delta f=\sum_{i=1}^N\frac{d^2f}{dx_i^2} [[/math]]

A function [math]f:\mathbb R^N\to\mathbb C[/math] satisfying [math]\Delta f=0[/math] will be called harmonic.

Observe that this generalizes what we know in two dimensions, from chapter 8. In order to discuss this, let us first recall the handful of results that we know from there, in [math]N[/math] dimensions, with the comment of course that chapter 8 was a physics chapter, and now that we learned mathematics in [math]N[/math] dimensions, you can check all that stuff, which was coming a bit in advance, and correct all that stuff is. First is the validity of the wave equation in [math]N[/math] dimensions, which is as follows, [math]v \gt 0[/math] being the propagation speed:

[[math]] \ddot{\varphi}=v^2\Delta\varphi [[/math]]

Then second is the validity of the heat equation in [math]N[/math] dimensions, which is as follows, with [math]\alpha \gt 0[/math] being the thermal diffusivity of the medium:

[[math]] \dot{\varphi}=\alpha\Delta\varphi [[/math]]

And finally, we had as well some mathematics in chapter 8, but mostly 2-dimensional, and not very useful for our purposes here. Let us recall however a key computation, that we did in [math]N[/math] dimensions, stating that the fundamental radial solutions of [math]\Delta f=0[/math] are as follows, with the [math]\log[/math] at [math]N=2[/math] basically coming from [math]\log'=1/x[/math]:

[[math]] f(x)=\begin{cases} ||x||^{2-N}&(N\neq 2)\\ \log||x||&(N=2) \end{cases} [[/math]]

Not bad for a start, all this. In analogy now with the one complex variable theory that we know from chapter 6 and chapter 8, we have the following result:

Theorem

The harmonic functions in [math]N[/math] dimensions obey to the same general principles as the holomorphic functions, namely:

The plain mean value formula.
The boundary mean value formula.
The maximum modulus principle.
The Liouville theorem.

Show Proof

This is something quite tricky, the idea being as follows:

(1) Regarding the plain mean value formula, here the statement is that given an harmonic function [math]f:X\to\mathbb C[/math], and a ball [math]B[/math], the following happens:

[[math]] f(x)=\int_Bf(y)dy [[/math]]

In order to prove this, we can assume that [math]B[/math] is centered at [math]0[/math], of radius [math]r \gt 0[/math]. If we denote by [math]\chi_r[/math] the characteristic function of this ball, nomalized as to integrate up to 1, in terms of the convolution operation from chapter 7, we want to prove that we have:

[[math]] f=f*\chi_r [[/math]]

For doing so, let us pick a number [math]0 \lt s \lt r[/math], and a solution [math]w[/math] of the following equation, supported on [math]B[/math], which can be constructed explicitly:

[[math]] \Delta w=\chi_r-\chi_s [[/math]]

By using the properties of the convolution operation [math]*[/math] from chapter 7, we have:

[[math]] \begin{eqnarray*} f*\chi_r-f*\chi_s &=&f*(\chi_r-\chi_s)\\ &=&f*\Delta w\\ &=&\Delta f*w\\ &=&0 \end{eqnarray*} [[/math]]

Thus [math]f*\chi_r=f*\chi_s[/math], and by letting now [math]s\to0[/math], we get [math]f*\chi_r=f[/math], as desired.

(2) Regarding the boundary mean value formula, here the statement is that given an harmonic function [math]f:X\to\mathbb C[/math], and a ball [math]B[/math], with boundary [math]\gamma[/math], the following happens:

[[math]] f(x)=\int_\gamma f(y)dy [[/math]]

But this follows as a consequence of the plain mean value formula in (1), with our two mean value formulae, the one there and the one here, being in fact equivalent, by using annuli and radial integration for the proof of the equivalence, in the obvious way.

(3) Regarding the maximum modulus principle, the statement here is that any holomorphic function [math]f:X\to\mathbb C[/math] has the property that the maximum of [math]|f|[/math] over a domain is attained on its boundary. That is, given a domain [math]D[/math], with boundary [math]\gamma[/math], we have:

[[math]] \exists x\in\gamma\quad,\quad |f(x)|=\max_{y\in D}|f(y)| [[/math]]

But this is something which follows again from the mean value formula in (1), first for the balls, and then in general, by using a standard division argument.

(4) Finally, regarding the Liouville theorem, the statement here is that an entire, bounded harmonic function must be constant:

[[math]] f:\mathbb R^N\to\mathbb C\quad,\quad\Delta f=0\quad,\quad |f|\leq M\quad\implies\quad f={\rm constant} [[/math]]

As a slightly weaker statement, again called Liouville theorem, we have the fact that an entire harmonic function which vanishes at [math]\infty[/math] must vanish globally:

[[math]] f:\mathbb R^N\to\mathbb C\quad,\quad\Delta f=0\quad,\quad\lim_{x\to\infty}f(x)=0\quad\implies\quad f=0 [[/math]]

But can view these as a consequence of the mean value formula in (1), because given two points [math]x\neq y[/math], we can view the values of [math]f[/math] at these points as averages over big balls centered at these points, say [math]B=B_x(R)[/math] and [math]C=B_y(R)[/math], with [math]R \gt \gt 0[/math]:

[[math]] f(x)=\int_Bf(z)dz\quad,\quad f(y)=\int_Cf(z)dz [[/math]]

Indeed, the point is that when the radius goes to [math]\infty[/math], these averages tend to be equal, and so we have [math]f(x)\simeq f(y)[/math], which gives [math]f(x)=f(y)[/math] in the limit, as desired.

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General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].