Nice way of thinking about the Laplace operator... but what's the proof?

Solution 1:

Given a point $p$ in the domain of the function $f$, consider the function which is the average value $A(r)$ of $f$ over the sphere $S_r$ of radius $r$ centred at $p$. If $S^{n-1}$ is the unit sphere in $\mathbb R^n$ then you can write the function as:

$$A(r) = \int_{S^{n-1}} f(p+rx)dx$$

I'm integrating with respect to the standard content on the sphere, but rescaled so that the sphere has unit content.

Differentiating $A(r)$ with respect to $r$ gives

$$\frac{dA}{dr} = \int_{S^{n-1}} x\cdot\nabla f(p+rx)dx$$

in the above equation $x \in S^{n-1}$ is a unit vector, and the $\cdot$ is dot product of vectors.

The above can be interpreted as a flux integral, so Gauss's theorem about flux integrals applies giving

$$\frac{dA}{dr} = r\int_{D^n} \nabla^2 f(p+rx)dx$$

now $x$ is a variable for the unit ball $D^n$.

So this tells you the first few terms of the Taylor expansion for the function $A$, in particular $A(0)=f(p)$, $A'(0)=0$, and so on with the next terms being some multiple of $\nabla^2f(p)$. A quick calculation says it should be $\frac{\nabla^2(f(p))}{n}$ but perhaps I've been too sketchy.