In this post, I will prove Gauss MVT for harmonic functions.
Its statement is given as follows.
Note |p B_r| denotes the surface area and |B_r| denotes the volume. One can explicitly compute its measure and obtain
First I will prove the simplest case, 1-dimension and develop that idea to general dimension. They have same logical process.
First, Recall divergence theorem for 1-d.
Then Simply Let $F = grad u$ where $u$ is harmonic, we can do following computation.
Now integrate up to $R$
Reording this we have equation 1.1.
Now Let's do the general case
Here we do exactly same procedure as before. [Divergence theorem and radial differentiation]