I'm going through Axler's proof of the mean value property, and I'm a little puzzled. Since this proof is in the first few pages of his book, I thought I'd ask for clarification before going further into the book.
He begins by fixing $0 < \epsilon < 1$ and defining $\Omega = \{x \in \mathbb{R}^{n} : \epsilon < |x| < 1\}$. Letting $u(x)$ be harmonic and $v(x) = |x|^{2-n}$, he says we should apply Green's identity under these conditions. Recall Green's identity is:
$$\int_{\Omega} (u \,\Delta{v} - v\, \Delta{u})\,dV = \int_{\partial\Omega} (u \,\nabla{v} - v \,\nabla{u})\,ds$$
From this he deduces that
$$0 = (2-n) \int_S u\, ds - (2-n)\epsilon^{1-n} \int_{\epsilon S} u\, ds - \int_S \frac{\partial u}{\partial n} \, ds -\epsilon^{2-n} \int_{\epsilon S} \frac{\partial u}{\partial n} \, ds.$$
but I don't see how to actually compute this. I've tried it myself, but there's obviously something I am missing. For instance, he has 4 terms, but I would figure there'd only be 3 since $|x|^{2-n} \Delta{u} = 0$ since $u$ is harmonic.
Other than being mystified by the computation, I'm not sure what he means by $\epsilon{S}$. Here, $S$ refers to $S^{n-1}$, but I'm not sure that $\epsilon{S}$ represents. How does it even come about?