Consider the following the theorem in the classical PDE book of Evans (chapter 2):
(Part of the strong maximum principle) Let $U$ a open set in $R^n$ and $u \in C^2 (U) \cap C(\overline{U})$, with $\Delta u = 0$ in $U$.
If $U$ is connected and there exists a point $x_0 \in U$ such that $$ u(x_0) = \displaystyle\max_{\overline{U}} u$$ then $u$ is constant within $U$.
Part of the proof:
Suppose there exists a point $x_0 \in U$ with $u(x_0) = M = \displaystyle\max_{\overline{U}} u . $ Then for $0 < r < \mbox{dist} (x_0 , \partial U)$, the mean value property asserts $$ M = u(x_0) = \displaystyle\frac{\displaystyle1}{|B(x_0, r)|}\int_{B(x_0,r) } u \ dy \leq M.$$
Then $$u = M\quad\text{in}\quad B(x_0 , r)\tag{$*$}.$$
I dont understand the equality in $(*)$. If I be non rigorous, for me is clear to see the equality in $(*)$. But I dont know how to prove the equality... Someone can give me a hint about how to prove the equality in $(*)$?
Thanks in advance!