My attempt:
Hi, there! I have known how to prove the above statement when $u\in C^2(U)$, however, I have question about proving the above statement. Because it is $u\in C^2(U^{+}) \cap C(\overline{ U^+})$.
It is really bothering me and I really need help. Thanks a lot.:)
It is sufficient to verify the mean value property for sufficient small balls for every $x\in \Omega^+ \cup T \cup \Omega^-$. This is clearly true for $x$ such that $x_n>0$, since we start with a function that is harmonic in $\Omega ^+$. By reflection, the function is also harmonic on $\Omega ^-$. Finally, when $x_n=0$, the integral over the upper half ball cancels that over the lower half ball. so we found $U$ is harmonic on the domain $\Omega^+ \cup T \cup \Omega^-$