# Schwarz Reflection Principle for Harmonic Functions

Given $\Omega \subset \mathbb{R}^n$ define $\Omega^+ = \Omega \cap \{x_n>0\}$ and $\Omega^0$, $\Omega^-$ analogously let $u \in C^2(\bar{\Omega}^+)$ be harmonic and such that $\frac{\partial u}{\partial x_n}=0$. Then we want to show that the reflection of $u$, i.e. $\tilde{u}=u(x,-x_n)$ for $(x,x_n) \in \Omega^-$, is harmonic on $\Omega^+ \cup \Omega^0 \cup \Omega^-$.

The mean value property seems like the easiest way to go about showing this since it easily gives us that the reflection is harmonic on $\Omega^-$.

How do I use $\frac{\partial u}{\partial x_n}=0$ to conclude that the reflection must be harmonic on the whole of $\Omega^+ \cup \Omega^0 \cup \Omega^-$ though?

The condition $\frac{\partial u}{\partial x_n}=0$ is put in place just to ensure that the extended function is smooth along the hyperplane $x_n=0$. Simply consider the easy example where $n=1$ and $u(x)=kx$ for $x>0$ where $k$ is some constant. In order to have a differentiable extension obtained by reflection you must have $k=0$.