Schwarz reflection principle for harmonic functions

Remarks

Proof idea: $U$ is continuous on $\Omega$ and harmonic off $\sigma$, so only points of $\sigma$ need care. Take a closed ball centred at a point of $\sigma$ inside $\Omega$, and let $v$ be the Poisson integral of $U$ over its boundary sphere. The boundary data are odd in $x_n$, so $v$ is odd in $x_n$ and vanishes on the hyperplane. On each half ball, $U - v$ is harmonic and has boundary values $0$, so $U = v$ by the maximum principle. Example: $u(x) = x_n$ on the upper half-space vanishes on $x_n = 0$, and its reflection is $U(x) = x_n$ on all of $\R^n$, which is harmonic. The hypothesis $u = 0$ on $\sigma$ is needed: for $u \equiv 1$ the odd reflection jumps from $1$ to $-1$ across the hyperplane and is not continuous. For the plane, this principle and the Schwarz reflection principle for holomorphic functions imply each other through harmonic conjugates.

Going deeper

For a student

The reflection principle is a tool to extend harmonic functions across flat pieces of the boundary. It shows, for example, that a harmonic function on a half-ball that vanishes on the flat part is real analytic up to that flat part. Combined with the Kelvin transform, the same idea works for reflection in a sphere. The method of images for half-spaces rests on the same odd reflection.