 Harmonic odd reflection across a hyperplane

A harmonic function on one side of a hyperplane that is continuous up to an open portion of the hyperplane and zero there extends harmonically by odd reflection. One local proof takes a ball centered on the boundary and assigns odd continuous boundary data on its sphere. The Poisson integral on the ball is odd, so it vanishes on the equatorial disk. Uniqueness by the maximum principle for harmonic functions identifies it with the original function on the half-ball. Thus the odd extension is harmonic near each boundary point. A globally bounded zero-boundary harmonic function on a half-space is consequently zero by the Liouville theorem for harmonic functions.