= 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>.
Back to article page
