= Even reflection across a homogeneous Neumann boundary
{title2=$\widetilde u(x,y)=u(|x|,y)$}
If a <harmonic function> on one side of a straight boundary has zero <normal derivative> there, even <reflection> extends it harmonically across the boundary. Locally take the boundary to be $x=0$ and assume $u$ and its first <derivatives> extend continuously to its open segment. Define $\widetilde u(x,y)=u(|x|,y)$. The zero value of $u_x$ makes the first <derivatives> match across the join. Applying <Green's first identity> on the two sides against a smooth compactly supported test function leaves no interface term, so the reflected function is weakly harmonic and hence harmonic by <Weyl lemma>.
Reflecting across both edges of a right-angle corner gives a <harmonic function> on a punctured neighbourhood when both <normal derivatives> vanish. If the original function is bounded at the corner, the <removable singularity for a bounded harmonic function> makes the extension harmonic there too. Thus a maximum on such an edge or corner can be tested using the interior <strong maximum principle for harmonic functions>.
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