Even reflection across a homogeneous Neumann boundary ($\tilde{u}(x,y)=u(|x|,y)$)

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 $\tilde{u}(x,y)=u(|x|,y)$. 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.