Harmonic functions with zeros on two lines

Solution 1:

The idea to use the reflection principle is good. When a harmonic function vanishes on a line (say, the real axis), redefining it in the lower half-plane as $\tilde (x,y)=-u(x,-y)$ also yields a harmonic function. Since $u=\tilde u$ in the upper half-plane, we conclude that $u\equiv \tilde u$, that is, $u(x,y)\equiv -u(x,-y)$. The same applies to reflections about other lines.

As a corollary: if a harmonic function vanishes on two lines $L_1$ and $L_2$, then it also vanished on the line $L_3$ that is obtained by reflecting $L_1$ about $L_2$. The process of reflection can be repeated. If the angle between $L_1$ and $L_2$ is not a rational multiple of $\pi$, then repeated reflections produce a dense set of lines on which the function vanishes. Therefore, it vanishes identically.

If the angle between $L_1$ and $L_2$ is a rational multiple of $\pi$, say $m\pi/n$, then such harmonic function exists, and can be obtained from the holomorphic function $z^n$.