Find harmonic $u : \mathbb{R}^2 \to \mathbb{R}$ such that $u(x, 0) = u_y(0,0) = 0.$
Without the last condition, we have $u = y.$ I'm trying to prove that if in addition $u>0$ on the upper half-plane and $u<0$ on the lower half-plane (some redundancy as we can show $u(x,y) = -u(x,-y)$), no such $u$ exists. In order to do so, I need to remove the last condition, find a function that works, and try to imagine what goes wrong when you add it back in.