suppose $u$ and $v$ are real harmonic function in a plane region $\omega$

Question

Suppose $u$ and $v$ are real harmonic function in a plane region $\omega$. Under what conditions is $uv$ harmonic?

Answer 1

Let $f=uv$. Then we have

$f_{xx}+f_{yy}=u(v_{xx}+v_{yy})+v(u_{xx}+u_{yy})+2(u_xv_x+u_yv_y)=2(u_xv_x+u_yv_y)$.

Your turn !