Thread: Problem Of The Week # 356 - Jan 21, 2020

1. Here is this week's POTW:

-----
Let $\phi_1$ and $\phi_2$ be harmonic functions on a bounded domain $\Omega \subset \mathbb R^3$ such that $\phi_1 \frac{\partial \phi_1}{\partial n} + \phi_2 \frac{\partial \phi_2}{\partial n} = \phi_2 \frac{\partial \phi_1}{\partial n} + \phi_1 \frac{\partial \phi_2}{\partial n}\quad \text{on}\quad \partial \Omega$
Prove that $\phi_1 = \phi_2$ everywhere in $\Omega$. [The operator $\frac{\partial}{\partial n}$ denotes the normal derivative on $\partial \Omega$.]

-----