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- Jun 20, 2014

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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 $\partial \Omega$.]

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