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It suppose to be a easy task. But I couldn't solve it (I guess I can't learn much analysis). If $ u $ is harmonic, in the middle of my problem, I need to prove that the integral $ \int _ {\partial \Omega } \displaystyle\frac{\partial u}{\partial v} d s_x =0 $ THis $ v $ denotes the unit exterior normal.

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2 Answers 2

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Using Green's first identity,

$$ \int_{\partial\Omega}\frac{\partial u}{\partial \nu}\,ds = \int_{\Omega} (\Delta u + \nabla 1 \cdot \nabla u)\,dV = 0, $$ since $\Delta u = 0$ by assumption and $\nabla 1 = \mathbf{0}$.

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It is just a consequence of Divergence theorem, isn't it? You have that u is harmonic, so $$ \Delta u =\nabla\cdot \nabla u=0. $$ Now integrate over $\Omega$ and apply the divergence theorem.

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