Let $f$ and $g$ be distinct real-valued harmonic functions, which not merely differ by a constant or are not merely multiples of each other. Also, assume that the first order partial derivatives of the two functions do not vanish identically. Is it true that in such a case the expression $f_xg_x+f_yg_y+f_zg_z$ will attain positive as well as negative values?
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1We have $$\nabla^2 (fg) = \sum_{i} (fg){x_i,x_i} = \sum{i} \left(f_{x_i,x_i}g + fg_{x_i,x_i} 2 f_{x_i} g_{x_i}\right) = g \nabla^2 f + f \nabla^2 g + 2 (\vec{\nabla} f) \cdot (\vec{\nabla} g)$$ Since $f,g$ are harmonic, we have $$\nabla^2 (fg) = 2 (\vec{\nabla} f) \cdot (\vec{\nabla} g)$$ – Adhvaitha Jan 24 '16 at 08:43
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@Leg Thanks for pointing that out. But I still cannot see how $\Delta (fg)$ cannot be always $\ge 0$ or always $\le 0$. – vnd Jan 24 '16 at 08:57
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It depends on the chosen domain. If your expression is $>0$ at some point it will be strictly positive in a full neighborhood of this point. – Christian Blatter Jan 24 '16 at 09:21