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Let $u\in C^2(\overline{B(0,1)})$ be harmonic $\Delta u = 0$ and not constant and let the function $u$ attain its maximum at $e_1=(1,0\dots0)\in\partial B(0,1)$. The task is to prove that $\partial _{x_1}u(e_1)>0$.

MY ATTEMPT: since the function is harmonic and hence smooth, it follows immediately $\partial _{x_1} u\geq 0$ due to maximum principle. Then I tried to use the approach that is used to prove maximum principle of harmonic functions in general, i.e. to add to the function some term that would yield sharp inequality, for example:

$$w(x)=u(x)+\varepsilon \exp(-\lambda|x|^2) + C.$$

However, I still can't see why this approach wouldn't yield a constant function $u$.

Thank you in advance!

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    What is the domain? harmonic where? the maximum over what set? why should $\partial_{x_1}u(e_1)$ exist? – 2017-01-30
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    Sorry for that, you are right I didn't specify the domain. The domain is the closed ball $\overline{B(0,1)}$ and the function $u$ is in$C^2(\overline{B(0,1)})$, hence the partial derivative exists. – 2017-01-30

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