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This question has been discussed before: here and here

If we're given a harmonic function in a region that's continuous on the boundary (say we have a smooth boundary), such that the function and it's normal derivative are both zero on a segment of the boundary, then the conclusion is that the function must be identically zero.

My question is: in the second link above, someone gave a solution that I am 99% satisfied with minus the fact that one fact (?), which is crucial to the proof, is given without proof:

If $u$ is $C^1$ and satisfies $\int_C \frac{\partial u}{\partial \nu}=0$ for all sufficiently small circles $C$ centered at $x$, then $u$ is harmonic at $x$.

I'm more or less convinced that this must be true. It is also very similar to the converse to the mean value theorem - that a continuous function satisfying the mean value property (say at a point $x$) must in fact be harmonic at $x$. But could someone provide me with a rigorous proof?

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I found a PDF in which there is a proof of the following: $$\text{Let } \Omega \subset \mathbb{R}^n \text{ and } \varphi \in C(\Omega) \text{ satisfy the mean value property.} \\ \text{ Then } \varphi \text{ is smooth and harmonic in } \Omega.$$ It's Theorem 8 of the 2nd page of http://www.math.harvard.edu/~canzani/math253/Lecture5.pdf

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    This is relatively well established - I'm trying to look for the proof for the analogue for $d\phi/d\nu$ i.e. when $d\phi/d\nu$ satisfies the mean value property. – 2017-02-03
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    Isn't the last part of the proof of the theorem what you were looking for? – 2017-02-04
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    hmm now that you mention it, it does look like the last part would bring me closer to a proof than before. but i don't think it's complete quite yet because the last part works after it's been shown that $\phi$ is smooth (and hence it makes sense to consider $\Delta \phi$; before we show smoothness, we only have $C^1$ from the hypothesis). And the proof showing that $\phi$ is smooth uses the mean value property (the standard one). But having said all that, it does seem plausible that one could tweak the proof for smoothness so that it involves normal derivatives somehow. – 2017-02-04