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?
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