Problem of Harmonic function.

Question

If H is a harmonic function on an unit disk; And $H=0$ on $R_1\cup R_2$, here $R_1, R_2$ are radius of $D(0,1)$. The angle between $R_1$ and $ R_2$ is $r\pi$; here $r\in (0,1]$. If $r$ is an irrational number then is $H$ identically zero on $D(0,1)$?

---

I think if $r$ is irrational then $H\equiv0$ on unit disk $D$; and following lemma can help us to prove it.

Lemma: If $a,b$ are two semilines starting at the same point $O$ there exists $u \not\equiv 0$ is a harmonic function vanishing on $a,b$ if and only if there exists $r \in \mathbb{Q}$ such that the angle between a and b is $r\pi.$

Proof of the Lemma: Using the following property of harmonic function we can prove this lemma.

1. A harmonic function vanishing on an open set is identically zero.

2. (Schwarz Reflection Principle) If a harmonic function is defined in a neighborhood of a line segment contained in one of the halfplanes determined by that segment and it is continued to zero on that segment, then it can be extended harmonically to the symmetric region by the given line segment.

Answer 1

Assuming a semiline need only exist within the domain of the function (I am unsure of the definition), we have the following proof:

Note that $R_1$ and $R_2$ are two semilines starting from the same point such that the angle between them is not a rational multiple of $\pi$, and that $H$ is harmonic on the unit disk, which is a neighborhood of these two segments. By our lemma, no non-zero harmonic function vanishes on $R_1 \cup R_2$. We conclude that $H$ must be identically zero.

Answer 2

The Maximum and Minimum Principle of Harmonic functions imply that the answer of your question is positive.