A question on Harmonic functions.

Question

$f$ and $g$ are two given continuous functions on $\overline {D(0,1)}$) and both $f$, $g:\overline {D(0,1)}\to\Bbb R$ are harmonic on unit disc $D(0,1)$. Now, If $f\equiv g$ on some $C=\{e^{i\theta} :a\lt\theta\lt b\}$ where $a,b \in[0,2\pi]$ and $a\lt b$. Does this information implies that $f\equiv g$ on $\overline {D(0,1)}$? Thanks in advance.

Answer 1

No, the identity theorem for harmonic functions isn't that strong.

For every continuous function $h \colon \partial D(0,1) \to \mathbb{R}$, there is a unique function $\overline{h}$ that is harmonic in $D(0,1)$ and continuous on $\overline{D(0,1)}$ with boundary values $h$.

So we can choose $h$ vanishing on the arc $\{e^{i\theta} : a \leqslant \theta \leqslant b\}$ but not identically, if $[a,b] \neq [0,2\pi]$, and then $g = f + \overline{h}$ coincides with $f$ on the specified arc, but not everywhere.

If $[a,b] = [0,2\pi]$, then $f$ and $g$ must be identical.