Let $u\in C(2\overline{\mathbb{D}})$ be harmonic in $\mathbb{D}$, and also harmonic inside the annulus $\{1<z<2\}$. Suppose $v\in C(2\overline{\mathbb{D}})$ is another function that is harmonic in $2\mathbb{D}$ . Given that $u=v$ in $\overline{\mathbb{D}}$ , can we conclude that $u\equiv v$ in $2\mathbb{D}$.
P.S $2\mathbb{D}$ simply means the disk of radius two centered at the origin.
$$v(z)\equiv 0\quad \mbox{and}\quad u(z)=\cases{0& for $ 0\leq |z|\leq 1$\cr \log|z| & for $1\leq|z|\leq 2$.}$$