Let $\Omega \subset \mathbb{R}^2$ be an annular (bounded and connected) domain with inner and outer boundary $\Gamma_1$ and $\Gamma_2$, respectively. It is known that the PDE system $$ \begin{eqnarray} \Delta u &=& 0 &\ \text{in}\ \Omega\\ u &=& f &\ \text{on}\ \Gamma_1\\ \frac{\partial u}{\partial \nu} &=& 0 &\ \text{on}\ \Gamma_2\\ \end{eqnarray} $$ has a unique solution on, say $H^1(\Omega)$, provided $\Omega$ and $f$ have sufficient regularity.

Suppose $w$ is harmonic in $\mathbb{R}^2\setminus(\Gamma_1 \cup \Gamma_2)$. If I can show that $w=0$ on $\Gamma_1$ and $\partial_\nu w = 0$ on $\Gamma_2$, then I can conclude (by uniqueness of the above PDE system with homogenous Dirichlet boundary condition) that $w=0$ on $\Omega$.

Now, my concern is as follows.

**First Question** With the same assumption on $w$, if I can show that $w = \partial_\nu w = 0$ on $\Gamma \subset \Gamma_1$, $\Gamma \neq \emptyset$, then can I conclude that $w=0$ in $\Omega$?

**Second Question** Suppose, again, $w$ is harmonic in $\mathbb{R}^2\setminus(\Gamma_1 \cup \Gamma_2)$. Do the conditions $w = 0$ on $\Gamma_1$ and $\partial_\nu w = 0$ on $\Gamma_2$ also imply that $\partial_\nu w + a w = 0 \ \text{on}\ \Gamma_2$. What if I only have $w = \partial_\nu w = 0$ on $\Gamma \subset \Gamma_1$, is it also true that $\partial_\nu w + a w = 0 \ \text{on}\ \Gamma_2$?

My thoughts:

For the first question, I am not sure if it holds, but if it is, I think it follows from Holmgren's theorem.

For the second one, I think it only holds when $a>0$. Am I correct? If not, can you please point out what might be the problem.

Thank you.