I have a question on a harmonic function and the boundary behavior.

Let $\mathbb{U} \subset \mathbb{C}$ be a unit disk. We denote by $\overline{\mathbb{U}}$ the closure of $\mathbb{U}$ in $\mathbb{C}$.

We have a reflected Brownian motion $X=(\{X_t\}_{t \ge0}, \{P_x\}_{x \in \overline{U}})$ on $\overline{\mathbb{U}}$. Let $B:=B(a,r)$ be an open disk centered at $a \in \mathbb{C}$ with radius $r>0$ and $C$ a closed disk such that $C \subset \mathbb{U} \cap B$.

$u(x):=P_{x}(\sigma_{C}>\tau_{B})$ is a harmonic function with respect to $X$ on $(\overline{\mathbb{U}}\cap B) \setminus C$, which satisfies $\lim_{x \to \partial C}u(x)=0$. Here, we define

\begin{align*} \sigma_C=\inf\{t>0 \mid X_t \in C\},\\ \tau_B=\inf\{t>0 \mid X_t \notin B\}. \end{align*}

In other words, $u$ is a positive harmonic function on $ (\mathbb{U} \cap B) \setminus C$ with the Neumann boundary condition on $\partial \mathbb{U} \cap B$ and the Dirichlet boundary condition on $\partial C$.

**Question**

How $u(x)$ behave as $x \to \partial C$?

I am intersted in the rate of convergence of $\lim_{x \to \partial C}u(x)=0$.

Can we construct a nice positive harmonic function on $\mathbb{U} \setminus C$ with the Neumann boundary condition on $\partial \mathbb{U}$ and the Dirichlet boundary condition on $\partial C$?? If we know the behavior near $\partial C$, we should be able to obtain the behavior of $u$ near $\partial C$ by the boundary Harnack inequality.