Let $\lambda_1$ be the first positive eigenvalue of the following problem in the unit semiball $\mathbb{B}_+^n = \{x \in \mathbb{R}^n : \vert x \vert \leq 1 \text{ and } x_n \geq 0 \}$:
\begin{cases} \Delta u + \lambda u = 0 \, \quad \text{on } \, \mathbb{B}^n \\ u = 0 \, \quad \text{on } \, \mathbb{S}_+^{n-1} \\ \frac{\partial u}{\partial N} = 0 \, \quad \text{on } \, \mathbb{B}^n \cap \{x_n = 0\} \end{cases}
Here, $\mathbb{S}_+^{n-1}$ is the upper hemisphere of the unit sphere and $N = -e_n$ is the outward unit normal to $\mathbb{B}^n \cap \{x_n = 0\}$.
My question: is there an explicit formula for the inverse operator $(\Delta + \lambda_1)^{-1}$?
In more concrete terms: let $g \in C^{0,\alpha}(\mathbb{B}_+^n)$ and suppose that $g$ is orthogonal to the first eigenfunction of the above problem. Is there an explicit formula for the unique $f \in C^{2,\alpha}(\mathbb{B}_+^n)$ such that $(\Delta + \lambda_1)f = g$? In other words, is there an explicit Green function for the problem above?
If not for the semiball, is there such a formula for the whole ball $\mathbb{B}^n$?