For some purpose, I have to calculate the Green's function $G$ such that $\Delta_\text{SdS} G=\delta$ for $\Delta_\text{SdS}$ the Laplacian in the four-dimensional Euclidean Schwarzschild-de-Sitter metric. Usually one goes as follows (I am working in a physics context, hence the notations): \begin{align} G(x,y)=&\int_0^\infty ds\,\langle x |e^{s\Delta_\text{SdS}} |y \rangle \\ =&\int_0^\infty ds \left( \frac{1}{16\pi^2 s^2} \Delta^{1/4}(x,y) \,e^{-\frac{\sigma(x,y)}{2s}} \sum_{n=0}^\infty s^n a_n(x,y)\right). \tag{1} \end{align} Where $\Delta$ is the van Vleck-Morette determinant, $\sigma(x,y)$ the squared geodesic distance from $x$ to $y$, and $a_n$ the Seeley-DeWitt coefficients.
I would have had to simply evaluate this expression at $x=y$ for my goal if my space-time hadn't boundaries... I can't find any literature on the evaluation of Green's functions in general Riemannian manifold with boundaries. More precisely my boundary is the horizon of the black hole of the SdS metric, thus the interior of the horizon is cut out from the manifold. And my boundary condition is $G(x_\partial,y)=0$ for $x_\partial$ on the boundary.
So how should I proceed in order to take into account the boundary condition in this case and the boundary contributions? I know the formula for the trace of the heat kernel in a Riemannian manifold with boundaries, but I find nothing on the internet for Green's functions...