Green's function for Laplace equation in the exterior of disks

Question

For two disks in $\mathbb{R}^2$, namely, $D_1(0)$ and $D_1(a)$ for some $a=(a_1,a_2)$ such that these two disks are non-overlapping. How can we find the Green function for the Dirichlet Laplace equation $\Delta u=0$ in the domain $\mathbb{R}^2 \setminus (\overline{D_1(0)} \cup \overline{D_1(a)})$? For a single disk, namely the domain $\mathbb{R}^2 \setminus \overline{D_1(0)}$, I think the Green function in this case reads $$ G(x,y) = \dfrac{1}{2\pi} \left[ -\log|x-y| + \log\left|x|y|-\dfrac{y}{|y|}\right| \right]$$ for $x,y \in \mathbb{R}^2$.

Answer 1

There is a Moebius transformation $\psi(z)$ of the Riemann sphere that maps the unit disk $D_1(0)$ to itself, and maps $D_1(a)$ to the complement of $D_R(0)$ for some $R>1$. To find $\psi$, first apply inversion $z \mapsto 1/z=\tilde{z}$, then compose it with $\tilde{z} \mapsto (\tilde{z}-s)/(1-s\tilde{z})=w$ for suitable $s$, and then apply one final inversion $w \mapsto 1/w$. Since Green's function is conformally invariant, it suffices to find the Green's function for the annulus. This is described in Appendix V of [1]. That paper also cites the earlier references [2] and [3].

[1]Appendix V in Engliš, Miroslav, and Jaak Peetre. "A Green's function for the annulus." Annali di Matematica Pura ed Applicata 171, no. 1 (1996): 313-377. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.630.1641&rep=rep1&type=pdf

[2] R. Courant - D. Hilbert, Methoden der Mathematischen Physik I. 3. Aufl. Springer-Verlag, Berlin - Heidelberg - New York, 1968. Page 335-337

[3] H. Villat, Le probl`eme de Dirichlet dans une aire annulaire. Rend. Circ. Mat. Palermo 33 (1912), 134-175.