How to solve the two-dimensional Laplace equation in this unbounded and multiply-connected domain?

Question

I have the Laplace equation $$\nabla^2 \phi(x,y) = 0$$ where $\phi$ is a function defined on the domain $\mathbb{R^2}\setminus(C_1 \cup C_2)$.

$C_1$ and $C_2$ are two circles of radius $r = 0.5$ and whose centers are $(0,1)$ and $(0,-1)$.

The boundary conditions are $\phi = 0$ on the bottom circumference, $\phi = 1$ on the upper circumference, and $\phi \rightarrow 0.5$ at infinity.

For reference, this problem is related to the calculation of the potential field in a two-wire transmission line. My question is the following: is it solvable analytically and, if so, how does one do it? I suppose this necessitates some sort of expansion into orthogonal functions, but what coordinate system would be the most appropriate?

Answer 1

Hint: The function $$ \phi(x,y)=\frac{1}{2}+k\ln\left(\frac{x^2+(y+a)^2}{x^2+(y-a)^2}\right) $$ is a solution to the Laplace equation satisfying $\phi\to\frac{1}{2}$ at infinity and such that the level curves $\phi(x,y)=c$ are circumferences with center on the $y$-axis. To solve your problem, find $a$ and $k$ such that $\phi=1$ on the upper circumference (by symmetry, $\phi=0$ on the bottom circumference).

Answer 2

The easiest way is use the result "harmonic iff locally real part of an analytic function" from complex analysis and transform the domain into something easier to work with.

So identifying $\mathbb{R}^2$ with $\mathbb{C}$, we choose centers $\pm\lambda i$ and send them to $0,\infty$ by a Mobius transformation $T$. We want the resulting boundary $C_1',C_2'$ to be concentric circles of radii $\rho,1/\rho$ centered at $0$, (this can be achieved by $T(0)=1$) so then we can just "read off" $$ \phi(z)=\frac{\log\lvert T(z)\rvert+\log\rho}{2\log\rho} $$ by uniqueness.

So it remains to find $\lambda$, which means solving $$ \frac{\lambda-\frac12}{\lambda+\frac12}= \frac{\frac32-\lambda}{\lambda+\frac32}=\rho $$ so $\lambda=\frac{\sqrt{3}}2$ and $\rho=2+\sqrt3$.