Although I have read a lot of questions and answers here, this is my first time actually posting. Feel free to suggest needed edits.
My question is the following (in a simplified setting). All this is in $3$-d. Let's say we have two spheres with their centres distant of $d$, of radii $r$ and $1 - r$.
Denoting $C(r)$ the capacity of the set we're looking at, it's obvious that $C(\frac{1}{2})$ is a local extremum since $C$ is symmetric around $\frac{1}{2}$.
I suspect that the capacity of the union of the spheres is minimal when $r = \frac{1}{2}$. In fact I have made a few tests on my computer that would suggest so.
Does anyone have any idea of how I could prove that?
Maybe someone could point to a way to show $C$ is convex?
Another way to see this might lie on the remark that when $d = \infty$, $C$ is constant equal to $r + (1-r) = 1$, and that when $d = 0$, $C(r) = \max(r, 1-r)$, of which the minimum is attained at $\frac{1}{2}$. Therefore, one might assume that the cases $0<d<\infty$ is some kind of combination of the two above... But I haven't managed to formalize that either.
I have tried looking at Brownian motions started on a sphere surrounding the two spheres and playing with conditioning, without success. The more ''analytic" formulations of capacity are a little obscure to me, and I would prefer not having to dig into that if possible.
Thank you,
EDIT: This is another possibly interesting remark:
The capacity is the total energy at equilibrium in space : \begin{equation} \int_{\mathbb{R}^3 \setminus D} |\nabla u|^2 \end{equation}
where $u$ is the capacitary potential, ie the solution of the Dirichlet problem with boundary conditions $u|_{\partial D} = 1$ and $u = 0$ at infinity.
It's likely that in my problem, when the spheres have same radius, the symmetry should make that potential close to constant in the space between the spheres, which in turn should translate in a small capacity.