For a metric space $(E,\rho)$ the $a$-capacity is defined as $$\mathrm{Cap}_{a}(E)=\left[\inf\left\{\int \int \frac{d\mu(x) \, d\mu(y)}{\rho(x,y)^{a}}:\mu\text{ probability measures on }E\right\}\right]^{-1}.$$
Newtonian Capacity: Set $E=\mathbb{R}^{d}$ with $d\geq 3$ and $a=d-2$ i.e.
$$\mathrm{Cap}_{d-2}(\mathbb{R}^{d})=\left[\inf \left\{\int \int \frac{d\mu(x) \, d\mu(y)}{\rho(x,y)^{d-2}}:\mu\text{ probability measures on }\mathbb{R}^{d}\right\}\right]^{-1}.$$
How can I compute the Newtonian capacity of Borel sets $A\subset \mathbb{R}^{d}$? Specifically, of intervals, discs?
Since there can be infinite measures, how can we be sure we found the inf?
Thanks
To begin with, the formulas you have define the minimal energy; the capacity is the reciprocal of that.
For a general set, there is no way to compute its Newtonian capacity analytically. One of rare exceptions is the ball $B=\{x : |x|\le R\}$. Based on the facts that the minimizing measure (i) exists; (ii) is unique; (iii) is supported on $\partial B$; one can conclude that it is the Lebesgue measure, by rotational symmetry.
But more often, the computation of capacity goes through the potential of the minimizing (equilibrium) measure, called the equilibrium potential:
$$U^\mu(x) = \int \frac{d\mu(y)} {|x-y|^{d-2}}$$ One can show that $U^\mu$ is harmonic on the complement of $E$ (this is not hard), decays like $|x|^{2-d}$ at infinity (this is clear), and is constant a.e. on $E$ (this is a nontrivial variational argument). Then the problem is to find such a function, i.e., solve a boundary value problem for the Laplacian. For nice sets (again, with a lot of symmetry) one can sometimes do this by separation of variables.
Once such $U$ is found, normalize it so that $\lim_{|x|\to\infty}|x|^{d-2} U(x)=1$; this amounts to normalizing the corresponding measure $\mu$. Then the value taken by $U$ on $E$ is exactly the capacity of $E$, since integration over $y$ will produce this value.
Recommended reading: Foundations of modern potential theory by Landkof, especially Chapter II. In section II.3 the author lists the values of the Newtonian capacity of several families of sets in $\mathbb R^3$. For example, the ball of radius $r$ has capacity $a/\pi$; the disk of radius $r$ has capacity $2a/\pi^2$; a line segment has capacity $0$.