V(x)<+\infty\}$ and $\Sigma_V$ is not log-polar.

4.  Let $$\label{cvdef}
     c_V:=I_V(\mu_V)-\int_{{\mathbb R}^2}\frac{V(x)}{2}d\mu_V(x)$$ denote the Robin constant. Then we have the following Frostman variational inequalities : $$\begin{aligned}
     \label{Frost1}& U^{\mu_V}(x)+\frac{V(x)}{2}\geq c_V \quad \text{q.e. on } {\mathbb R}^2,\\
    \label{Frost2} & U^{\mu_V}(x)+\frac{V(x)}{2}\leq c_V \quad \text{for all } x\in\Sigma_V.\end{aligned}$$

In particular we have $\displaystyle U^{\mu_V}(x)+\frac{V(x)}{2}=c_V$ q.e. on $\Sigma_V$.

As explained in [@Hardy], the hypothesis of Theorem \[equi\] can be usefully transported to the sphere $\mathcal{S}$ in ${\mathbb R}^3$ centred at $(0,0,1/2)$ with radius $1/2$, by the inverse stereographic projection $T:{\mathbb R}^2\to \mathcal{S}$ defined by $$T(x_1,x_2)=\left(\frac{x_1}{1+|x|^2}, \frac{x_2}{1+|x|^2},\frac{|x|^2}{1+|x|^2} \right), \text{for any } x=(x_1,x_2)\in{\mathbb R}^2.$$ We know that $T$ is a conformal homeomorphism from ${\mathbb R}^2$ to $\mathcal{S}\backslash \{N\}$ where $N :=(0,0,1)$ is the North pole of $\mathcal{S}$.

The procedure is as follows: Given $V:{\mathbb R}^2\to{\mathbb R}$, we may define (see [@Hardy]) $\mathcal V:\mathcal S\to {\mathbb R}$ by letting $$\label{transport}\mathcal V(T(x)) = V(x) - \log(1+|x|^2), \quad \mathcal V(N) = \liminf_{|x|\to +\infty} \{V(x) - \log(1+|x|^2)\}.$$ Then (see [@Hardy]), $V$ satisfies the hypothesis of Theorem \[equi\] if and only if $\mathcal V$ is a lowersemicontinuous function on $\mathcal S$ which is finite on a nonpolar set. Therefore, in this case, the minimum of $$I_{\mathcal V}(\mu) := \iint_{\mathcal S\times\mathcal S}\left( -\log\|x-y\| +\frac{\mathcal 