Equilibrium problem with an external field

Remarks

The weighted energy equals $I(\mu) + 2\int Q\, d\mu$, where $I(\mu)$ is the logarithmic energy. So the field $Q$ pushes the charge away from where $Q$ is large. For $w = 1$ on a compact set $E$ this is the problem of the equilibrium measure of $E$. Gauss considered the analogous problem for the Newtonian kernel in space. The weighted logarithmic version in the plane is the subject of the book of Saff and Totik (1997). Existence and uniqueness are known under the hypotheses above: $V_w$ is finite, there is a unique minimiser $\mu_w$, and $S_w$ is compact with positive capacity. The minimiser is characterised by Frostman-type conditions on its logarithmic potential $U^{\mu_w}$. For a constant $F_w$, one has $U^{\mu_w} + Q \ge F_w$ quasi-everywhere on $E$ and $U^{\mu_w} + Q \le F_w$ on $S_w$. Finding $S_w$ for a given weight is the hard part, and it is done explicitly only for special weights. Example: on $E = \R$ with $w(x) = e^{-x^2}$, the support is $[-1, 1]$ and $\mu_w$ has the semicircle density $\frac{2}{\pi}\sqrt{1 - x^2}$.