Frostman's theorem

Remarks

Proof idea: suppose that $U^\nu < V(K)$ on a compact subset of $K$ of positive capacity, or $U^\nu > V(K)$ on a set of positive $\nu$-measure. Then moving a little mass of $\nu$ lowers the energy, which contradicts minimality. This gives $U^\nu \ge V(K)$ on $K$ outside a set of capacity zero and $U^\nu \le V(K)$ on the support of $\nu$. The maximum principle for potentials then extends the upper bound from the support of $\nu$ to all of $\C$. A separate argument shows that the support of $\nu$ lies in $\partial K$.

Example: for the unit circle, $\nu = d\theta / 2\pi$ and $U^\nu(z) = -\log \max(\abs{z}, 1)$. This is $0 = V(K)$ on the closed unit disc and negative outside, as the theorem says. The exceptional set $E$ cannot be dropped in general: it can contain boundary points that are irregular for the Dirichlet problem, where the potential is strictly below $V(K)$.

Going deeper

For a student

Frostman's theorem is the bridge between the energy definition of capacity and the Green function. On the unbounded component $\Omega$ of $\C \setminus K$, the function $g(z) = V(K) - U^\nu(z)$ is the Green function of $\Omega$ with pole at infinity. The exceptional set $E$ is linked to the boundary points that are irregular for the Dirichlet problem. The Kellogg–Evans theorem says that such points form a set of capacity zero. The theorem is also the basic tool for computing capacities: a probability measure on $K$ whose potential is constant on all of $K$ must be the equilibrium measure.