Wiener's criterion

Remarks

The factor $2^{k(n-2)}$ normalizes the capacity of $E_k$ by the capacity of a ball of radius $2^{-k}$, which is a constant times $2^{-k(n-2)}$. So the criterion says that the complement of $\Omega$ is not too thin near $\xi$, measured shell by shell. Any ratio $\lambda \in (0, 1)$ in place of $\frac12$ gives an equivalent criterion, and constant factors in the normalization of capacity do not matter. Example: suppose the complement of $\Omega$ contains a cone with vertex $\xi$. Then each $E_k$ contains a ball of radius comparable to $2^{-k}$, so every term of the series is bounded below and $\xi$ is regular. Example: if $\xi$ is an isolated point of $\R^n \setminus \Omega$, then $E_k$ is empty for large $k$, the series converges, and $\xi$ is irregular. In the plane there is an analogous criterion with logarithmic capacity.

Going deeper

For a student

Wiener's criterion is the complete answer to the question of regularity for the Laplacian. It explains the exterior cone condition, the irregularity of the Lebesgue spine, and the irregularity of single points for $n \ge 3$. The logarithmic version gives the same in the plane. It also shows that regularity depends only on the complement of $\Omega$ near $\xi$, and it extends to a large class of elliptic equations.