A potential-theoretic condition that characterizes when a boundary point of a domain is regular for the Dirichlet problem for the Laplace operator (or certain linear uniformly elliptic operators): roughly, the point is regular precisely when a capacity-based series or integral computed on complements of shrinking neighborhoods diverges (or converges, depending on normalization).

Principle

Regularity at a boundary point is governed not by ordinary measure but by capacity: the fine geometric size of the omitted complement near the point, measured by Newtonian or logarithmic capacity, determines whether harmonic functions attain prescribed boundary values there.

Demonstration

In R^n, n≥3, given a point x0 on ∂Ω consider concentric annuli of radii 2^{-k} around x0; the Wiener criterion asserts x0 is regular if and only if the series sum_k cap( (B(x0,2^{-k})ackslash Ω) ) / 2^{-k(n-2)} diverges. In the plane the same idea uses logarithmic capacity and an appropriate logarithmic normalization.

Misapplication

Replacing capacity by Lebesgue measure or Hausdorff measure of the complement near the point can give false conclusions: a set of zero measure can still have positive capacity and affect regularity, so measure-theoretic criteria are insufficient in general.

Consequence

When the Wiener criterion is satisfied at x0, the solution produced by Perron's method attains the prescribed continuous boundary value at x0; equivalently, the classical Dirichlet problem is pointwise solvable at that boundary point.

Reversal

The converse statement identifies irregular points: if the corresponding capacity series converges (or the integral is finite under the appropriate convention), then the point is irregular and some continuous boundary data cannot be attained there by harmonic functions.

Boundary

Applies primarily to the Laplacian and to uniformly elliptic linear operators in divergence or nondivergence form under standard regularity of coefficients; the precise form of capacity and normalization depends on dimension and operator, and the criterion does not directly apply to general nonlinear PDEs without modification.

Semantic Tension

Competes with the Barrier Condition approach: Wiener gives a capacitary, quantitative condition often amenable to verification, whereas barrier methods construct explicit superharmonic functions; both characterize the same regularity under classical hypotheses but emphasize different structures.

Synthesis

The Wiener criterion ties a geometric-measure quantity (capacity of complementary sets near a boundary point) to analytic boundary regularity: divergence of a capacity series implies existence of harmonic extensions and coincides, under standard hypotheses, with other characterizations such as the existence of barriers and success of the Perron method.