A boundary point at which a prescribed boundary-value problem (for example the Dirichlet problem for Laplace's equation) fails to attain the given boundary value in the classical limit; limiting values of solutions do not equal the boundary datum at that point.

Principle

Attainment of boundary data by harmonic or elliptic solutions depends on a local capacity or barrier condition; points failing the criterion (for Laplace's equation, those that fail the Wiener test) are irregular because the influence of the boundary value is negligible in the fine topology.

Demonstration

In potential theory, the Wiener criterion gives a capacitary series test: certain thin sets near a boundary point make the point irregular for the Dirichlet problem. Concrete examples include boundary spikes or fractal boundary portions that cause non-attainment.

Misapplication

Assuming every continuous boundary datum is achieved pointwise everywhere on the boundary for elliptic PDEs; this leads to wrong expectations about pointwise convergence and probabilistic interpretations (e.g. Brownian hitting).

Consequence

Irregular points force use of generalized boundary concepts (fine limit, nontangential limits, Perron solutions) and care in probabilistic representations; they identify where classical boundary conditions fail to control interior behaviour.

Reversal

A regular boundary point is one where the Dirichlet solution does attain the prescribed continuous boundary value; probabilistically, Brownian motion almost surely hits the boundary in arbitrarily small neighborhoods of that point.

Boundary

Applies to boundary-value problems for elliptic operators and notions of pointwise attainment; excludes topological singularities unrelated to boundary-value attainment and alternative boundary conditions such as Neumann or Robin unless specified.

Semantic Tension

Close to the geometric notion of corner singularity but distinct: irregularity is a potential-theoretic property about limit attainment, while corner singularity is about differentiability and local behaviour of solutions.

Synthesis

An irregular boundary point is a location where the usual mechanism for propagating boundary data into the domain fails; characterizing such points via capacity or probabilistic tests clarifies when classical Dirichlet conditions are inadequate and which generalized notions must be used.