A geometric singularity occurring where two or more boundary segments meet at a nontrivial angle; the local geometry forces a loss of regularity for solutions of boundary-value problems (for example elliptic PDEs) near that point.

Principle

The mismatch between smooth boundary parametrization and the angle at the meeting point prevents representation of solutions by globally smooth expansions; singular powers or angular eigenmodes appear and reduce Sobolev or Hölder regularity according to the opening angle.

Demonstration

For the Laplace equation in a planar sector of opening angle ω, harmonic functions admit series terms like r^{kπ/ω} sin(kπθ/ω); the smallest positive exponent π/ω governs the leading singular behaviour near the corner.

Misapplication

Treating a corner as if the boundary were smooth and applying standard elliptic regularity estimates uniformly up to the corner, which leads to incorrect error bounds in numerical methods.

Consequence

Correct analysis requires weighted function spaces, local singular expansions, mesh refinement near the corner in finite element methods, or explicit extraction of singular modes to restore approximation order.

Reversal

If the boundary is smoothed so that the meeting angle tends to π (a smooth transition) the corner singularity disappears and classical regularity up to the boundary is regained.

Boundary

This term refers to piecewise-smooth boundaries meeting at an angle; it excludes cusp singularities produced by tangential degeneracy, interior singularities not tied to the boundary, and removable irregularities fixed by boundary data.

Semantic Tension

Often confused with irregular boundary points from potential theory: the corner singularity is geometric and concerns regularity of PDE solutions, while potential-theoretic irregularity concerns attainment of boundary values and probabilistic hitting properties.

Synthesis

A corner singularity is the geometric mechanism by which angular boundary geometry forces specific non-smooth modal behaviour of solutions; recognising the angle-driven singular exponents allows one to choose adapted function spaces, refine discretizations, or subtract singular parts to recover accurate analysis.