A localized breakdown of regularity of solutions to partial differential equations that is concentrated along an edge (a one-dimensional boundary locus) where standard elliptic or parabolic estimates fail and solution asymptotics develop non-smooth angular dependence.

Principle

When geometry or boundary conditions create a codimension-two locus (an edge) the local behavior separates into radial/edge directions and transverse angular modes; elliptic regularity is governed by the spectrum of an associated transverse operator, producing power-law or logarithmic asymptotics along the edge.

Demonstration

Consider the Laplace equation on a three-dimensional polyhedral domain with a re-entrant edge: near a point on the edge solutions expand in r^{λ} times angular eigenfunctions on the cross-sectional wedge, where some λ have negative real part or non-integer values, producing reduced Sobolev regularity along the edge.

Misapplication

Treating an edge singularity as a removable point singularity and applying standard unweighted H^k estimates across the edge; or smoothing numerical meshes only transversely while leaving the edge unresolved, which yields spurious convergence and incorrect error rates.

Consequence

Correct identification forces use of weighted Sobolev or Kondrat'ev spaces, adapted mesh refinement and modified boundary layer analyses; it explains loss of derivatives in trace theorems along edges and finite-dimensional obstructions to regularity.

Reversal

Replacing the sharp edge by a C^∞ rounded fillet or enforcing boundary conditions that eliminate problematic transverse eigenmodes removes the edge singularity and restores classical local regularity.

Boundary

Applies to PDEs on domains whose boundary contains edges (polyhedra, manifolds with corners of codimension two); does not describe isolated interior singularities caused purely by distributional sources or global resonance phenomena that are not localized at a geometric edge.

Semantic Tension

Competes with the notion of conical (point) singularity: both produce power-law asymptotics, but an edge is extended and couples longitudinal regularity to transverse spectral data, whereas a cone singularity is localized to a single point.

Synthesis

An edge singularity is the phenomenon where an extended boundary locus forces solution modes determined by a transverse spectral problem, producing characteristic non-smooth asymptotics along the edge and necessitating weighted function spaces and boundary-layer-aware analysis.