A point-like boundary singularity occurring at a cone tip or vertex where local geometry is modeled by a product of a radial variable and a compact link (the cone cross-section), and where solutions exhibit asymptotics determined by the spectral problem on that link.

Principle

Local separation into radial behavior and angular (link) eigenmodes causes solution expansions in powers r^{λ} (possibly with logarithmic factors); the allowed exponents λ are roots of a characteristic equation coming from the transverse operator on the link, so regularity is spectral in origin.

Demonstration

Solve the Laplace equation in a planar sector or on an n-dimensional cone: near the tip, harmonic functions decompose into r^{λ} times eigenfunctions on the circular or spherical link; non-integer λ produce fractional Sobolev regularity and, for some λ, non-integrable singularities at the vertex.

Misapplication

Assuming classical Hölder or standard Sobolev regularity at the cone tip and ignoring the link spectrum when constructing parametrices; or treating cone tips like smooth boundary points in numerical discretizations without mesh grading toward the tip.

Consequence

Leads to the use of Mellin transform methods, weighted Kondrat'ev spaces, and explicit matching of asymptotic terms; affects existence/uniqueness results and requires boundary modification or compatibility conditions for well-posedness in standard spaces.

Reversal

Smoothing the tip to a C^∞ neighborhood or imposing boundary conditions that remove the offending transverse eigenvalues eliminates the conical singularity and restores classical local expansions.

Boundary

Relevant when the domain or manifold has an isolated conical point (manifold with isolated conical singularity) or when coefficients create an effective cone; does not describe extended edge singularities or singularities produced by rough coefficients away from a geometric tip.

Semantic Tension

Often conflated with corner or edge singularities; the tension is that cones are 0-dimensional singular loci whose spectral theory is captured by the link, whereas corners or edges involve different codimensions and coupling of directions.

Synthesis

A conical singularity is the pointwise failure of classical regularity at a cone tip: the asymptotic behavior is encoded by eigenvalues on the cross-section link, necessitating Mellin techniques and weighted function spaces to capture exact solution structure.