The inability to extend a solution of a PDE defined in a domain to a larger set (typically across part of the boundary) in the same continuity or regularity class; this may be caused by boundary singularities, lack of trace regularity, capacity obstructions, or fundamental incompatibilities with analytic continuation.

Principle

Extension requires both local control of behaviour near the boundary (e.g., boundedness, integrability, control of oscillation) and global compatibility conditions; failure of any required trace or removable-singularity condition prevents extension in the target class.

Demonstration

A harmonic function in the unit disc that behaves like a multiple of the Poisson kernel concentrating at a boundary point may fail to have a finite nontangential limit there, so it cannot be continuously extended across that boundary point; similarly, solutions with singularities approaching the boundary may not admit Sobolev extensions.

Misapplication

Confusing nonextendability with nonuniqueness is a misapplication: a solution may be unique inside the domain but still fail to extend across the boundary. Another misapplication is assuming analyticity implies extendability across arbitrary boundary sets without verifying boundary regularity.

Consequence

Nonextendability restricts the use of global continuation arguments, prevents certain boundary regularity bootstraps, and may force the use of localized or weaker solution concepts; it also impacts inverse problems and control, where extension properties are often assumed.

Reversal

Extendability (the reversal) means the solution admits a continuation across the boundary with the same qualitative regularity, which typically implies stronger trace properties, removable singularities, or presence of barriers and sufficient capacity near the boundary.

Boundary

This concept applies to various regularity classes (continuous, C^k, analytic, Sobolev); the precise obstruction and terminology depend on the chosen class and the PDE: an obstruction in the continuous category may be removable in a weaker Sobolev sense, and vice versa.

Semantic Tension

Tension arises between the notions of removable singularity and essential obstruction: some singular behaviors can be removed by redefinition on a set of capacity zero, whereas others are genuine obstructions to any extension in the class; deciding which is which requires fine potential-theoretic tools.

Synthesis

Nonextendability synthesizes geometric, analytic and measure-theoretic phenomena: failure to extend a solution across the boundary reflects singularities, lack of trace or capacity conditions, and ties back to boundary regularity notions (Wiener, barriers) and to whether Perron's or variational constructions can be continued.