Capacitylike

Capacitylike is an informal term used to describe mathematical objects or functionals that behave similarly to capacities in potential theory. It is not a standard, widely adopted term in textbooks, but it appears in discussions where a notion of “size” or “negligibility” is needed without relying on a traditional measure.

A capacitylike functional C is typically defined on subsets of a topological space and aims to capture the idea that some sets are small or negligible for certain purposes. Common features include monotonicity (A ⊆ B implies C(A) ≤ C(B)), C(∅) = 0, and continuity properties that mirror those of classical capacities, such as continuity from above on decreasing sequences of compact sets. In many contexts, C is subadditive (C(A ∪ B) ≤ C(A) + C(B)) and respects outer regularity on open sets or inner regularity on compact sets. A set A with C(A) = 0 is called capacityzero and is often considered negligible for the problems governed by the capacitylike functional.

Examples and relationships to established notions include the classical Newtonian capacity, p-capacity in Sobolev spaces, and Choquet capacities, which provide concrete instances that inspire capacitylike definitions in other areas. In analysis and geometric measure theory, capacitylike notions help characterize thin or removable sets, energy minimization, and boundary behavior of solutions to partial differential equations.

Usage of the term is context-dependent and typically requires an explicit definition within a given framework. The concept serves as a flexible shorthand for properties that resemble those of capacities while being tailored to specific applications.