capacitary

Capacitary is an adjective relating to capacity, in particular the mathematical notion of capacity as a size or influence of a set in potential theory, as well as to notions arising in physics and engineering that involve storing or transmitting charge. In mathematics, capacitary concepts describe how a set can carry or affect electrical or harmonic energy rather than how large it is in the usual geometric sense.

In potential theory, capacity provides a way to quantify the “size” of a set with respect to its ability to influence potential functions. For a compact set K in Euclidean space, one defines capacity via an energy minimization problem: the capacity is the infimum of the Dirichlet energy ∫|∇u|^2 over functions u that take prescribed values on K (often u ≥ 1 on K and vanish at infinity). The associated capacitary potential is the energy-minimizing function, which is harmonic outside K and encodes the set’s capacity. Capacitary quantities can be extended to more general sets and to other spaces, with variations such as p-capacity using L^p energies.

Related concepts include capacitary measures and capacitary distributions, which represent the distribution of charge or energy that realizes a given capacity. Capacitary quantities play roles in problems concerning removable singularities, boundary regularity, and probabilistic questions related to hitting probabilities of stochastic processes.

The term is distinct from capacitive in common electrical engineering usage, where capacitive describes devices or effects involving a capacitor, while capacitary emphasizes the potential-theoretic or energy-based aspects of capacity. See also capacity (potential theory) and capacity (electrostatics).