Unlike, say, Lebesgue measure, which measures a set's volume or physical extent, capacity is a mathematical analogue of a set's ability to hold electrical charge.
The potential energy is computed with respect to an idealized ground at infinity for the harmonic or Newtonian capacity, and with respect to a surface for the condenser capacity.
The notion of capacity of a set and of "capacitable" set was introduced by Gustave Choquet in 1950: for a detailed account, see reference (Choquet 1986).
Let Σ be a closed, smooth, (n − 1)-dimensional hypersurface in n-dimensional Euclidean space
, n ≥ 3; K will denote the n-dimensional compact (i.e., closed and bounded) set of which Σ is the boundary.
More precisely, let u be the harmonic function in the complement of K satisfying u = 1 on Σ and u(x) → 0 as x → ∞.
Thus u is the Newtonian potential of the simple layer Σ.
To wit, let Sr denote the sphere of radius r about the origin in
It can be obtained via a Green's function as with x a point exterior to S, and when
It is related to the capacity as The variational definition of capacity over the Dirichlet energy can be re-expressed as with the infimum taken over all positive Borel measures
Solutions to a uniformly elliptic partial differential equation with divergence form are minimizers of the associated energy functional subject to appropriate boundary conditions.
The capacity of a set E with respect to a domain D containing E is defined as the infimum of the energy over all continuously differentiable functions v on D with v(x) = 1 on E; and v(x) = 0 on the boundary of D. The minimum energy is achieved by a function known as the capacitary potential of E with respect to D, and it solves the obstacle problem on D with the obstacle function provided by the indicator function of E. The capacitary potential is alternately characterized as the unique solution of the equation with the appropriate boundary conditions.