In mathematics, the capacity of a set in Euclidean space is a measure of that set's "size". 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. More precisely, it is the capacitance of the set: the total charge a set can hold while maintaining a given potential energy. 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.
Let ? be a closed, smooth, (n − 1)-dimensional hypersurface in n-dimensional Euclidean space Rn, n >= 3; K will denote the n-dimensional compact (i.e., closed and bounded) set of which ? is the boundary. Let S be another (n − 1)-dimensional hypersurface that encloses ?: in reference to its origins in electromagnetism, the pair (?, S) is known as a condenser. The condenser capacity of ? relative to S, denoted C(?, S) or cap(?, S), is given by the surface integral
C(?, S) can be equivalently defined by the volume integral
over all continuously-differentiable functions v on D with v(x) = 1 on ? and v(x) = 0 on S.
Heuristically, the harmonic capacity of K, the region bounded by ?, can be found by taking the condenser capacity of ? with respect to infinity. More precisely, let u be the harmonic function in the complement of K satisfying u = 1 on ? and u(x) -> 0 as x -> ?. Thus u is the Newtonian potential of the simple layer ?. Then the harmonic capacity (also known as the Newtonian capacity) of K, denoted C(K) or cap(K), is then defined by
If S is a rectifiable hypersurface completely enclosing K, then the harmonic capacity can be equivalently rewritten as the integral over S of the outward normal derivative of u:
The harmonic capacity can also be understood as a limit of the condenser capacity. To wit, let Sr denote the sphere of radius r about the origin in Rn. Since K is bounded, for sufficiently large r, Sr will enclose K and (?, Sr) will form a condenser pair. The harmonic capacity is then the limit as r tends to infinity:
The harmonic capacity is a mathematically abstract version of the electrostatic capacity of the conductor K and is always non-negative and finite: 0 C(K) < +?.
The characterization of the capacity of a set as the minimum of an energy functional achieving particular boundary values, given above, can be extended to other energy functionals in the calculus of variations.
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.