# Capacity of a set

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.

## Historical note

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).

## Definitions

### Condenser capacity

Let Σ 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 Σ is the boundary. Let S be another (n − 1)-dimensional hypersurface that encloses Σ: in reference to its origins in electromagnetism, the pair (Σ, S) is known as a condenser. The condenser capacity of Σ relative to S, denoted C(Σ, S) or cap(Σ, S), is given by the surface integral

$C(\Sigma, S) = - \frac1{(n - 2) \sigma_{n}} \int_{S'} \frac{\partial u}{\partial \nu}\,\mathrm{d}\sigma',$

where:

$\frac{\partial u}{\partial \nu} (x) = \nabla u (x) \cdot \nu (x)$
is the normal derivative of u across S′; and
• σn = 2πn⁄2 ⁄ Γ(n ⁄ 2) is the surface area of the unit sphere in ℝn.

C(Σ, S) can be equivalently defined by the volume integral

$C(\Sigma, S) = \frac1{(n - 2) \sigma_{n}} \int_{D} | \nabla u |^{2}\mathrm{d}x.$

The condenser capacity also has a variational characterization: C(Σ, S) is the infimum of the Dirichlet's energy functional

$I[v] = \frac1{(n - 2) \sigma_{n}} \int_{D} | \nabla v |^{2}\mathrm{d}x$

over all continuously-differentiable functions v on D with v(x) = 1 on Σ and v(x) = 0 on S.

### Harmonic/Newtonian capacity

Heuristically, the harmonic capacity of K, the region bounded by Σ, can be found by taking the condenser capacity of Σ with respect to infinity. 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. Then the harmonic capacity (also known as the Newtonian capacity) of K, denoted C(K) or cap(K), is then defined by

$C(K) = \int_{\mathbb{R}^n\setminus K} |\nabla u|^2\mathrm{d}x.$

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:

$C(K) = \int_S \frac{\partial u}{\partial\nu}\,\mathrm{d}\sigma.$

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 (Σ, Sr) will form a condenser pair. The harmonic capacity is then the limit as r tends to infinity:

$C(K) = \lim_{r \to \infty} C(\Sigma, S_{r}).$

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) < +∞.

## Generalizations

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.

### Divergence form elliptic operators

Solutions to a uniformly elliptic partial differential equation with divergence form

$\nabla \cdot ( A \nabla u ) = 0$

are minimizers of the associated energy functional

$I[u] = \int_D (\nabla u)^T A (\nabla u)\,\mathrm{d}x$

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.