Capacity of a set
# Capacity of a set

In mathematics, the capacity of a set in Euclidean space is a measure of the "size" of that set. 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 $ℝ^n,$ 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

    $C(\Sigma ,S)=-{\frac {1}{(n-2)\sigma _{n}}}\int _{S'}{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma ',$

where:

* u is the unique harmonic function defined on the region D between Σ and S with the boundary conditions u(x) = 1 on Σ and u(x) = 0 on S;
* S′ is any intermediate surface between Σ and S;
* ν is the outward unit normal field to S′ and

       ${\frac {\partial u}{\partial \nu }}(x)=\nabla u(x)\cdot \nu (x)$
    is the normal derivative of u across S′; and
* $σ_n$ = $2π^{n⁄2}$ ⁄ Γ(n ⁄ 2) is the surface area of the unit sphere in $ℝ^n.$

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

    $C(\Sigma ,S)={\frac {1}{(n-2)\sigma _{n}}}\int _{D}|\nabla u|^{2}\mathrm {d} x.$

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

    $I[v]={\frac {1}{(n-2)\sigma _{n}}}\int _{D}|\nabla v|^{2}\mathrm {d} x$

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 $S_r$ denote the sphere of radius r about the origin in $ℝ^n.$ Since K is bounded, for sufficiently large r, $S_r$ will enclose K and (Σ, $S_r)$ 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.

