# 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 Template:Harv.

## 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

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

- is the normal derivative of
*u*across*S*′; and

*σ*_{n}= 2*π*^{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

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

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

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 *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:

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

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.

## See also

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}. The second edition of these lecture notes, revised and enlarged with the help of S. Ramaswamy, re–typeset, proof read once and freely available for download.

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|CitationClass=citation }}, available from Gallica. A historical account of the development of capacity theory by its founder and one of the main contributors; an English translation of the title reads: "The birth of capacity theory: reflections on a personal experience".

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- {{#invoke:citation/CS1|citation

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