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

{\displaystyle \mathbb {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

(

Σ

,

S

)

=

−

1

(

n

−

2

)

σ

n

∫

S

′

∂

u

∂

ν

d

σ

′

,

{\displaystyle 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

{\displaystyle \nu }

is the outward unit normal field to S′ and

∂

u

∂

ν

(

x

)

=

∇

u

(

x

)

⋅

ν

(

x

)

{\displaystyle {\frac {\partial u}{\partial \nu }}(x)=\nabla u(x)\cdot \nu (x)}

is the normal derivative of u across S′; and

σn = 2π n/2 / Γ(n ⁄ 2) is the surface area of the unit sphere in

R

n

{\displaystyle \mathbb {R} ^{n}}

.

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

C

(

Σ

,

S

)

=

1

(

n

−

2

)

σ

n

∫

D

|

∇

u

|

2

d

x

.

{\displaystyle 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

]

=

1

(

n

−

2

)

σ

n

∫

D

|

∇

v

|

2

d

x

{\displaystyle 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 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.