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

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

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 is the outward unit normal field to S′ and

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

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

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

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

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

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

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. 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 or Newtonian capacity of K, denoted C(K) or cap(K), is then defined by

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

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:

${\displaystyle C(K)=\int _{S}{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma .}$

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 ${\displaystyle \mathbb {R} ^{n}}$. 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:

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

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) < +∞.

The Wiener capacity or Robin constant W(K) of K is given by

${\displaystyle C(K)=e^{-W(K)}}$

Logarithmic capacity

In two dimensions, the capacity is defined as above, but dropping the factor of ${\displaystyle (n-2)}$ in the definition:

${\displaystyle C(\Sigma ,S)=-{\frac {1}{2\pi }}\int _{S'}{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma '={\frac {1}{2\pi }}\int _{D}|\nabla u|^{2}\mathrm {d} x}$

This is often called the logarithmic capacity, the term logarithmic arises, as the potential function goes from being an inverse power to a logarithm in the ${\displaystyle n\to 2}$ limit. This is articulated below. It may also be called the conformal capacity, in reference to its relation to the conformal radius.

Properties

The harmonic function u is called the capacity potential, the Newtonian potential when ${\displaystyle n\geq 3}$ and the logarithmic potential when ${\displaystyle n=2}$. It can be obtained via a Green's function as

${\displaystyle u(x)=\int _{S}G(x-y)d\mu (y)}$

with x a point exterior to S, and

${\displaystyle G(x-y)={\frac {1}{|x-y|^{n-2}}}}$

when ${\displaystyle n\geq 3}$ and

${\displaystyle G(x-y)=\log {\frac {1}{|x-y|}}}$

for ${\displaystyle n=2}$.

The measure ${\displaystyle \mu }$ is called the capacitary measure or equilibrium measure. It is generally taken to be a Borel measure. It is related to the capacity as

${\displaystyle C(K)={\int <!--\; \int \; -->}_{S}d\mathrm{\mu <!--\; \mu \; -->}(y)=\mathrm{\mu <!--\; \mu \; -->}(S}$Zdroj:https://en.wikipedia.org?pojem=Energy_functional
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