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 $\mathbb {R} ^{n}$ , $n \geq 3; K$ 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 $\mathbb {R} ^{n}$ .

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 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

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

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

C(K)=e^{-W(K)}

Logarithmic capacity

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

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 $n\to 2$ 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 $n\geq 3$ and the logarithmic potential when $n=2$ . It can be obtained via a Green's function as

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

with x a point exterior to S, and

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

when $n\geq 3$ and

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

for $n=2$ .

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

C(K)=\int _{S}d\mu (y)=\mu (S)

The variational definition of capacity over the Dirichlet energy can be re-expressed as

C(K)=\left[\inf _{\lambda }E(\lambda )\right]^{-1}

with the infimum taken over all positive Borel measures $\lambda$ concentrated on K, normalized so that $\lambda (K)=1$ and with $E(\lambda )$ is the energy integral

E(\lambda )=\int \int _{K\times K}G(x-y)d\lambda (x)d\lambda (y)

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.

References

Brélot, Marcel (1967) [1960], Lectures on potential theory (Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy.) (PDF), Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Mathematics., vol. 19 (2nd ed.), Bombay: Tata Institute of Fundamental Research, MR 0259146, Zbl 0257.31001. 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.
Choquet, Gustave (1986), "La naissance de la théorie des capacités: réflexion sur une expérience personnelle", Comptes rendus de l'Académie des sciences. Série générale, La Vie des sciences (in French), 3 (4): 385–397, MR 0867115, Zbl 0607.01017, 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".
Doob, Joseph Leo (1984), Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften, vol. 262, Berlin–Heidelberg–New York: Springer-Verlag, pp. xxiv+846, ISBN 0-387-90881-1, MR 0731258, Zbl 0549.31001
Littman, W.; Stampacchia, G.; Weinberger, H. (1963), "Regular points for elliptic equations with discontinuous coefficients", Annali della Scuola Normale Superiore di Pisa – Classe di Scienze, Serie III, 17 (12): 43–77, MR 0161019, Zbl 0116.30302, available at NUMDAM.
Ransford, Thomas (1995), Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge: Cambridge University Press, ISBN 0-521-46654-7, Zbl 0828.31001
Solomentsev, E. D. (2001) [1994], "Capacity", Encyclopedia of Mathematics, EMS Press
Solomentsev, E. D. (2001) [1994], "Robin constant", Encyclopedia of Mathematics, EMS Press
Solomentsev, E. D. (2001) [1994], "Energy of measures", Encyclopedia of Mathematics, EMS Press

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