Capacity of A Set

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## Historical note

## Definitions

### Condenser capacity

### Harmonic/Newtonian capacity

## Generalizations

### Divergence form elliptic operators

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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**.

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).

Let ? 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 ? is the boundary. Let *S* be another (*n* − 1)-dimensional hypersurface that encloses ?: in reference to its origins in electromagnetism, the pair (?, *S*) is known as a condenser. The **condenser capacity** of ? relative to *S*, denoted *C*(?, *S*) or cap(?, *S*), is given by the surface integral

where:

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

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

*?*_{n}= 2*?*^{n/2}/ ?(*n*/ 2) is the surface area of the unit sphere in R^{n}.

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

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

over all continuously-differentiable functions *v* on *D* with *v*(*x*) = 1 on ? and *v*(*x*) = 0 on *S*.

Heuristically, the harmonic capacity of *K*, the region bounded by ?, can be found by taking the condenser capacity of ? with respect to infinity. More precisely, let *u* be the harmonic function in the complement of *K* satisfying *u* = 1 on ? and *u*(*x*) -> 0 as *x* -> ?. Thus *u* is the Newtonian potential of the simple layer ?. 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 (?, *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*) < +?.

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.

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.

- 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., No. 19 (2nd ed.), Bombay: Tata Institute of Fundamental Research, pp. ii+170+iv, 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,**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,**28**, Cambridge: Cambridge University Press, ISBN 0-521-46654-7, Zbl 0828.31001 - Solomentsev, E. D. (2001) [1994], "Capacity of a set",
*Encyclopedia of Mathematics*, EMS Press

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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