Capacity of a set

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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. The modern definition of condenser capacity given below generalizes earlier special cases such as harmonic capacity and Newtonian capacity.

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[edit] Definitions

[edit] Condenser capacity

Let Σ be a closed, smooth, (n − 1)-dimensional hypersurface in n-dimensional Euclidean space Rn, 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(\Sigma, S) = - \frac1{(n - 2) \sigma_{n}} \int_{S'} \frac{\partial u}{\partial \nu},

where:

\frac{\partial u}{\partial \nu} (x) = \nabla u (x) \cdot \nu (x)
is the normal derivative of u across S′; and
  • σn = 2πn⁄2 ⁄ Γ(n ⁄ 2) is the surface area of the unit sphere in Rn.

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

C(\Sigma, S) = \frac1{(n - 2) \sigma_{n}} \int_{D} | \nabla u |^{2}.

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

I[v] = \frac1{(n - 2) \sigma_{n}} \int_{D} | \nabla v |^{2}

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

[edit] Harmonic/Newtonian capacity

Heuristically, the harmonic capacity of K, the region bounded by Σ, is found by taking the condenser capacity of Σ with respect to S as S "tends to infinity". More formally, let Sr denote the sphere of radius r about the origin in Rn. Since K is bounded, for sufficiently large r, Sr will enclose K and (Σ, Sr) will form a condenser pair. The harmonic capacity (also known as the Newtonian capacity) of K, denoted C(K) or cap(K), is then defined by taking 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) < +∞.

[edit] References

  • Brélot, Marcel (1967). Lectures on potential theory (Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy.), Second edition, revised and enlarged with the help of S. Ramaswamy., Tata Institute of Fundamental Research Lectures on Mathematics, No. 19, Bombay: Tata Institute of Fundamental Research, pp. ii+170+iv.  MR0259146
  • Doob, Joseph Leo (1984). Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 262. New York: Springer-Verlag, pp. xxiv+846. ISBN 0-387-90881-1.  MR731258

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