Capacity of a set

Inmathematics,thecapacity of a setinEuclidean spaceis a measure of the "size" of that set. Unlike, say,Lebesgue measure,which measures a set'svolumeor physical extent, capacity is a mathematical analogue of a set's ability to holdelectrical charge.More precisely, it is thecapacitanceof the set: the total charge a set can hold while maintaining a givenpotential energy.The potential energy is computed with respect to an idealized ground at infinity for theharmonicorNewtonian capacity,and with respect to a surface for thecondenser capacity.

Historical note

The notion of capacity of a set and of "capacitable" set was introduced byGustave Choquetin 1950: for a detailed account, see reference (Choquet 1986).

Definitions

Condenser capacity

Let Σ be aclosed,smooth, (n− 1)-dimensionalhypersurfaceinn-dimensional Euclidean space$\mathbb {R} ^{n}$,n≥ 3;Kwill denote then-dimensionalcompact(i.e.,closedandbounded) set of which Σ is theboundary.LetSbe another (n− 1)-dimensional hypersurface that encloses Σ: in reference to its origins inelectromagnetism,the pair (Σ,S) is known as acondenser.Thecondenser capacityof Σ relative toS,denotedC(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:

• uis the uniqueharmonic functiondefined on the regionDbetween Σ andSwith theboundary conditionsu(x) = 1 on Σ andu(x) = 0 onS;
• S′is any intermediate surface between Σ andS;
• νis the outwardunit normalfieldtoS′and

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

is thenormal derivativeofuacrossS′;and

• σ_n= 2π^n⁄2⁄ Γ(n⁄ 2) is the surface area of theunit spherein$\mathbb {R} ^{n}$.

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 avariational characterization:C(Σ,S) is theinfimumof theDirichlet's energyfunctional

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

over allcontinuously differentiable functionsvonDwithv(x) = 1 on Σ andv(x) = 0 onS.

Harmonic capacity

Heuristically,the harmonic capacity ofK,the region bounded by Σ, can be found by taking the condenser capacity of Σ with respect to infinity. More precisely, letube the harmonic function in the complement ofKsatisfyingu= 1 on Σ andu(x) → 0 asx→ ∞. Thusuis theNewtonian potentialof the simple layer Σ. Then theharmonic capacityorNewtonian capacityofK,denotedC(K) or cap(K), is then defined by

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

IfSis arectifiable hypersurfacecompletely enclosingK,then the harmonic capacity can be equivalently rewritten as the integral overSof the outward normal derivative ofu:

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

The harmonic capacity can also be understood as a limit of the condenser capacity. To wit, letS_rdenote thesphereof radiusrabout the origin in$\mathbb {R} ^{n}$.SinceKis bounded, for sufficiently larger,S_rwill encloseKand (Σ,S_r) will form a condenser pair. The harmonic capacity is then thelimitasrtends to infinity:

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

The harmonic capacity is a mathematically abstract version of theelectrostatic capacityof the conductorKand is always non-negative and finite: 0 ≤C(K) < +∞.

TheWiener capacityorRobin constantW(K)ofKis 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 thelogarithmic capacity,the termlogarithmicarises, 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 theconformal capacity,in reference to its relation to theconformal radius.

Properties

The harmonic functionuis called thecapacity potential,theNewtonian potentialwhen$n\geq 3$and thelogarithmic potentialwhen$n=2$.It can be obtained via aGreen's functionas

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

withxa point exterior toS,and

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

when$n\geq 3$and

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

for$n=2$.

Themeasure$\mu$is called thecapacitary measureorequilibrium measure.It is generally taken to be aBorel measure.It is related to the capacity as

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

The variational definition of capacity over theDirichlet energycan be re-expressed as

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

with the infimum taken over all positive Borel measures$\lambda$concentrated onK,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 anenergy functionalachieving particular boundary values, given above, can be extended to other energy functionals in thecalculus of variations.

Divergence form elliptic operators

Solutions to a uniformlyelliptic partial differential equationwith 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 setEwith respect to a domainDcontainingEis defined as theinfimumof the energy over allcontinuously differentiable functionsvonDwithv(x) = 1 onE;andv(x) = 0 on the boundary ofD.

The minimum energy is achieved by a function known as thecapacitary potentialofEwith respect toD,and it solves theobstacle problemonDwith the obstacle function provided by theindicator functionofE.The capacitary potential is alternately characterized as the unique solution of the equation with the appropriate boundary conditions.

See also

• Analytic capacity– number that denotes how big a certain bounded analytic function can becomePages displaying wikidata descriptions as a fallback
• Capacitance– Ability of a body to store an electrical charge
• Newtonian potential– Green's function for Laplacian
• Potential theory– Harmonic functions as solutions to Laplace's equation
• Choquet theory– Area of functional analysis and convex analysis

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,MR0259146,Zbl0257.31001.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,MR0867115,Zbl0607.01017,available fromGallica.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,ISBN0-387-90881-1,MR0731258,Zbl0549.31001
• 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,MR0161019,Zbl0116.30302,available atNUMDAM.
• Ransford, Thomas (1995),Potential theory in the complex plane,London Mathematical Society Student Texts, vol. 28, Cambridge:Cambridge University Press,ISBN0-521-46654-7,Zbl0828.31001
• 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