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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]].  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 {{harv|Choquet|1986}}.


==Definitions==
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===Condenser capacity===
 
Let Σ be a [[closed surface|closed]], smooth, (''n''&nbsp;&minus;&nbsp;1)-[[dimension]]al [[hypersurface]] in ''n''-dimensional Euclidean space ℝ<sup>''n''</sup>, ''n'' ≥ 3; ''K'' will denote the ''n''-dimensional [[compact space|compact]] (i.e., [[closed set|closed]] and [[bounded set|bounded]]) set of which Σ is the [[boundary (topology)|boundary]]. Let ''S'' be another (''n'' &minus; 1)-dimensional hypersurface that encloses Σ: in reference to its origins in [[electromagnetism]], the pair (Σ,&nbsp;''S'') is known as a [[capacitor|condenser]]. The '''condenser capacity''' of Σ relative to ''S'', denoted ''C''(Σ,&nbsp;''S'') or cap(Σ,&nbsp;''S''), is given by the surface integral
 
:<math>C(\Sigma, S) = - \frac1{(n - 2) \sigma_{n}} \int_{S'} \frac{\partial u}{\partial \nu}\,\mathrm{d}\sigma',</math>
 
where:
 
* ''u'' is the unique [[harmonic function]] defined on the region ''D'' between Σ and ''S'' with the [[boundary condition]]s ''u''(''x'')&nbsp;=&nbsp;1 on Σ and ''u''(''x'') = 0 on ''S'';
* ''S''&prime; is any intermediate surface between Σ and ''S'';
* ''ν'' is the outward [[unit normal]] [[vector field|field]] to ''S''&prime; and
 
::<math>\frac{\partial u}{\partial \nu} (x) = \nabla u (x) \cdot \nu (x)</math>
 
:is the [[normal derivative]] of ''u'' across ''S''&prime;; and
* ''σ''<sub>''n''</sub>&nbsp;=&nbsp;2''π''<sup>''n''⁄2</sup>&nbsp;⁄&nbsp;Γ(''n''&nbsp;⁄&nbsp;2) is the surface area of the [[unit sphere]] in ℝ<sup>''n''</sup>.
 
''C''(Σ,&nbsp;''S'') can be equivalently defined by the volume integral
 
:<math>C(\Sigma, S) = \frac1{(n - 2) \sigma_{n}} \int_{D} | \nabla u |^{2}\mathrm{d}x.</math>
 
The condenser capacity also has a [[calculus of variations|variational characterization]]: ''C''(Σ,&nbsp;''S'') is the [[infimum]] of the [[Dirichlet's energy]] [[functional (mathematics)|functional]]
 
:<math>I[v] = \frac1{(n - 2) \sigma_{n}} \int_{D} | \nabla v |^{2}\mathrm{d}x</math>
 
over all [[smooth function|continuously-differentiable functions]] ''v'' on ''D'' with ''v''(''x'')&nbsp;=&nbsp;1 on Σ and ''v''(''x'')&nbsp;=&nbsp;0 on ''S''.
 
===Harmonic/Newtonian capacity===
 
[[Heuristic]]ally, 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''&nbsp;=&nbsp;1 on Σ and ''u''(''x'')&nbsp;→&nbsp;0 as ''x''&nbsp;→&nbsp;∞.  Thus ''u'' is the [[Newtonian potential]] of the simple layer Σ. Then the '''harmonic capacity''' (also known as the '''Newtonian capacity''') of ''K'', denoted ''C''(''K'') or cap(''K''), is then defined by
 
:<math>C(K) = \int_{\mathbb{R}^n\setminus K} |\nabla u|^2\mathrm{d}x.</math>
 
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'':
 
:<math>C(K) = \int_S \frac{\partial u}{\partial\nu}\,\mathrm{d}\sigma.</math>
 
The harmonic capacity can also be understood as a limit of the condenser capacity. To wit, let ''S''<sub>''r''</sub> denote the [[sphere]] of radius ''r'' about the origin in ℝ<sup>''n''</sup>. Since ''K'' is bounded, for sufficiently large ''r'', ''S''<sub>''r''</sub> will enclose ''K'' and (Σ,&nbsp;''S''<sub>''r''</sub>) will form a condenser pair.  The harmonic capacity is then the [[Limit of a function|limit]] as ''r'' tends to infinity:
 
:<math>C(K) = \lim_{r \to \infty} C(\Sigma, S_{r}).</math>
 
The harmonic capacity is a mathematically abstract version of the [[electrostatic capacity]] of the conductor ''K'' and is always non-negative and finite: 0&nbsp;≤&nbsp;''C''(''K'')&nbsp;&lt;&nbsp;+∞.
 
==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
:<math> \nabla \cdot ( A \nabla u ) = 0 </math>
are minimizers of the associated energy functional
:<math>I[u] = \int_D (\nabla u)^T A (\nabla u)\,\mathrm{d}x</math>
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 [[smooth function|continuously-differentiable functions]] ''v'' on ''D'' with ''v''(''x'')&nbsp;=&nbsp;1 on ''E''; and ''v''(''x'')&nbsp;=&nbsp;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.
 
==See also==
*[[Capacitance]]
*[[Newtonian potential]]
*[[Potential theory]]
 
==References==
* {{citation
| last = Brélot
| first = Marcel
| author-link = Marcel Brélot
| title = Lectures on potential theory (Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy.)
| series = Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Mathematics.
| volume = No. 19
| edition = 2nd
| publisher = Tata Institute of Fundamental Research
| location = Bombay
| year = 1967
| origyear = 1960
| pages = ii+170+iv
| url = http://www.math.tifr.res.in/~publ/ln/tifr19.pdf
| 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.
*{{Citation
  | last = Choquet
  | first = Gustave
  | author-link = Gustave Choquet
  | title = La naissance de la théorie des capacités: réflexion sur une expérience personnelle
  | journal = [[Comptes rendus de l'Académie des sciences|Comptes rendus de l'Académie des sciences. Série générale, La Vie des sciences]]
  | volume = 3
  | issue = 4
  | pages = 385–397
  | year = 1986
  | language = French
  | url = http://gallica.bnf.fr/ark:/12148/bpt6k54708101/f85
  | 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".
*{{citation
| last = Doob
| first = Joseph Leo
| author-link = Joseph Leo Doob
| title = Classical potential theory and its probabilistic counterpart
| series = Grundlehren der Mathematischen Wissenschaften
| volume = 262
| publisher = Springer-Verlag
| location = Berlin–[[Heidelberg]]–New York
| year = 1984
| pages = xxiv+846
| isbn = 0-387-90881-1
| mr = 731258
| zbl = 0549.31001
}}
*{{Citation
| last = Littman
| first = W.
| author-link = Walter Littman
| last2 = Stampacchia
| first2 = G.
| author2-link = Guido Stampacchia
| last3 = Weinberger
| first3 = H.
| author3-link = Hans Weinberger
| year = 1963
| title = Regular points for elliptic equations with discontinuous coefficients
| journal = Annali della Scuola Normale Superiore di Pisa – Classe di Scienze
| series = Serie III
| volume = 17
| issue = 12
| pages = 43–77
| url = http://www.numdam.org/item?id=ASNSP_1963_3_17_1-2_43_0
| mr = 161019
| zbl = 0116.30302
}}, available at [http://www.numdam.org NUMDAM].
* {{citation | last=Ransford | first=Thomas | title=Potential theory in the complex plane | series=London Mathematical Society Student Texts | volume=28 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-46654-7 | zbl=0828.31001 }}
* {{springer
| id = c/c020280
| title = Capacity of a set
| last = Solomentsev
| first = E. D.
}}
 
[[Category:Potential theory]]

Revision as of 17:38, 15 February 2014


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