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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'''.
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| ==Historical note==
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| 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}}.
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| ==Definitions==
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| ===Condenser capacity===
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| Let Σ be a [[closed surface|closed]], smooth, (''n'' − 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'' − 1)-dimensional hypersurface that encloses Σ: in reference to its origins in [[electromagnetism]], the pair (Σ, ''S'') is known as a [[capacitor|condenser]]. The '''condenser capacity''' of Σ relative to ''S'', denoted ''C''(Σ, ''S'') or cap(Σ, ''S''), is given by the surface integral
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| :<math>C(\Sigma, S) = - \frac1{(n - 2) \sigma_{n}} \int_{S'} \frac{\partial u}{\partial \nu}\,\mathrm{d}\sigma',</math>
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| where:
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| * ''u'' is the unique [[harmonic function]] defined on the region ''D'' between Σ and ''S'' with the [[boundary condition]]s ''u''(''x'') = 1 on Σ and ''u''(''x'') = 0 on ''S'';
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| * ''S''′ is any intermediate surface between Σ and ''S'';
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| * ''ν'' is the outward [[unit normal]] [[vector field|field]] to ''S''′ and
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| ::<math>\frac{\partial u}{\partial \nu} (x) = \nabla u (x) \cdot \nu (x)</math>
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| :is the [[normal derivative]] of ''u'' across ''S''′; and
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| * ''σ''<sub>''n''</sub> = 2''π''<sup>''n''⁄2</sup> ⁄ Γ(''n'' ⁄ 2) is the surface area of the [[unit sphere]] in ℝ<sup>''n''</sup>.
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| ''C''(Σ, ''S'') can be equivalently defined by the volume integral
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| :<math>C(\Sigma, S) = \frac1{(n - 2) \sigma_{n}} \int_{D} | \nabla u |^{2}\mathrm{d}x.</math>
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| The condenser capacity also has a [[calculus of variations|variational characterization]]: ''C''(Σ, ''S'') is the [[infimum]] of the [[Dirichlet's energy]] [[functional (mathematics)|functional]]
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| :<math>I[v] = \frac1{(n - 2) \sigma_{n}} \int_{D} | \nabla v |^{2}\mathrm{d}x</math>
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| over all [[smooth function|continuously-differentiable functions]] ''v'' on ''D'' with ''v''(''x'') = 1 on Σ and ''v''(''x'') = 0 on ''S''.
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| ===Harmonic/Newtonian capacity===
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| [[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'' = 1 on Σ and ''u''(''x'') → 0 as ''x'' → ∞. 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
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| :<math>C(K) = \int_{\mathbb{R}^n\setminus K} |\nabla u|^2\mathrm{d}x.</math> | |
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| 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'':
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| :<math>C(K) = \int_S \frac{\partial u}{\partial\nu}\,\mathrm{d}\sigma.</math>
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| 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, ''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:
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| :<math>C(K) = \lim_{r \to \infty} C(\Sigma, S_{r}).</math>
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| 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'') < +∞.
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| ==Generalizations==
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| 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]].
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| ===Divergence form elliptic operators===
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| Solutions to a uniformly [[elliptic partial differential equation]] with divergence form
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| :<math> \nabla \cdot ( A \nabla u ) = 0 </math>
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| are minimizers of the associated energy functional
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| :<math>I[u] = \int_D (\nabla u)^T A (\nabla u)\,\mathrm{d}x</math>
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| subject to appropriate boundary conditions.
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| 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'') = 1 on ''E''; and ''v''(''x'') = 0 on the boundary of ''D''.
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| 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.
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| ==See also==
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| *[[Capacitance]]
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| *[[Newtonian potential]]
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| *[[Potential theory]]
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| ==References==
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| * {{citation
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| | last = Brélot
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| | first = Marcel
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| | author-link = Marcel Brélot
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| | title = Lectures on potential theory (Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy.)
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| | series = Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Mathematics.
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| | volume = No. 19
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| | edition = 2nd
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| | publisher = Tata Institute of Fundamental Research
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| | location = Bombay
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| | year = 1967
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| | origyear = 1960
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| | pages = ii+170+iv
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| | url = http://www.math.tifr.res.in/~publ/ln/tifr19.pdf
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| | mr = 0259146
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| | zbl= 0257.31001
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| }}. 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.
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| *{{Citation
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| | last = Choquet
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| | first = Gustave
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| | author-link = Gustave Choquet
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| | title = La naissance de la théorie des capacités: réflexion sur une expérience personnelle
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| | journal = [[Comptes rendus de l'Académie des sciences|Comptes rendus de l'Académie des sciences. Série générale, La Vie des sciences]]
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| | volume = 3
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| | issue = 4
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| | pages = 385–397
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| | year = 1986
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| | language = French
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| | url = http://gallica.bnf.fr/ark:/12148/bpt6k54708101/f85
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| | mr = 0867115
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| | zbl = 0607.01017
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| }}, 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".
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| *{{citation
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| | last = Doob
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| | first = Joseph Leo
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| | author-link = Joseph Leo Doob
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| | title = Classical potential theory and its probabilistic counterpart
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| | series = Grundlehren der Mathematischen Wissenschaften
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| | volume = 262
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| | publisher = Springer-Verlag
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| | location = Berlin–[[Heidelberg]]–New York
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| | year = 1984
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| | pages = xxiv+846
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| | isbn = 0-387-90881-1
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| | mr = 731258
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| | zbl = 0549.31001
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| }}
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| *{{Citation
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| | last = Littman
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| | first = W.
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| | author-link = Walter Littman
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| | last2 = Stampacchia
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| | first2 = G.
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| | author2-link = Guido Stampacchia
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| | last3 = Weinberger
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| | first3 = H.
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| | author3-link = Hans Weinberger
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| | year = 1963
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| | title = Regular points for elliptic equations with discontinuous coefficients
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| | journal = Annali della Scuola Normale Superiore di Pisa – Classe di Scienze
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| | series = Serie III
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| | volume = 17
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| | issue = 12
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| | pages = 43–77
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| | url = http://www.numdam.org/item?id=ASNSP_1963_3_17_1-2_43_0
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| | mr = 161019
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| | zbl = 0116.30302
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| }}, available at [http://www.numdam.org NUMDAM].
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| * {{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 }}
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| * {{springer
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| | id = c/c020280
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| | title = Capacity of a set
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| | last = Solomentsev
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| | first = E. D.
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| }}
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| [[Category:Potential theory]]
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