Capacity of a set

<h2 id="historical-note">Historical note</h2>
The notion of capacity of a set and of "capacitable" set was introduced by <a href="/facts/Gustave_Choquet/anvF1pPY">Gustave Choquet</a> in 1950: for a detailed account, see reference (Choquet 1986).

<h2 id="definitions">Definitions</h2>
<h3>Condenser capacity</h3>
Let Σ be a <a href="/facts/Closed_surface/jK1b1KIG">closed</a>, smooth, (n − 1)-<a href="/facts/Dimension/0ktmUyac">dimensional</a> <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a> in n-dimensional Euclidean space 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
, n ≥ 3; K will denote the n-dimensional <a href="/facts/Compact_space/d0cgXJH7">compact</a> (i.e., <a href="/facts/Closed_set/upYnRTUj">closed</a> and <a href="/facts/Bounded_set/yCldjtoy">bounded</a>) set of which Σ is the <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a>. Let S be another (n − 1)-dimensional hypersurface that encloses Σ: in reference to its origins in <a href="/facts/Electromagnetism/oAkmALHT">electromagnetism</a>, the pair (Σ, S) is known as a <a href="/facts/Capacitor/hyowqv7n">condenser</a>. The condenser capacity of Σ relative to S, denoted C(Σ, S) or cap(Σ, S), is given by the surface integral

C
        (
        Σ
        ,
        S
        )
        =
        −
        
          
            1
            
              (
              n
              −
              2
              )
              
                σ
                
                  n
                
              
            
          
        
        
          ∫
          
            
              S
              ′
            
          
        
        
          
            
              ∂
              u
            
            
              ∂
              ν
            
          
        
        
        
          d
        
        
          σ
          ′
        
        ,
      
    
    {\displaystyle C(\Sigma ,S)=-{\frac {1}{(n-2)\sigma _{n}}}\int _{S'}{\frac {\partial u}{\partial \nu }}\,\mathrm {d} \sigma ',}

where:

<ul><li>u is the unique <a href="/facts/Harmonic_function/H6pg4dMT">harmonic function</a> defined on the region D between Σ and S with the <a href="/facts/Boundary_condition/QEfmUqmP">boundary conditions</a> u(x) = 1 on Σ and u(x) = 0 on S;</li>
<li>S′ is any intermediate surface between Σ and S;</li>
<li>ν is the outward <a href="/facts/Unit_normal/jhtLIhcu">unit normal</a> <a href="/facts/Vector_field/ccFNuPPp">field</a> to S′ and</li></ul>

∂
              u
            
            
              ∂
              ν
            
          
        
        (
        x
        )
        =
        ∇
        u
        (
        x
        )
        ⋅
        ν
        (
        x
        )
      
    
    {\displaystyle {\frac {\partial u}{\partial \nu }}(x)=\nabla u(x)\cdot \nu (x)}

is the <a href="/facts/Normal_derivative/Fjc3JSVn">normal derivative</a> of u across S′; and
<ul><li>σn = 2πn⁄2 ⁄ Γ(n ⁄ 2) is the surface area of the <a href="/facts/Unit_sphere/5UeoTcug">unit sphere</a> in 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
.</li></ul>
C(Σ, S) can be equivalently defined by the volume integral

C
        (
        Σ
        ,
        S
        )
        =
        
          
            1
            
              (
              n
              −
              2
              )
              
                σ
                
                  n
                
              
            
          
        
        
          ∫
          
            D
          
        
        
          |
        
        ∇
        u
        
          
            |
          
          
            2
          
        
        
          d
        
        x
        .
      
    
    {\displaystyle C(\Sigma ,S)={\frac {1}{(n-2)\sigma _{n}}}\int _{D}|\nabla u|^{2}\mathrm {d} x.}

The condenser capacity also has a <a href="/facts/Calculus_of_variations/Ya5Lsma3">variational characterization</a>: C(Σ, S) is the <a href="/facts/Infimum/oXCKVopz">infimum</a> of the <a href="/facts/Dirichlet%2527s_energy/M31YfPKF">Dirichlet's energy</a> <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a>

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

over all <a href="/facts/Smooth_function/mffL4ch2">continuously differentiable functions</a> v on D with v(x) = 1 on Σ and v(x) = 0 on S.

<h3>Harmonic capacity</h3>
<a href="/facts/Heuristic/BcBEkv9c">Heuristically</a>, 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 <a href="/facts/Newtonian_potential/gWpfnW6G">Newtonian potential</a> of the simple layer Σ. Then the harmonic capacity or Newtonian capacity of K, denoted C(K) or cap(K), is then defined by

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

If S is a <a href="/facts/Rectifiable_set/nylBklGT">rectifiable hypersurface</a> completely enclosing K, then the harmonic capacity can be equivalently rewritten as the integral over S of the outward normal derivative of u:

C
        (
        K
        )
        =
        
          ∫
          
            S
          
        
        
          
            
              ∂
              u
            
            
              ∂
              ν
            
          
        
        
        
          d
        
        σ
        .
      
    
    {\displaystyle 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, let Sr denote the <a href="/facts/Sphere/jywYLNM2">sphere</a> of radius r about the origin in 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
. Since K is bounded, for sufficiently large r, Sr will enclose K and (Σ, Sr) will form a condenser pair. The harmonic capacity is then the <a href="/facts/Limit_of_a_function/HFbQ69e5">limit</a> as r tends to infinity:

C
        (
        K
        )
        =
        
          lim
          
            r
            →
            ∞
          
        
        C
        (
        Σ
        ,
        
          S
          
            r
          
        
        )
        .
      
    
    {\displaystyle C(K)=\lim _{r\to \infty }C(\Sigma ,S_{r}).}

The harmonic capacity is a mathematically abstract version of the <a href="/facts/Capacitance/0pH046TL">electrostatic capacity</a> of the conductor K and is always non-negative and finite: 0 ≤ C(K) < +∞.
The Wiener capacity or Robin constant W(K) of K is given by

C
        (
        K
        )
        =
        
          e
          
            −
            W
            (
            K
            )
          
        
      
    
    {\displaystyle C(K)=e^{-W(K)}}

<h3>Logarithmic capacity</h3>
In two dimensions, the capacity is defined as above, but dropping the factor of 
 
 
 
 (
 n
 −
 2
 )
 
 
 {\displaystyle (n-2)}
 
 in the definition:

C
        (
        Σ
        ,
        S
        )
        =
        −
        
          
            1
            
              2
              π
            
          
        
        
          ∫
          
            
              S
              ′
            
          
        
        
          
            
              ∂
              u
            
            
              ∂
              ν
            
          
        
        
        
          d
        
        
          σ
          ′
        
        =
        
          
            1
            
              2
              π
            
          
        
        
          ∫
          
            D
          
        
        
          |
        
        ∇
        u
        
          
            |
          
          
            2
          
        
        
          d
        
        x
      
    
    {\displaystyle 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 the logarithmic capacity, the term logarithmic arises, as the potential function goes from being an inverse power to a logarithm in the 
 
 
 
 n
 →
 2
 
 
 {\displaystyle n\to 2}
 
 limit. This is articulated below. It may also be called the conformal capacity, in reference to its relation to the <a href="/facts/Conformal_radius/LTc0horh">conformal radius</a>.

<h2 id="properties">Properties</h2>
The harmonic function u is called the capacity potential, the <a href="/facts/Newtonian_potential/gWpfnW6G">Newtonian potential</a> when 
 
 
 
 n
 ≥
 3
 
 
 {\displaystyle n\geq 3}
 
 and the logarithmic potential when 
 
 
 
 n
 =
 2
 
 
 {\displaystyle n=2}
 
. It can be obtained via a <a href="/facts/Green%2527s_function/rKxKKPcp">Green's function</a> as

u
        (
        x
        )
        =
        
          ∫
          
            S
          
        
        G
        (
        x
        −
        y
        )
        d
        μ
        (
        y
        )
      
    
    {\displaystyle u(x)=\int _{S}G(x-y)d\mu (y)}

with x a point exterior to S, and

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

when 
 
 
 
 n
 ≥
 3
 
 
 {\displaystyle n\geq 3}
 
 and

G
        (
        x
        −
        y
        )
        =
        log
        ⁡
        
          
            1
            
              
                |
              
              x
              −
              y
              
                |
              
            
          
        
      
    
    {\displaystyle G(x-y)=\log {\frac {1}{|x-y|}}}

for 
 
 
 
 n
 =
 2
 
 
 {\displaystyle n=2}
 
.
The <a href="/facts/Measure_(mathematics)/yCq7zlI4">measure</a> 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is called the capacitary measure or equilibrium measure. It is generally taken to be a <a href="/facts/Borel_measure/8cYR8Ysw">Borel measure</a>. It is related to the capacity as

C
        (
        K
        )
        =
        
          ∫
          
            S
          
        
        d
        μ
        (
        y
        )
        =
        μ
        (
        S
        )
      
    
    {\displaystyle C(K)=\int _{S}d\mu (y)=\mu (S)}

The variational definition of capacity over the <a href="/facts/Dirichlet_energy/M31YfPKF">Dirichlet energy</a> can be re-expressed as

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

with the infimum taken over all positive Borel measures 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 concentrated on K, normalized so that 
 
 
 
 λ
 (
 K
 )
 =
 1
 
 
 {\displaystyle \lambda (K)=1}
 
 and with 
 
 
 
 E
 (
 λ
 )
 
 
 {\displaystyle E(\lambda )}
 
 is the energy integral

E
        (
        λ
        )
        =
        ∫
        
          ∫
          
            K
            ×
            K
          
        
        G
        (
        x
        −
        y
        )
        d
        λ
        (
        x
        )
        d
        λ
        (
        y
        )
      
    
    {\displaystyle E(\lambda )=\int \int _{K\times K}G(x-y)d\lambda (x)d\lambda (y)}

<h2 id="generalizations">Generalizations</h2>
The characterization of the capacity of a set as the minimum of an <a href="/facts/Energy_functional/2v0ElfB9">energy functional</a> achieving particular boundary values, given above, can be extended to other energy functionals in the <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a>.

<h3>Divergence form elliptic operators</h3>
Solutions to a uniformly <a href="/facts/Elliptic_partial_differential_equation/pQx2wB7V">elliptic partial differential equation</a> with divergence form

∇
        ⋅
        (
        A
        ∇
        u
        )
        =
        0
      
    
    {\displaystyle \nabla \cdot (A\nabla u)=0}

are minimizers of the associated energy functional

I
        [
        u
        ]
        =
        
          ∫
          
            D
          
        
        (
        ∇
        u
        
          )
          
            T
          
        
        A
        (
        ∇
        u
        )
        
        
          d
        
        x
      
    
    {\displaystyle I[u]=\int _{D}(\nabla u)^{T}A(\nabla u)\,\mathrm {d} x}

subject to appropriate boundary conditions.
The capacity of a set E with respect to a domain D containing E is defined as the <a href="/facts/Infimum/oXCKVopz">infimum</a> of the energy over all <a href="/facts/Smooth_function/mffL4ch2">continuously differentiable functions</a> 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 <a href="/facts/Obstacle_problem/XrRH7c0K">obstacle problem</a> on D with the obstacle function provided by the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of E. The capacitary potential is alternately characterized as the unique solution of the equation with the appropriate boundary conditions.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Analytic_capacity/CWAoWnlt">Analytic capacity</a> – number that denotes how big a certain bounded analytic function can becomePages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Capacitance/0pH046TL">Capacitance</a> – Ability of a body to store an electrical charge</li>
<li><a href="/facts/Newtonian_potential/gWpfnW6G">Newtonian potential</a> – Green's function for Laplacian</li>
<li><a href="/facts/Potential_theory/3sXGIcDA">Potential theory</a> – Harmonic functions as solutions to Laplace's equation</li>
<li><a href="/facts/Choquet_theory/vhTyXcTu">Choquet theory</a> – Area of functional analysis and convex analysis</li></ul>

<ul><li>Brélot, Marcel (1967) [1960], <a href="http://www.math.tifr.res.in/~publ/ln/tifr19.pdf">Lectures on potential theory (Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy.)</a> (PDF), Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Mathematics., vol. 19 (2nd ed.), Bombay: Tata Institute of Fundamental Research, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0259146">0259146</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0257.31001">0257.31001</a>. 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.</li>
<li><a href="/facts/Gustave_Choquet/anvF1pPY">Choquet, Gustave</a> (1986), <a href="http://gallica.bnf.fr/ark:/12148/bpt6k54708101/f85">"La naissance de la théorie des capacités: réflexion sur une expérience personnelle"</a>, <a href="/facts/Comptes_rendus_de_l%2527Acad%25C3%25A9mie_des_sciences/9yR9XTN5">Comptes rendus de l'Académie des sciences. Série générale, La Vie des sciences</a> (in French), 3 (4): 385–397, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0867115">0867115</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0607.01017">0607.01017</a>, available from <a href="/facts/Gallica/4TNyWsxG">Gallica</a>. 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".</li>
<li><a href="/facts/Joseph_Leo_Doob/HLcsS24G">Doob, Joseph Leo</a> (1984), <a href="https://archive.org/details/classicalpotenti0000doob/page/">Classical potential theory and its probabilistic counterpart</a>, Grundlehren der Mathematischen Wissenschaften, vol. 262, Berlin–<a href="/facts/Heidelberg/jfbzRTo0">Heidelberg</a>–New York: Springer-Verlag, pp. <a href="https://archive.org/details/classicalpotenti0000doob/page/">xxiv+846</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90881-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0731258">0731258</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0549.31001">0549.31001</a></li>
<li>Littman, W.; <a href="/facts/Guido_Stampacchia/Gw5ATqXt">Stampacchia, G.</a>; <a href="/facts/Hans_Weinberger/nthhhKLT">Weinberger, H.</a> (1963), <a href="http://www.numdam.org/item?id=ASNSP_1963_3_17_1-2_43_0">"Regular points for elliptic equations with discontinuous coefficients"</a>, Annali della Scuola Normale Superiore di Pisa – Classe di Scienze, Serie III, 17 (12): 43–77, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0161019">0161019</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0116.30302">0116.30302</a>, available at <a href="http://www.numdam.org">NUMDAM</a>.</li>
<li>Ransford, Thomas (1995), <a href="https://archive.org/details/potentialtheoryi0000rans">Potential theory in the complex plane</a>, London Mathematical Society Student Texts, vol. 28, Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-46654-7, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0828.31001">0828.31001</a></li>
<li>Solomentsev, E. D. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Capacity">"Capacity"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Solomentsev, E. D. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Robin_constant">"Robin constant"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Solomentsev, E. D. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Energy_of_measures">"Energy of measures"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>

Capacity of a set open-in-new

Capacity of a set