In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the capacity of a set in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
is a measure of the "size" of that set. Unlike, say,
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
, which measures a set's
volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
or physical extent, capacity is a mathematical analogue of a set's ability to hold
electrical charge
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
. More precisely, it is the
capacitance
Capacitance is the ability of an object to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related ...
of the set: the total charge a set can hold while maintaining a given
potential energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
. 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 .
Definitions
Condenser capacity
Let Σ be a
closed, smooth, (''n'' − 1)-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al
hypersurface in ''n''-dimensional Euclidean space
, will denote the ''n''-dimensional
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
(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
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, 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
:
where:
* ''u'' is the unique
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that i ...
defined on the region ''D'' between Σ and ''S'' with the
boundary conditions ''u''(''x'') = 1 on Σ and ''u''(''x'') = 0 on ''S'';
* ' is any intermediate surface between Σ and ''S'';
* ''ν'' is the outward
unit normal field to ' and
::
:is the
normal derivative of ''u'' across '; and
* ''σ''
''n'' = 2''π''
''n''⁄2 ⁄ Γ(''n'' ⁄ 2) is the surface area of the
unit sphere
In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
in
.
''C''(Σ, ''S'') can be equivalently defined by the volume integral
:
The condenser capacity also has a
variational characterization: ''C''(Σ, ''S'') is the
infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
of the
Dirichlet's energy functional
:
over all
continuously differentiable functions ''v'' on ''D'' with ''v''(''x'') = 1 on Σ and ''v''(''x'') = 0 on ''S''.
Harmonic capacity
Heuristic
A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...
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 or Newtonian capacity of ''K'', denoted ''C''(''K'') or cap(''K''), is then defined by
:
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'':
:
The harmonic capacity can also be understood as a limit of the condenser capacity. To wit, let ''S''
''r'' denote the
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
of radius ''r'' about the origin in
. Since ''K'' is bounded, for sufficiently large ''r'', ''S''
''r'' will enclose ''K'' and (Σ, ''S''
''r'') will form a condenser pair. The harmonic capacity is then the
limit as ''r'' tends to infinity:
:
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'') < +∞.
The Wiener capacity or Robin constant ''W(K)'' of ''K'' is given by
:
Logarithmic capacity
In two dimensions, the capacity is defined as above, but dropping the factor of
in the definition:
:
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
limit. This is articulated below. It may also be called the conformal capacity, in reference to its relation to the
conformal radius.
Properties
The harmonic function ''u'' is called the capacity potential. It is called the
Newtonian potential when
and called the logarithmic potential when
. It can be obtained via a
Green's function
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if L is a linear dif ...
''G'' as
:
with ''x'' a point exterior to ''S'', where ''G'' is defined as
:
when
and
:
for
.
The
measure is called the capacitary measure or equilibrium measure. It is generally taken to be a
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
...
. It is related to the capacity as
:
The variational definition of capacity over the
Dirichlet energy can be re-expressed as
:
with the infimum taken over all positive Borel measures
concentrated on ''K'', normalized so that
and with
is the energy integral
:
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
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of f ...
.
Divergence form elliptic operators
Solutions to a uniformly
elliptic partial differential equation
In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which gene ...
with divergence form
:
are minimizers of the associated energy functional
:
subject to appropriate boundary conditions.
The capacity of a set ''E'' with respect to a domain ''D'' containing ''E'' is defined as the
infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
of the energy over all
continuously differentiable functions ''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
obstacle problem on ''D'' with the obstacle function provided by the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
of ''E''. The capacitary potential is alternately characterized as the unique solution of the equation with the appropriate boundary conditions.
See also
*
*
*
*
*
References
* . 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.
*, 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".
*
*, available a
NUMDAM
*
*
*
*
{{Authority control
Potential theory