Capacity
of a set
A set function arising in potential theory as the analogue of the physical concept of the electrostatic capacity.
Let and
be two smooth closed hypersurfaces in a Euclidean space
,
, with
enclosing
. Such a system is called a condenser
. Let
be the harmonic function in the domain
between
and
taking the value 1 on
and
on
. The condenser capacity
is the number
![]() | (1) |
![]() |
where is the area of the unit sphere in
,
is the derivative in the direction of the outward normal to an arbitrary intermediate hypersurface
lying between
and
and enclosing
,
is the area element on
, and
is the volume element. Alternatively, the condenser capacity
may be defined as the infimum of the integrals
![]() |
in the class of all continuously-differentiable functions in
that take the values 1 and 0 on
and
, respectively. If
is a sphere with centre at the origin and sufficiently large radius
, then, letting
in (1), one obtains the capacity of the compact set
bounded by
, also called the harmonic capacity of
or the Newtonian capacity of
:
![]() |
which always satisfies .
is the analogue of the electrostatic capacity of the isolated conductor
.
In the case of the plane a condenser
is a system of two non-intersecting smooth simple closed curves
and
with
enclosing
. Let
be the harmonic function in the domain
between
and
taking the value 1 on
and
on
. The condenser capacity
is the number
![]() |
where is the element of arc length of a curve
lying between
and
and enclosing
. Let
be a circle with centre at the origin and sufficiently large radius
; then letting
in the formula,
![]() |
gives the Wiener capacity, or the Robin constant, of the compact set bounded by
; the Wiener capacity can take any value
. The logarithmic capacity, also called the harmonic capacity or the conformal capacity, is more often used:
![]() | (2) |
it varies between .
The capacity of a compact set bounded by a hypersurface
for
may also be defined rather differently. Let
be the capacitary, or equilibrium, potential of this compact set (cf. Capacity potential), that is, the function harmonic everywhere outside
, regular at infinity and taking the value 1 on
. Then
![]() | (3) |
![]() |
where is the exterior of
. Formula (3) shows that the capacity
is a positive measure, distributed on
and such that the Newtonian potential of the simple layer generated by this measure coincides precisely with the capacitary potential
, that is,
![]() |
The measure is called the capacitary, or equilibrium, measure.
In the class of all positive Borel measures on
such that
, the capacitary measure
minimizes the energy integral
![]() | (4) |
In other words, the capacity can be defined by the formula
, where the infimum is taken over the class of all positive measures
concentrated on
and normalized by the condition
.
For , because of the singular behaviour of the logarithmic potential at infinity, the construction given above for the capacitary potential is possible only for a condenser, for example, for
, by means of the Green function
for the interior
of the circle
, in the form
![]() | (5) |
where the capacitary potential coincides in
with the harmonic function
introduced earlier for
. The capacity defined by formula (5) is sometimes called the Green capacity; this construction is possible for any
. The formula
,
, for
gives the Wiener capacity of the compact set
, and the energy integral
![]() | (6) |
is now not always positive.
The capacity of an arbitrary compact set ,
, may be defined by means of the above property of minimum energy:
![]() |
where the integrals are computed as in formula (4). For
this leads to the definition of the Wiener capacity of an arbitrary compact set:
![]() |
where the energy is computed as in formula (6). The transition to the logarithmic capacity is effected by the formula
.
An equivalent method is the construction of a capacitary potential for an arbitrary compact set
. For
it may be defined as the largest of the potentials
of the positive measures
concentrated on
for which
. The measure
generating
is the capacity measure,
. For
, the construction of the capacitary potential is done as above for a condenser
by means of the Green function for the disc
. The capacity
of a compact set is then obtained by limit transition, as in formula (2).
If , then
. For
, the equations
and
are equivalent. Compact sets of capacity zero play the same role in potential theory as sets of measure zero in integration theory. For example, the equation
on
holds everywhere with the possible exception of a set of points belonging to some compact set of capacity zero. The potential of any positive measure concentrated on a compact set
of capacity zero is unbounded. In addition, for any compact set
of capacity zero, there exists a positive measure
, concentrated on
, such that
for
and
for
, that is, any compact set of capacity zero is a polar set.
Properties of capacitary potentials and capacities of compact sets: 1) The mappings and
are increasing, that is,
implies that
everywhere and
; 2) these mappings are continuous on the right, that is, for any fixed
and any
there exists an open set
such that if a compact set
satisfies
, then
everywhere, and
; and 3)
and
are strongly subadditive as functions of
, that is,
![]() |
![]() |
If is an open set lying in a ball
, then, by definition,
. For an arbitrary set
, the inner capacity
is defined as the least upper bound
over all compact sets
. The outer capacity
is defined as the greatest lower bound
over all open sets
. A set
is called capacitable if
. All Borel sets, and even all analytic sets, in
are capacitable. Thus, the capacity
is a set function invariant under motions, but, however, not additive. The fact that the capacity
of a set
is zero is a very important property of
. In many problems of potential theory sets of capacity zero in the above sense may be neglected. For example, the following strong maximum principle is valid. Let
be a subharmonic function bounded from above on a domain
,
,
; let
hold for all
, with the possible exception of a set
with
,
. Then
everywhere in
, and equality, even at a single point, is possible only if
.
The concept of a capacity can be generalized in various directions. Starting from the concept of a capacitary potential and a capacitary measure or an energy, theories of capacity have been constructed for non-Newtonian potentials, such as for Bessel potentials, non-linear potentials, Riesz potentials, and others (cf. Bessel potential; Non-linear potential; Riesz potential). In particular, these constructions enable one to vary the concept of a set of capacity zero in accordance with various problems of mathematical physics and analysis (see [6]).
According to G. Choquet, a capacity in an abstract separable topological space is defined axiomatically as a numerical set function
satisfying the following axioms: it is increasing, continuous on the right and strongly subadditive. A rather different approach to the theory of capacity in abstract spaces
can be found within the framework of the general axiomatics of abstract potential theory (cf. Potential theory, abstract) or the theory of harmonic spaces (cf. Harmonic space). In an abstract theory of capacities, a fundamental result is Choquet's theorem, stating that
-analytic sets, i.e. continuous images of sets of type
in some compact space, are capacitable.
In general, in various problems of function theory, mainly concerning approximation in specific classes of functions, it turns out to be useful to introduce an appropriate notion of capacity. For example, the concept of analytic capacity is of great importance in questions of approximation by analytic functions. Let be a compact set in the complex
-plane, let
be an analytic function outside
,
,
; the analytic capacity
is the number
![]() |
where is a contour enclosing
and the supremum is taken over all
satisfying the stated conditions.
References
[1] | N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) |
[2] | M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) |
[3] | G. Pólya, G. Szegö, "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press (1951) |
[4] | M. Brélot, "Lectures on potential theory" , Tata Inst. (1960) |
[5] | C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) |
[6] | L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967) |
[7] | A.G. Vitushkin, "Uniform approximation by holomorphic functions" J. Soviet Math. , 5 : 5 (1976) pp. 607–611 Itogi Nauk. i Tekhn. Sovrem. Pobl. Mat. , 4 (1975) pp. 5–12 |
Comments
For capacities within the framework of harmonic spaces see [a1]; [a2], [a3] contain among other things discussions of the Robin constant, where [a3] focuses on the relation with analytic functions in domains in . Recently one has started to study capacities in
in relation with obtaining bounds on the growth of analytic functions on domains in
, cf. [a4] and the references given there.
References
[a1] | C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) |
[a2] | J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 |
[a3] | M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975) |
[a4] | J. Korevaar, "Green functions, capacities, polynomial approximation numbers and applications in real and complex analysis" Nieuw Archief voor Wiskunde IV , 4 : 2 (1986) pp. 133–153 |
Capacity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Capacity&oldid=28160