Capacity

of a set

A set function arising in potential theory as the analogue of the physical concept of the electrostatic capacity.

Let $ S $ and $ S ^ {*} $ be two smooth closed hypersurfaces in a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 3 $, with $ S ^ {*} $ enclosing $ S $. Such a system is called a condenser $ ( S , S ^ {*} ) $. Let $ u (x) $ be the harmonic function in the domain $ D $ between $ S $ and $ S ^ {*} $ taking the value 1 on $ S $ and $ 0 $ on $ S ^ {*} $. The condenser capacity $ C ( S , S ^ {*} ) $ is the number

$$ \tag{1 } C ( S , S ^ {*} ) = - \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ {S ^ \prime } \frac{\partial u (x) }{\partial n } d \sigma = $$

$$ = \ \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ { D } | \mathop{\rm grad} u (x) | ^ {2} d \omega , $$

where $ \sigma _ {n} = 2 \pi ^ {n/2} / \Gamma ( n / 2 ) $ is the area of the unit sphere in $ \mathbf R ^ {n} $, $ \partial u / \partial n $ is the derivative in the direction of the outward normal to an arbitrary intermediate hypersurface $ S ^ \prime $ lying between $ S $ and $ S ^ {*} $ and enclosing $ S $, $ d \sigma $ is the area element on $ S ^ \prime $, and $ d \omega $ is the volume element. Alternatively, the condenser capacity $ C ( S , S ^ {*} ) $ may be defined as the infimum of the integrals

$$ \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ { D } | \mathop{\rm grad} v (x) | ^ {2} d \omega $$

in the class of all continuously-differentiable functions $ v (x) $ in $ D $ that take the values 1 and 0 on $ S $ and $ S ^ {*} $, respectively. If $ S ^ {*} = S ( 0 , R ) $ is a sphere with centre at the origin and sufficiently large radius $ R $, then, letting $ R \rightarrow \infty $ in (1), one obtains the capacity of the compact set $ K $ bounded by $ S $, also called the harmonic capacity of $ K $ or the Newtonian capacity of $ K $:

$$ C (K) = \lim\limits _ {R \rightarrow \infty } \ C ( S , S ( 0 , R ) ) , $$

which always satisfies $ 0 \leq C (K) < \infty $. $ C (K) $ is the analogue of the electrostatic capacity of the isolated conductor $ K $.

In the case of the plane $ \mathbf R ^ {2} $ a condenser $ ( L , L ^ {*} ) $ is a system of two non-intersecting smooth simple closed curves $ L $ and $ L ^ {*} $ with $ L ^ {*} $ enclosing $ L $. Let $ u (x) $ be the harmonic function in the domain $ D $ between $ L $ and $ L ^ {*} $ taking the value 1 on $ L $ and $ 0 $ on $ L ^ {*} $. The condenser capacity $ C ( L , L ^ {*} ) $ is the number

$$ C ( L , L ^ {*} ) = - \frac{1}{2 \pi } \int\limits _ {L ^ \prime } \frac{\partial u (x) }{\partial n } ds = \frac{1}{2 \pi } \int\limits _ { D } | \mathop{\rm grad} u (x) | ^ {2} d \omega , $$

where $ ds $ is the element of arc length of a curve $ L ^ \prime $ lying between $ L $ and $ L ^ {*} $ and enclosing $ L $. Let $ L ^ {*} = S ( 0 , R ) $ be a circle with centre at the origin and sufficiently large radius $ R $; then letting $ R \rightarrow \infty $ in the formula,

$$ W (K) = \lim\limits _ {R \rightarrow \infty } \ \left [ \frac{1}{C ( L , S ( 0 , R ) ) } - \mathop{\rm ln} R \right ] , $$

gives the Wiener capacity, or the Robin constant, of the compact set $ K $ bounded by $ L $; the Wiener capacity can take any value $ - \infty < W (K) < \infty $. The logarithmic capacity, also called the harmonic capacity or the conformal capacity, is more often used:

$$ \tag{2 } C (K) = e ^ {-W(K)} = \ \lim\limits _ {R \rightarrow \infty } R e ^ {- 1 / {C ( L , S ( 0 , R ) ) } } , $$

it varies between $ 0 \leq C (K) < \infty $.

The capacity of a compact set $ K $ bounded by a hypersurface $ S $ for $ n \geq 3 $ may also be defined rather differently. Let $ v _ {K} (x) $ be the capacitary, or equilibrium, potential of this compact set (cf. Capacity potential), that is, the function harmonic everywhere outside $ K $, regular at infinity and taking the value 1 on $ S $. Then

$$ \tag{3 } C (K) = - \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ { S } \frac{\partial v _ {K} (x) }{\partial n } d \sigma = $$

$$ = \ \frac{1}{( n - 2 ) \sigma _ {n} } \int\limits _ {D ^ \prime } | \mathop{\rm grad} v _ {K} (x) | ^ {2} d \omega , $$

where $ D ^ \prime $ is the exterior of $ S $. Formula (3) shows that the capacity $ C (K) $ is a positive measure, distributed on $ S $ and such that the Newtonian potential of the simple layer generated by this measure coincides precisely with the capacitary potential $ v _ {K} (x) $, that is,

$$ \left . \begin{array}{l} v _ {K} (x) = \int\limits _ { S } \frac{d \mu (y) }{| x - y | ^ {n-2} } ,\ x \in D ^ \prime ; \\ C (K) = \int\limits _ { S } d \mu (y) = \mu (S). \\ \end{array} \right \} $$

The measure $ \mu $ is called the capacitary, or equilibrium, measure.

In the class of all positive Borel measures $ \lambda $ on $ K $ such that $ \lambda (K) = \mu (S) = C (K) $, the capacitary measure $ \mu $ minimizes the energy integral

$$ \tag{4 } E ( \lambda ) = {\int\limits \int\limits } _ {K \times K } \frac{d \lambda (x) d \lambda (y) }{| x - y | ^ {n-2} } . $$

In other words, the capacity $ C (K) $ can be defined by the formula $ C (K) = 1 / \inf E ( \lambda ) $, where the infimum is taken over the class of all positive measures $ \lambda $ concentrated on $ K $ and normalized by the condition $ \lambda (K) = 1 $.

For $ n = 2 $, 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 $ ( L , S ( 0 , R ) ) $, by means of the Green function $ G ( x , y ) $ for the interior $ \Delta $ of the circle $ S ( 0 , R) $, in the form

$$ \tag{5 } \left . \begin{array}{l} u _ {K} ( x ; S ( 0 , R ) ) = \int\limits _ { L } G ( x , y ) d \mu (y) ,\ x \in D ; \\ C ( L , S ( 0 , R ) ) = \int\limits _ { L } d \mu (y) = \ \mu (L) , \\ \end{array} \right \} $$

where the capacitary potential $ u _ {K} ( x ; S ( 0 , R ) ) $ coincides in $ D $ with the harmonic function $ u (x) $ introduced earlier for $ ( L , S ( 0 , R ) ) $. The capacity defined by formula (5) is sometimes called the Green capacity; this construction is possible for any $ n \geq 2 $. The formula $ W (K) = 1 / \inf E ( \lambda ) $, $ \lambda (K) = 1 $, for $ n = 2 $ gives the Wiener capacity of the compact set $ K $, and the energy integral

$$ \tag{6 } E ( \lambda ) = {\int\limits \int\limits } _ {K \times K } \mathop{\rm ln} { \frac{1}{| x-y | } } d \lambda (x) d \lambda (y) $$

is now not always positive.

The capacity of an arbitrary compact set $ K \subset \mathbf R ^ {n} $, $ n \geq 3 $, may be defined by means of the above property of minimum energy:

$$ C (K) = \frac{1}{\inf E ( \lambda ) } ,\ \ \lambda (K) = 1 ,\ \lambda \geq 0 , $$

where the integrals $ E ( \lambda ) $ are computed as in formula (4). For $ n = 2 $ this leads to the definition of the Wiener capacity of an arbitrary compact set:

$$ W (K) = \frac{1}{\inf E ( \lambda ) } ,\ \ \lambda (K) = 1 ,\ \lambda \geq 0 , $$

where the energy $ E ( \lambda ) $ is computed as in formula (6). The transition to the logarithmic capacity is effected by the formula $ C (K) = e ^ {-W(K)} $.

An equivalent method is the construction of a capacitary potential $ v _ {K} (x) $ for an arbitrary compact set $ K $. For $ n \geq 3 $ it may be defined as the largest of the potentials $ U _ \lambda (x) $ of the positive measures $ \lambda $ concentrated on $ K $ for which $ U _ \lambda (x) \leq 1 $. The measure $ \mu $ generating $ v _ {K} (x) $ is the capacity measure, $ \mu (K) = C (K) $. For $ n = 2 $, the construction of the capacitary potential is done as above for a condenser $ ( K , S ( 0 , R ) ) $ by means of the Green function for the disc $ \Delta $. The capacity $ C (K) $ of a compact set is then obtained by limit transition, as in formula (2).

If $ v _ {K} (x) = 0 $, then $ C (K) = 0 $. For $ n = 2 $, the equations $ C (K) = 0 $ and $ W (K) = + \infty $ 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 $ v _ {K} (x) = 1 $ on $ K $ 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 $ K $ of capacity zero is unbounded. In addition, for any compact set $ K $ of capacity zero, there exists a positive measure $ \nu $, concentrated on $ K $, such that $ U _ \nu (x) = + \infty $ for $ x \in K $ and $ U _ \nu (x) < + \infty $ for $ x \notin K $, that is, any compact set of capacity zero is a polar set.

Properties of capacitary potentials and capacities of compact sets: 1) The mappings $ K \rightarrow v _ {K} (x) $ and $ K \rightarrow C (K) $ are increasing, that is, $ K _ {1} \subset K _ {2} $ implies that $ v _ {K _ {1} } (x) \leq v _ {K _ {2} } (x) $ everywhere and $ C ( K _ {1} ) \leq C ( K _ {2} ) $; 2) these mappings are continuous on the right, that is, for any fixed $ x \in \mathbf R ^ {n} $ and any $ \epsilon > 0 $ there exists an open set $ \omega $ such that if a compact set $ K ^ \prime $ satisfies $ K \subset K ^ \prime \subset \omega $, then $ v _ {K ^ \prime } (x) - v _ {K} (x) < \epsilon $ everywhere, and $ C ( K ^ \prime ) - C (K) < \epsilon $; and 3) $ v _ {K} (x) $ and $ C (K) $ are strongly subadditive as functions of $ K $, that is,

$$ v _ {K _ {1} \cup K _ {2} } (x) + v _ {K _ {1} \cap K _ {2} } (x) \leq v _ {K _ {1} } (x) + v _ {K _ {2} } (x) , $$

$$ C ( K _ {1} \cup K _ {2} ) + C ( K _ {1} \cap K _ {2} ) \leq C ( K _ {1} ) + C ( K _ {2} ) . $$

If $ G $ is an open set lying in a ball $ B = B ( 0 , R ) $, then, by definition, $ C (G) = C (B) - C ( \overline{B}\; \setminus G ) $. For an arbitrary set $ E $, the inner capacity $ \underline{C} (E) $ is defined as the least upper bound $ \underline{C} (E) = \sup C (K) $ over all compact sets $ K \subset E $. The outer capacity $ \overline{C}\; (E) $ is defined as the greatest lower bound $ \overline{C}\; (E) = \inf C (G) $ over all open sets $ G \subset E $. A set $ E $ is called capacitable if $ \overline{C}\; (E) = \underline{C} (E) = C (E) $. All Borel sets, and even all analytic sets, in $ \mathbf R ^ {n} $ are capacitable. Thus, the capacity $ C (E) $ is a set function invariant under motions, but, however, not additive. The fact that the capacity $ C (E) $ of a set $ E $ is zero is a very important property of $ E $. 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 $ w (x) $ be a subharmonic function bounded from above on a domain $ G \subset \mathbf R ^ {n} $, $ n \geq 3 $, $ \infty \notin G $; let $ \lim\limits _ {x \rightarrow y } \sup w (x) \leq M $ hold for all $ y \in \partial G $, with the possible exception of a set $ E $ with $ C (E) = 0 $, $ \infty \notin E $. Then $ w (x) \leq M $ everywhere in $ G $, and equality, even at a single point, is possible only if $ w (x) \equiv M $.

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 $ X $ is defined axiomatically as a numerical set function $ K \rightarrow C (K) $ 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 $ X $ 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 $ K $- analytic sets, i.e. continuous images of sets of type $ K _ {\sigma \delta } $ 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 $ K $ be a compact set in the complex $ z $- plane, let $ f (z) $ be an analytic function outside $ K $, $ | f (z) | \leq 1 $, $ f ( \infty ) = 0 $; the analytic capacity $ \gamma (K) $ is the number

$$ \gamma (K) = \sup \ \left | \frac{1}{2 \pi } \int\limits _ {L ^ \prime } f (z) d z \right | , $$

where $ L ^ \prime $ is a contour enclosing $ K $ and the supremum is taken over all $ f (z) $ satisfying the stated conditions.

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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 $ \mathbf C $. Recently one has started to study capacities in $ \mathbf C ^ {n} $ in relation with obtaining bounds on the growth of analytic functions on domains in $ \mathbf C ^ {n} $, cf. [a4] and the references given there.

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