6.1 Capacity

Capacity is a key concept in potential theory that measures a set's ability to hold electric charge or conduct electricity. It's defined using potentials, measures, and energies, providing insights into the size and structure of sets in relation to their potential-theoretic properties.

Capacity has important connections to regularity, polarity, and Hausdorff measures. It helps characterize sets' boundary behavior, size, and structure in terms of potential theory. Understanding capacity is crucial for analyzing electric fields, solving boundary value problems, and studying geometric properties of sets.

Definition of capacity

• Capacity is a fundamental concept in potential theory that measures the size or content of a set in a way that is compatible with the underlying potential-theoretic structure
• It provides a way to quantify the ability of a set to hold an electric charge or the effectiveness of a set as a conductor of electricity
• Capacity is defined using potential-theoretic notions such as potentials, measures, and energies, and it has important connections to other concepts in potential theory, such as regularity and polarity

Capacity of a set

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• The capacity of a set $E$ in a metric space $X$ is defined as $\operatorname{cap}(E) = \sup\{\mu(E) : \mu \in \mathcal{M}(X), \mathcal{E}(\mu) \leq 1\}$, where $\mathcal{M}(X)$ is the space of Borel measures on $X$ and $\mathcal{E}(\mu)$ is the energy of the measure $\mu$
• Intuitively, the capacity of a set measures the maximum amount of charge that can be placed on the set while keeping the energy of the charge distribution bounded
• The capacity of a set depends on the underlying metric space and the potential kernel used to define the energy of measures

Capacity of a compact set

• For a compact set $K$ in a locally compact metric space $X$, the capacity can be equivalently defined as $\operatorname{cap}(K) = (\inf\{\mathcal{E}(\mu) : \mu \in \mathcal{M}(K), \mu(K) = 1\})^{-1}$
• This definition highlights the connection between capacity and the minimum energy problem for measures supported on the set
• The capacity of a compact set is always finite and can be computed using potential-theoretic techniques such as the Riesz representation theorem and the Frostman lemma

Capacity of an open set

• For an open set $U$ in a locally compact metric space $X$, the capacity is defined as $\operatorname{cap}(U) = \sup\{\operatorname{cap}(K) : K \subset U, K \text{ compact}\}$
• The capacity of an open set is the supremum of the capacities of all compact subsets contained in the open set
• The capacity of an open set can be infinite, depending on the size and geometry of the set and the properties of the underlying metric space

Properties of capacity

• Capacity has several important properties that make it a useful tool in potential theory and related fields
• These properties reflect the underlying potential-theoretic structure and provide insights into the behavior of sets and measures in the context of potential theory
• The properties of capacity are often used in proofs and arguments involving potential-theoretic concepts such as regularity, polarity, and Hausdorff measures

Monotonicity of capacity

• Capacity is monotone: if $A \subset B$, then $\operatorname{cap}(A) \leq \operatorname{cap}(B)$
• This property reflects the intuitive idea that larger sets have a greater capacity to hold charge or conduct electricity
• Monotonicity is a fundamental property of capacity and is often used in proofs and arguments involving set inclusions and comparisons

Subadditivity of capacity

• Capacity is subadditive: for any countable collection of sets $\{E_i\}$, $\operatorname{cap}(\bigcup_i E_i) \leq \sum_i \operatorname{cap}(E_i)$
• Subadditivity implies that the capacity of a union of sets is bounded above by the sum of the capacities of the individual sets
• This property is useful in estimating the capacity of complex sets and in proving the existence of sets with specific capacity properties

Continuity of capacity from above

• Capacity is continuous from above: if $\{E_i\}$ is a decreasing sequence of sets with $\bigcap_i E_i = E$, then $\lim_{i \to \infty} \operatorname{cap}(E_i) = \operatorname{cap}(E)$
• Continuity from above implies that the capacity of a decreasing intersection of sets is equal to the limit of the capacities of the sets
• This property is important in the study of capacitary potential and in the proof of the Choquet capacitability theorem

Continuity of capacity from below

• Capacity is continuous from below: if $\{K_i\}$ is an increasing sequence of compact sets with $\bigcup_i K_i = K$, then $\lim_{i \to \infty} \operatorname{cap}(K_i) = \operatorname{cap}(K)$
• Continuity from below implies that the capacity of an increasing union of compact sets is equal to the limit of the capacities of the sets
• This property is used in the construction of equilibrium measures and in the proof of the Wiener criterion for regular points

Capacity and potentials

• Capacity is closely related to the concept of potentials in potential theory
• Potentials are functions that describe the electric field or the gravitational field generated by a charge distribution or a mass distribution
• The connection between capacity and potentials provides a way to characterize the properties of sets and measures using potential-theoretic tools

Equilibrium potential

• The equilibrium potential of a compact set $K$ is the unique function $u_K$ that minimizes the energy integral $\int_X |∇u|^2 dx$ among all functions $u$ that are equal to 1 on $K$ and vanish at infinity
• The equilibrium potential represents the electric potential generated by the equilibrium charge distribution on the set $K$
• The equilibrium potential is related to the capacity of the set $K$ by the formula $\operatorname{cap}(K) = \int_X |∇u_K|^2 dx$

Equilibrium measure

• The equilibrium measure of a compact set $K$ is the unique probability measure $\mu_K$ that minimizes the energy $\mathcal{E}(\mu)$ among all probability measures supported on $K$
• The equilibrium measure represents the charge distribution on the set $K$ that generates the equilibrium potential
• The equilibrium measure is related to the capacity of the set $K$ by the formula $\operatorname{cap}(K) = \mathcal{E}(\mu_K)^{-1}$

Capacity and equilibrium measure

• The capacity of a compact set $K$ can be expressed in terms of the equilibrium measure $\mu_K$ as $\operatorname{cap}(K) = \mu_K(K)^2 / \mathcal{E}(\mu_K)$
• This formula highlights the connection between capacity and the energy of the equilibrium measure
• The equilibrium measure provides a way to construct sets with prescribed capacity and to study the properties of sets with known capacity

Capacity and energy of equilibrium measure

• The energy of the equilibrium measure $\mu_K$ of a compact set $K$ is related to the capacity of $K$ by the formula $\mathcal{E}(\mu_K) = \operatorname{cap}(K)^{-1}$
• This formula expresses the fact that sets with higher capacity have equilibrium measures with lower energy
• The energy of the equilibrium measure is a key quantity in potential theory and is used in the study of regularity, polarity, and other properties of sets

Capacity and regularity

• Capacity is closely related to the concept of regularity in potential theory
• A set is regular if it satisfies certain geometric and potential-theoretic conditions that ensure the existence and uniqueness of solutions to boundary value problems
• The connection between capacity and regularity provides a way to characterize the boundary behavior of potentials and to study the properties of sets in terms of their capacity

Regular sets

• A set $E$ is regular if every point of $E$ is regular, meaning that the solution to the Dirichlet problem on the complement of $E$ with continuous boundary data has a continuous extension to the boundary
• Regular sets have nice potential-theoretic properties, such as the existence and uniqueness of equilibrium measures and the continuity of potentials
• Examples of regular sets include balls, spheres, and smooth domains

Irregular sets

• A set $E$ is irregular if it contains irregular points, meaning points where the solution to the Dirichlet problem on the complement of $E$ with continuous boundary data does not have a continuous extension to the boundary
• Irregular sets have pathological potential-theoretic properties, such as the non-existence or non-uniqueness of equilibrium measures and the discontinuity of potentials
• Examples of irregular sets include sets with cusps, spikes, or other geometric irregularities

Criteria for regularity

• There are several criteria for determining the regularity of a set, such as the Wiener criterion and the Kellogg criterion
• The Wiener criterion states that a point is regular if and only if a certain series involving the capacity of balls centered at the point diverges
• The Kellogg criterion states that a point is regular if and only if the complement of the set satisfies an exterior cone condition at the point

Capacity and regularity of sets

• The capacity of a set is related to its regularity by the following result: a set $E$ is regular if and only if $\operatorname{cap}(E \setminus K) = 0$ for every compact subset $K$ of $E$
• This result implies that regular sets have full capacity, meaning that removing any compact subset does not change the capacity of the set
• On the other hand, irregular sets have subsets of positive capacity that are not regular, and removing these subsets can change the capacity of the set

Capacity and polarity

• Capacity is closely related to the concept of polarity in potential theory
• A set is polar if it has zero capacity, meaning that it is too small to hold any charge or to conduct electricity
• The connection between capacity and polarity provides a way to characterize the size and structure of sets in terms of their potential-theoretic properties

Polar sets

• A set $E$ is polar if $\operatorname{cap}(E) = 0$
• Polar sets are negligible from the point of view of potential theory, as they do not contribute to the electric field or the potential
• Examples of polar sets include finite sets, countable sets, and sets of Hausdorff dimension less than the dimension of the underlying space

Totally nonpolar sets

• A set $E$ is totally nonpolar if it is not polar and does not contain any nonempty polar subsets
• Totally nonpolar sets are the opposite of polar sets, as they have positive capacity and cannot be decomposed into smaller sets of zero capacity
• Examples of totally nonpolar sets include balls, spheres, and sets of positive Lebesgue measure

Criteria for polarity

• There are several criteria for determining the polarity of a set, such as the Hausdorff dimension criterion and the energy criterion
• The Hausdorff dimension criterion states that a set is polar if and only if its Hausdorff dimension is less than the dimension of the underlying space
• The energy criterion states that a set is polar if and only if it has zero Newtonian energy, meaning that the energy of any measure supported on the set is infinite

Capacity and polarity of sets

• The capacity of a set is related to its polarity by the following result: a set $E$ is polar if and only if $\operatorname{cap}(E) = 0$
• This result implies that polar sets are the smallest sets from the point of view of capacity, as they have zero capacity and cannot be further decomposed into smaller sets of positive capacity
• On the other hand, totally nonpolar sets have positive capacity and can be decomposed into smaller sets of positive capacity, which provides a way to study their structure and properties

Capacity and Hausdorff measures

• Capacity is closely related to the concept of Hausdorff measures in potential theory
• Hausdorff measures are a family of measures that generalize the notion of Lebesgue measure and provide a way to measure the size of sets in terms of their Hausdorff dimension
• The connection between capacity and Hausdorff measures provides a way to characterize the size and structure of sets in terms of their potential-theoretic and geometric properties

Hausdorff measures

• The Hausdorff measure of dimension $d$ of a set $E$ is defined as $\mathcal{H}^d(E) = \lim_{\delta \to 0} \inf \left\{ \sum_i r_i^d : E \subset \bigcup_i B(x_i, r_i), r_i < \delta \right\}$, where $B(x, r)$ denotes the ball of radius $r$ centered at $x$
• Hausdorff measures provide a way to measure the size of sets in terms of their Hausdorff dimension, which is a fractal dimension that captures the local scaling behavior of the set
• Examples of sets with known Hausdorff dimensions include lines (dimension 1), planes (dimension 2), and Cantor sets (dimension log 2 / log 3)

Capacity and Hausdorff dimension

• The capacity of a set is related to its Hausdorff dimension by the following result: if $E$ is a compact set in $\mathbb{R}^n$, then $\dim_H(E) = \sup \{ d : \exists \mu \in \mathcal{M}(E), I_d(\mu) < \infty \}$, where $I_d(\mu) = \int \int |x-y|^{-d} d\mu(x) d\mu(y)$ is the $d$-energy of $\mu$
• This result implies that sets with higher Hausdorff dimension have higher capacity, as they can support measures with finite energy in higher dimensions
• The Hausdorff dimension of a set provides a way to estimate its capacity and to study its potential-theoretic properties

Sets of zero capacity

• A set $E$ has zero capacity if and only if $\mathcal{H}^d(E) = 0$ for all $d > 0$
• Sets of zero capacity are polar and have no effect on the potential-theoretic properties of the underlying space
• Examples of sets of zero capacity include finite sets, countable sets, and sets of Hausdorff dimension 0

Sets of positive capacity

• A set $E$ has positive capacity if and only if there exists $d > 0$ such that $\mathcal{H}^d(E) > 0$
• Sets of positive capacity are nonpolar and have a nontrivial effect on the potential-theoretic properties of the underlying space
• Examples of sets of positive capacity include balls, spheres, and sets of positive Lebesgue measure

Analytic capacity

• Analytic capacity is a variant of capacity that is defined using analytic functions instead of potentials
• Analytic capacity measures the size of a set in terms of its ability to support bounded analytic functions
• The connection between analytic capacity and classical capacity provides a way to study the properties of sets in complex analysis and potential theory

Definition of analytic capacity

• The analytic capacity of a compact set $K$ in the complex plane is defined as $\gamma(K) = \sup \{ |f'(\infty)| : f \in H^\infty(\overline{\mathbb{C}} \setminus K), |f| \leq 1 \}$, where $H^\infty(\overline{\mathbb{C}} \setminus K)$ is the space of bounded analytic functions on the complement of $K$ and $f'(\infty) = \lim_{z \to \infty} z(f(z) - f(\infty))$
• Analytic capacity measures the size of a set in terms of the maximum modulus of the derivative at infinity of bounded analytic functions that vanish on the set
• Analytic capacity is a conformal invariant, meaning that it is invariant under conformal mappings of the complex plane

Properties of analytic capacity

• Analytic capacity shares many properties with classical capacity, such as monotonicity, subadditivity, and continuity from above
• However, analytic capacity also has some unique properties that distinguish it from classical capacity, such as the Vitushkin conjecture and the semiadditivity property
• The Vitushkin conjecture states that a compact set has zero analytic capacity if and only if it is removable for bounded analytic functions, while the semiadditivity property states that $\gamma(K_1 \cup K_2) \leq \gamma(K_1) + \gamma(K_2) + C \sqrt{\gamma(K_1) \gamma(K_2)}$ for some constant $C$

Analytic capacity and continuity

• A compact set $K$ has zero analytic capacity if and only if every bounded analytic function on the complement of $K$ has a continuous extension to the whole complex plane
• This result implies that sets of zero analytic capacity are removable for bounded analytic functions, meaning that they do not affect the behavior of the functions near the boundary
• On the other hand, sets of positive analytic capacity support bounded analytic functions that have singularities or discontinuities near the boundary

Analytic capacity and rectifiability

• A compact set $K$