Riesz external field problems on the hypersphere

Riesz external field problems on the hypersphere and optimal point separation

Abstract.

We consider the minimal energy problem on the unit sphere in the Euclidean space in the presence of an external field , where the energy arises from the Riesz potential (where is the Euclidean distance and is the Riesz parameter) or the logarithmic potential . Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range The proof uses a maximum principle for measures supported on . When is the Riesz -potential of a signed measure and , our results lead to explicit point-separation estimates for -Fekete points, which are -point configurations minimizing the Riesz -energy on with external field . In the hyper-singular case , the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.

Key words and phrases:
-subharmonic functions, balayage, minimal energy problems with external fields, Riesz spherical potentials
2000 Mathematics Subject Classification:
31B05 (31B15, 78A30)
*The research of this author was supported, in part, by an APART-Fellowship of the Austrian Academy of Sciences and by an Australian Research Council Discovery grant.
†The research of this author was supported, in part, by a Grants-in-Aid program of ORESP at IPFW and by a grant from the Simons Foundation no. 282207.
‡The research of this author was supported, in part, by the U. S. National Science Foundation under grant DMS-1109266 as well as by an Australian Research Council Discovery grant.

1. Introduction

Let be the unit sphere in , where denotes the Euclidean norm. Given a compact set , consider the class of unit positive Borel measures supported on . For the Riesz -potential and Riesz -energy of a measure are given, respectively, by

where is the so-called Riesz kernel. The -capacity of is then defined as for , where is the -energy of the set . A property is said to hold quasi-everywhere (q.e.), if the exceptional set has -capacity zero. When , there exists a unique minimizer , called the -equilibrium measure on , such that . For more details see [21, Chapter II].

Whenever (we shall use ), which occurs, for example, when and , we replace the Riesz kernel by the logarithmic kernel

(In this case we define .)

We shall refer to a lower semi-continuous function such that on a set of positive Lebesgue surface measure, as an external field. We note that the lower semi-continuity implies the existence of a finite such that for all . The weighted energy associated with is then given by

(1)

(The terminology “weighted energy” is used here to indicate the presence of an external field, and should not be confused with “weighted energy functionals”, where the Riesz -kernel is multiplied by a weight function . We leave the study of the external field problem for such generalized kernels for a future investigation.)

Definition 1.

The Riesz external field problem on the unit sphere for the external field is concerned with minimizing the weighted energy (1) among all Borel probability measures supported on . A measure with

is called an -extremal (or positive equilibrium) measure on associated with .

If we consider only measures supported on some compact subset with positive -capacity, then the minimizing measure is referred to as the -extremal measure on associated with and denoted by . In the particular case when , the measure on is just the normalized unit surface area measure on the sphere for which we use the symbol .

We shall also consider the discrete analogue of the above external field problem which is defined as follows.

Definition 2.

Let or . For a set of points the discrete weighted energy associated with is given by

(2)

Then the discrete external field problem on the sphere concerns the minimization

(3)

where denotes the cardinality of the set . A solution of the discretized minimization problem (3) is called an -point -Fekete set.

The existence of -Fekete sets is an easy consequence of the lower semi-continuity of the energy functional and the compactness of the unit sphere. Further, we remark that a standard argument establishes the following monotonicity property

We remark that the discrete problem has application to image processing, namely the half-toning of images based on electrostatic repulsion of printed dots in the presence of an image-driven external field; cf. Schmaltz et al. [30] and Gräf [14, Section 6.5.2].

The outline of the paper is as follows. In Section 2 we provide Frostman-type characterization theorems for the solution to the external field minimal energy problem on the sphere. This is facilitated by a new restricted maximum principle on the sphere which holds for the range (see Theorem 5). We also introduce the signed equilibrium measure and discuss its relation to the positive equilibrium measure. In Section 3 we establish that for a large class of external fields , the sequences of -point -Fekete sets are well-separated; that is, have separation distance of order (Theorems 14 and 16). In Section 4, for an external field due to a negative point charge, we provide a detailed analysis and give explicit representations of the signed equilibrium (Theorem 19) and the -extremal measure on (Theorem 20). This extends results in [2]. In Section 5 we rigorously characterize the -point -Fekete set for a general class of convex external fields and provide numerical results for the four point problem with Riesz external fields (Figures 2 and 4 illustrate the analysis). The proofs of our results are provided in Section 6.

2. Basic Properties and Characterization Theorems

In [10] the second and the third authors formulated the following Frostman-type proposition, which deals with the existence and uniqueness of the measure , as well as a criterion that characterizes in terms of its potential. The proof of this proposition follows closely the proof of [28, Theorem I.1.3]. It could also be derived as a particular case from the more general results in [32] (see especially Theorems 1 and 2, and Proposition 1 of that paper).

Proposition 3.

Let .1 For the minimal energy problem on with external field the following properties hold:

  • is finite.

  • There exists a unique -extremal measure associated with . Moreover, the support of this measure is contained in the compact set for some .

  • The measure satisfies the variational inequalities

    (4)
    (5)

    where

    (6)
  • Inequalities (4) and (5) completely characterize the -extremal measure in the sense that if is a measure with finite -energy such that for some constant we have

    (7)
    (8)

    then and .

Observe that, if the external field is continuous on , then the inequality in (7) holds everywhere on .

Remark.

Proposition 3 remains true if is replaced with any compact subset with . Notationally, the dependence on will be indicated by a subscript (e.g., , , etc.).

In the case when , [10, Theorem 1.3] analyzes further the characterization property from Proposition 3(d) by studying the supremum and the essential infimum of the weighted potential . Our first theorem extends this analysis to the larger range . To state the theorem we introduce the notation to denote the essential infimum of with respect to a set ; that is,

in other words, the infimum is taken quasi-everywhere.

Theorem 4.

Let , be an external field on , and be defined as in (6). For any measure we have

(9)
and
(10)

If equality holds in both inequalities, then .

For the restricted range , the proof of this theorem as given in [10] utilizes the principle of domination for Riesz potentials, which generally is stated for the parameter range and measures supported on any subsets of . (A restricted version of the principle of domination for was established in [10, Lemma 5.1].) Via a different approach that utilizes the following restricted maximum principle on the sphere, we are able to prove the result for in the extended range ; see Section 6.

Theorem 5 (Sphere Maximum Principle).

Let . Suppose is a positive measure with such that for some , the relation holds -almost everywhere on . Then holds everywhere on .

An essential part of the analysis of external field problems is the determination the -extremal (equilibrium) measure on associated with the external field and, in particular, its support. In principle, if the latter is known, the measure can be recovered by solving an integral equation for the weighted -potential of arising from the variational inequalities (4) and (5). A substantially easier problem is to find a (signed) measure that has constant weighted -potential everywhere on . The solution of this problem turns out to be useful in solving the harder problem. This motivates the study of the signed equilibrium measure associated with an external field which is defined as follows.

Definition 6.

Given a compact subset () and an external field , we call a signed measure supported on and of total charge a signed -equilibrium on associated with if its weighted Riesz -potential is constant on ; that is,

(11)

We note that if a signed equilibrium exists, then it is unique (see [2, Lemma 23]).

A remarkable connection exists to the Riesz analog of the Mhaskar-Saff -functional from classical logarithmic potential theory in the plane (see [23] and [28, Chapter IV, p. 194]).

Definition 7.

The -functional of a compact subset of positive -capacity is defined as

(12)

where is the -energy of and is the -equilibrium measure (without external field) on .

Let with . If the signed equilibrium on a compact set associated with exists, then integration of (11) with respect to shows that

(13)

The essential property of the -functional is the following (cf. [2, Theorem 9]).

Proposition 8.

Let with and be an external field on a compact subset with . Then the -functional is minimized for the support of the -extremal measure on associated with ; that is, for every compact subset with ,

Given a compact subset , the extended support of is defined by

(14)

The following theorem, which is the Riesz analog of [9, Theorem 2.6] and [18, Lemma 3], establishes a relation between the extended support of (by (5) this set contains the support of ) and the support of the positive part of the Jordan decomposition of the signed equilibrium on associated with .

Theorem 9.

Let and suppose that is an external field such that a signed -equilibrium on a spherical cap exists. Then

Furthermore, if , then .

Remark.

The theorem remains true if the -extremal measure on a compact subset with and signed -equilibria on compact subsets with are considered.

The characterization results for the shape of the support of the -extremal measure on associated with a rotational symmetric external field given in [2, Theorem 10] immediately carry over to the external fields with extended range.

Proposition 10.

Let with and the external field be rotationally invariant about the polar axis; that is, , where is the altitude of , . Suppose that is a convex function on . Then the support of the -extremal measure on is a spherical zone; namely, there are numbers such that

(15)

Moreover, if additionally is increasing, then and the support of is a spherical cap centered at the South Pole.

Next we focus on the discretized version of the Riesz external field problem given in Definition 1. Recall that the normalized counting measure associated with an -point set is defined as

where is the Dirac-delta measure with unit mass at . The continuous and discrete external field minimization problems are related in the following way.

Proposition 11.

Let or . Then

Furthermore, if is any sequence of -point -Fekete sets on (see Definition 2), then the sequence of the normalized counting measures associated with converges in the weak-star sense to the -extremal measure .

The proof follows from a standard argument and utilizes the uniqueness result stated in Proposition 3(b).

We are interested in determining sets that contain all the -Fekete sets. For this purpose it is useful to investigate the weighted -potential of the normalized counting measure which is defined as

(16)

As an application of Theorem 4 we deduce the following result.

Theorem 12.

Let . Let be a set of distinct points, and suppose that, for some constant , the associated weighted potential satisfies the inequality

(17)

Then (cf. (6))

(18)

Furthermore,

(19)

We point out that this is an extension of [10, Theorem 1.7], which, as with Theorem 4 above, was originally established for . As in [10, Corollary 1.9], Theorems 4 and 12 yield the following.

Corollary 13.

For , every -Fekete set is contained in the extended support .

We note that for most of the above theorems, marks the lower end of the stated range of the Riesz parameter . It turns out that the case is distinctive because new phenomena arise in the solution of the signed equilibrium problem, see Section 4. Moreover, for in the interval , the Riesz- kernel becomes strictly superharmonic when considered in the stereographic projection space of ; consequently maximum principles and domination principles do not apply.

3. Application to point separation

Good separation of points is generally associated with the stability of an approximation or interpolation method (e.g., by splines or radial basis functions (RBF)); cf., e.g., [12, 22, 29]. In this section we shall apply results from Section 2 (especially Theorem 9 and Corollary 13) to obtain explicit point separation estimates for sequences of -point -Fekete sets (cf. Definition 2) associated with a large class of external fields and establish that such sequences are “well-separated” in the following sense. Let

denote the minimum distance among the points in . Then a sequence , for all , is called well-separated if is of order as . (It suffices to show the existence of a constant such that

(20)

since cannot exceed the best-packing distance which is of order ; cf. [4].)

In the potential-theoretical and field-free setting () it has been known since Dahlberg [5] that Fekete point sets (harmonic case ) on a sufficiently smooth closed bounded -dimensional surface in that separates into two parts will form a well-separated sequence (but no explicit constant for the lower bound of has been given). Götz [13] studied the discrete external field problem on surfaces in where the energy functional is defined in terms of the Green function for a domain . His separation result generalizes Dahlberg’s result. Well-separation of minimal logarithmic energy configurations on in the field-free setting was first established by Rakhmanov et al. [26, 27], and with an improved constant by Dragnev [8]. For minimal Riesz -energy configurations on in the field-free case, well-separation was established by Kuijlaars et al. [20] for and by Dragnev and Saff [10] for . Damelin and Maymeskul [6] give a separation result of order , , which is of sharp order in the boundary case . It is expected but still unproven that minimal logarithmic and Riesz -energy () configurations on , , are well-separated. The references [8], [10] and [27] also provide an explicit constant in the lower estimate (20). It should be noted that [10] uses external fields to derive the desired separation estimates in the field-free setting. In the hyper-singular case , Kuijlaars and Saff [19] establish well-separation of minimal Riesz -energy configurations on .

We now present a generalization of [10, Theorem 1.5] to the case when an external field is present and given by a potential.

Theorem 14.

Let and for some signed measure with -a.e. finite Riesz -potential on . Assume the support of the negative part in the Jordan decomposition satisfies that

(21)

for some and that

(22)

Then any sequence of -Fekete sets on is well-separated; more precisely,

(23)

where

(24)

It is understood that for and we replace by and by .

Since the Riesz -energy of appearing in the separation constant in (24) is given by the formula

(25)

we have in the harmonic case that

(26)

and in the limiting case (and )

(27)

wheras for and

(28)
Remark.

Note that whenever the support of lies outside of , then both conditions (21) and (22) are satisfied by taking sufficiently close to . Also observe that as approaches , the constant approaches .

In case of and the above Theorem 14 yields a known result for the well-separation of -point minimal Riesz -energy configurations on ([6] but without explicit constants). Also with , and we recover the same separation result as obtained in [8]. Here we prove them separately (with explicit constants in the former case), since they will be used to establish the separation bounds when an externalf field () is given.

Proposition 15.

For and we have

(29)

for any -point Riesz -energy 2 minimizing configuration on , where

(30)

Observe that when . The first three values of are , , and . Curiously, is the ratio of the volume of the unit ball in divided by the surface area of the unit sphere in . (This constant also appears as the coefficient of the leading term in the asymptotic expansion of the -point minimal Riesz -energy as (cf. [19]).)

Finally, we present a well-separation result for sequences of -Fekete sets in the hyper-singular case . In this case the (strongly repellent) short-range interactions between points on the sphere ensure well-separation of minimizing configurations for any continuous external field on . In fact, it is enough that be integrable on some small subset of of positive surface area measure.

Theorem 16.

Let . Suppose there is a subset such that and the fixed external field is integrable over with respect to . Then there is a constant independent of such that

(31)

for any -point -Fekete set on .

Remark.

In case of varies with , there may be no single fixed subset satisfying the hypotheses in Theorem 16. However, one can still deduce well-separation by requiring the following: there is a sequence of subsets of such that for some , for all , and for some independent of ,

where denotes the minimum of over . These conditions are derived from the main inequality (82) and the estimate (83).

Remark.

For large classes of external fields (e.g., continuous external fields), inequality (82) and the estimate (83) can be made explicit which, in turn, yields an explicit constant in the separation estimate (31), as the following example illustrates.

Example 17.

Let . Consider the external field , , , due to a point source above the North Pole . Clearly, is continuous and thus integrable on . Thus Theorem 16 assures well-separation of -point -Fekete sets on . An explicit lower bound can be easily derived from (82) and (83). We find

and, consequently,

where

Note that as . The constants and are given in (76) and (78), respectively, and the representation of appears in (41).

4. Negatively charged external fields

In the following we consider external fields that are generated by negative sources. The required lower semi-continuity of implies that no negative singularities can be on the sphere but it may support a negative “continuous” charge distribution (with no discrete part relative to ). We give a detailed analysis for the Riesz external field

(32)

where , that is due to a negative point source at below the South Pole and which also provides the basis for more general axis-supported fields defined by superposition of point source fields. Our analysis thus extends and complements results in [2] where positive axis-supported external fields were considered.

Intuitively, a negative point source under the South Pole will “pull” charge towards the South Pole and if sufficiently strong will cause a negatively charged spherical cap around the North Pole to appear on a grounded sphere. Grounding of the sphere imposes constant weighted potential everywhere on . This naturally leads to the signed equilibrium problem on the whole sphere or on its parts, say, the spherical cap centered at the South Pole. On a positively charged isolated sphere a sufficiently strong negative field will produce a spherical cap around the North Pole that is free of charge. We are specifically interested in the charge distribution on the remaining part , that is the -extremal measure on associated with the external field and its support .

We will use the methods and results of [2]. An essential concept is the -balayage of a measure. Recall that given a measure and a compact set (of the sphere ), the balayage measure preserves the Riesz -potential of onto the set and diminishes it elsewhere (on the sphere ). Let denote the signed -equilibrium on associated with the external field . Then it can be expressed as

(33)

where

(34)

are the -balayage measures onto of the positive unit point charge at and the uniform measure on . The function , defined by

(35)

in terms of the -energy of , given in (25) and norms and , plays an important role in what follows. Indeed,

and using that and on by (34), at every

(36)

By Definition 6, and (13) relates to the -functional by means of , whereas Proposition 8 implies that the latter is minimized by the support of the -extremal measure on associated with the external field (32) which turns out to be a spherical cap . We will see that the unique minimum of in the interval will provide this critical parameter (see Theorem 20). Moreover, the remark following Theorem 19 provides the necessary and sufficient conditions (involving and therefore ) under which the signed -equilibrium measure on turns into the -extremal measure on associated with the external field (32).

Throughout, and denote the Gauss hypergeometric function and its regularized form 3 with series expansions

(37)

where and for is the Pochhammer symbol. We also recall that the incomplete Beta function and the Beta function are defined as

(38)

whereas the regularized incomplete Beta function is given by

(39)

First, we give the representation of the signed equilibrium on the whole sphere , which is well-known from elementary physics (cf. [15, p. 61]) in the classical Coulomb case, and provide a necessary and sufficient condition when it also is the -extremal measure on .

Proposition 18.

Let and . The signed -equilibrium on associated with the Riesz external field (32), where in fact , is given by

(40)

Furthermore,

where has the following representation:

(41)

Moreover, if , then if and only if

(42)