Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser
aa r X i v : . [ m a t h . C A ] F e b CONSTRAINED MINIMUM RIESZ AND GREEN ENERGYPROBLEMS FOR VECTOR MEASURES ASSOCIATED WITHA GENERALIZED CONDENSER
B. FUGLEDE AND N. ZORII
Abstract.
For a finite collection A = ( A i ) i ∈ I of locally closed sets in R n , n >
3, withthe sign ± α -Riesz kernel | x − y | α − n , α ∈ (0 , µ = ( µ i ) i ∈ I such that each µ i , i ∈ I , is carried by A i and normalized by µ i ( A i ) = a i ∈ (0 , ∞ ). We show that, though the closures ofoppositely charged plates may intersect each other even in a set of nonzero capacity, thisproblem has a solution λ ξ A = ( λ i A ) i ∈ I (also in the presence of an external field) if werestrict ourselves to µ with µ i ξ i , i ∈ I , where the constraint ξ = ( ξ i ) i ∈ I is properlychosen. We establish the sharpness of the sufficient conditions on the solvability thusobtained, provide descriptions of the weighted vector α -Riesz potentials of the solutions,single out their characteristic properties, and analyze the supports of the λ i A , i ∈ I . Ourapproach is based on the simultaneous use of the vague topology and an appropriatesemimetric structure defined in terms of the α -Riesz energy on a set of vector measuresassociated with A , as well as on the establishment of an intimate relationship betweenthe constrained minimum α -Riesz energy problem and a constrained minimum α -Greenenergy problem, suitably formulated. The results are illustrated by examples. Introduction
The purpose of this paper is to study minimum energy problems with external fields (alsoknown in the literature as weighted minimum energy problems or as
Gauss variationalproblems ) relative to the α -Riesz kernel κ α ( x, y ) = | x − y | α − n of order α ∈ (0 , | x − y | is the Euclidean distance between x, y ∈ R n , n >
3, and infimum is taken over classesof vector measures µ = ( µ i ) i ∈ I associated with a generalized condenser A = ( A i ) i ∈ I andnormalized by µ i ( A i ) = a i ∈ (0 , ∞ ), i ∈ I . More precisely, an ordered finite collection A of locally closed sets A i , i ∈ I , termed plates , with the sign s i = ± generalized condenser if the oppositely signed plates are mutually disjoint, while a vectormeasure µ = ( µ i ) i ∈ I is said to be associated with A if each µ i , i ∈ I , is a positive scalarRadon measure (charge) carried by A i . In accordance with an electrostatic interpretationof a condenser, we say that the interaction between the components µ i , i ∈ I , of such a µ is characterized by the matrix ( s i s j ) i,j ∈ I , so that the f -weighted α -Riesz energy of µ is defined by G κ α , f ( µ ) := X i,j ∈ I s i s j Z Z | x − y | α − n dµ i ( x ) dµ j ( y ) + 2 X i ∈ I Z f i dµ i , where f = ( f i ) i ∈ I , each f i : R n → [ −∞ , ∞ ] being a universally measurable function treatedas an external field acting on the charges carried by the A i .The difficulties appearing in the course of our investigation are caused by the fact that ashort-circuit may occur between A i and A j with s i s j = −
1, because these conductors mayhave zero Euclidean distance. See Theorem 5.2 below providing an example of a generalizedcondenser with no α -Riesz energy minimizer. It is therefore meaningful to ask what kindsof additional requirements on the objects in question will prevent this blow-up effect, andsecure that a solution to the corresponding f -weighted minimum α -Riesz energy problemdoes exist. We show that, though the closures of oppositely charged plates may intersecteach other even in a set of nonzero α -Riesz capacity, such minimum energy problem isnevertheless solvable ( no short-circuit occurs) if we restrict ourselves to µ with µ i ξ i , i ∈ I , where the constraint ξ = ( ξ i ) i ∈ I is properly chosen (see Sections 3.5 and 5.2 for aformulation of the constrained problem). Sufficient conditions for the existence of solutions λ ξ A = ( λ i A ) i ∈ I to the constrained minimum α -Riesz energy problem are established inTheorems 7.1 and 7.7; those conditions are shown in Theorem 7.2 to be sharp. Theuniqueness of solutions is studied in Lemma 3.12 and Corollary 3.13. We also providedescriptions of the f -weighted vector α -Riesz potentials of the solutions λ ξ A , single outtheir characteristic properties, and analyze the supports of the components λ i A , i ∈ I (Theorems 7.3, 7.5 and 7.6). The results are illustrated in Examples 10.1 and 10.2.In particular, let A = ( A , A ) be a generalized condenser with the positive plate A := D and the negative plate A := R n \ D , D being an (open connected) bounded domain in R n with m n ( D ) > m n is the n -dimensional Lebesgue measure, and let f = .Then inf G κ α , f ( µ ) over all µ = ( µ , µ ) associated with A and normalized by µ i ( A i ) = 1, i = 1 ,
2, is an actual minimum (although A ∩ Cℓ R n A = ∂D ) if we require additionallythat µ ξ := m n | D and µ ξ , where ξ is a positive Radon measure carried by A and possessing the property ξ > ( m n | D ) A (cf. Theorems 7.1 and 7.7). Here m n | D denotesthe restriction of m n on D , and ( m n | D ) A the α -Riesz balayage of m n | D onto A .The approach developed is mainly based on the simultaneous use of the vague topologyand an appropriate (semi)metric structure defined in terms of the α -Riesz energy on aset of vector measures associated with a generalized condenser (see Section 3.2 for a def-inition of such a (semi)metric structure ), as well as on the establishment of an intimaterelationship between the constrained minimum α -Riesz energy problem and a constrained A key observation behind that definition is the fact that there corresponds to every positive vectormeasure µ = ( µ i ) i ∈ I of finite energy associated with A a scalar ( signed ) Radon measure R µ = P i ∈ I s i µ i on R n , and the mapping R : µ R µ preserves the corresponding energy semimetric (see Theorem 3.5).This approach extends that from [23]–[26] where the closures of the oppositely charged plates were assumedto be mutually disjoint. ONDENSERS WITH TOUCHING PLATES 3 minimum α -Green energy problem, suitably formulated. Regarding the correspondingminimum α -Green energy problem, crucial to the arguments applied in its investigationis the perfectness of the α -Green kernel g αD on a domain D , established recently by theauthors [16], which amounts to the completeness in the topology defined by the energynorm k ν k g αD := p g αD ( ν, ν ) of the cone of all positive scalar Radon measures ν on D withfinite α -Green energy g αD ( ν, ν ) := RR g αD ( x, y ) dν ( x ) dν ( y ) < ∞ .2. Preliminaries
Let X be a locally compact (Hausdorff) space [4, Chapter I, Section 9, n ◦ M ( X ) the linear space of all real-valued scalar Radon measures µ on X ,equipped with the vague topology, i.e. the topology of pointwise convergence on the class C ( X ) of all continuous functions on X with compact support. We refer the reader to[2, 3, 12] for the theory of measures and integration on a locally compact space, to be usedthroughout the paper (see also [13] for a short survey). In all that follows the integrals areunderstood as upper integrals [2].For the purposes of the present study it is enough to assume that X is metrizable and countable at infinity , where the latter means that X can be represented as a countable unionof compact sets [4, Chapter I, Section 9, n ◦ M ( X ) satisfiesthe first axiom of countability [11, Remark 2.5], and the vague convergence is entirelydetermined by convergence of sequences. The vague topology on M ( X ) is Hausdorff;hence, a vague limit of any sequence in M ( X ) is unique (whenever it exists).We denote by µ + and µ − the positive and the negative parts in the Hahn–Jordan decom-position of a measure µ ∈ M ( X ), and by S µX = S ( µ ) its support. A measure µ ∈ M ( X ) issaid to be bounded if | µ | ( X ) < ∞ , where | µ | := µ + + µ − . Let M + ( X ) stand for the (convex,vaguely closed) cone of all positive µ ∈ M ( X ), and let Ψ( X ) consist of all lower semicon-tinuous (l.s.c.) functions ψ : X → ( −∞ , ∞ ], nonnegative unless X is compact. Lemma 2.1 (see e.g. [13, Section 1.1]) . For any ψ ∈ Ψ( X ) the map µ
7→ h ψ, µ i := R ψ dµ is vaguely l.s.c. on M + ( X ) . We define a (function) kernel κ ( x, y ) on X as a symmetric positive function from Ψ( X × X ).Given µ, µ ∈ M ( X ), we denote by κ ( µ, µ ) and κ ( · , µ ) the mutual energy and the potential relative to the kernel κ , respectively, i.e. κ ( µ, µ ) := Z Z κ ( x, y ) dµ ( x ) dµ ( y ) ,κ ( x, µ ) := Z κ ( x, y ) dµ ( y ) , x ∈ X. When speaking of a continuous numerical function we understand that the values are finite real numbers. When introducing notation about numerical quantities we assume the corresponding object on the rightto be well defined — as a finite real number or ±∞ . ONDENSERS WITH TOUCHING PLATES 4
Note that κ ( x, µ ) is well defined provided that κ ( x, µ + ) or κ ( x, µ − ) is finite, and then κ ( x, µ ) = κ ( x, µ + ) − κ ( x, µ − ). In particular, if µ ∈ M + ( X ) then κ ( x, µ ) is defined ev-erywhere and represents a l.s.c. positive function on X (see Lemma 2.1). Also observethat κ ( µ, µ ) is well defined and equal to κ ( µ , µ ) provided that κ ( µ + , µ +1 ) + κ ( µ − , µ − )or κ ( µ + , µ − ) + κ ( µ − , µ +1 ) is finite. For µ = µ , κ ( µ, µ ) becomes the energy κ ( µ, µ ). Let E κ ( X ) consist of all µ ∈ M ( X ) whose energy κ ( µ, µ ) is finite, which by definition means that κ ( µ + , µ + ), κ ( µ − , µ − ) and κ ( µ + , µ − ) are all finite, and let E + κ ( X ) := E κ ( X ) ∩ M + ( X ).Given a set Q ⊂ X , let M + ( Q ; X ) consist of all µ ∈ M + ( X ) concentrated on (or carried by ) Q , which means that X \ Q is locally µ -negligible, or equivalently that Q is µ -measurable and µ = µ | Q , where µ | Q = 1 Q · µ is the trace (restriction) of µ on Q [3, Section 5, n ◦
2, Exemple].(Here 1 Q denotes the indicator function of Q .) If Q is closed then µ is concentrated on Q if and only if it is supported by Q , i.e. S ( µ ) ⊂ Q . It follows from the countability of X atinfinity that the concept of local µ -negligibility coincides with that of µ -negligibility; andhence µ ∈ M + ( Q ; X ) if and only if µ ∗ ( X \ Q ) = 0, µ ∗ ( · ) being the outer measure of a set.Write E + κ ( Q ; X ) := E κ ( X ) ∩ M + ( Q ; X ), M + ( Q, q ; X ) := { µ ∈ M + ( Q ; X ) : µ ( Q ) = q } and E + κ ( Q, q ; X ) := E κ ( X ) ∩ M + ( Q, q ; X ), where q ∈ (0 , ∞ ).Among the variety of potential-theoretic principles investigated for example in the com-prehensive work by Ohtsuka [21] (see also the references therein), in the present study weshall only need the following two: • A kernel κ is said to satisfy the complete maximum principle (introduced by Cartanand Deny [7]) if for any µ ∈ E + κ ( X ) and ν ∈ M + ( X ) such that κ ( x, µ ) κ ( x, ν ) + cµ -a.e., where c > X . • A kernel κ is said to be positive definite if κ ( µ, µ ) > µ ∈ M ( X ) for which the energy is well defined; and such κ is said to be strictlypositive definite , or to satisfy the energy principle if in addition κ ( µ, µ ) > µ = 0. Unless explicitly stated otherwise, in all that follows we assume a kernel κ to satisfy theenergy principle . Then E κ ( X ) forms a pre-Hilbert space with the inner product κ ( µ, µ )and the energy norm k µ k κ := p κ ( µ, µ ) (see [13]). The (Hausdorff) topology on E κ ( X )defined by the norm k · k κ is termed strong .In contrast to [14, 15] where capacity has been treated as a functional acting on positivenumerical functions on X , in the present study we use the (standard) concept of capacityas a set function. Thus the ( inner ) capacity of a set Q ⊂ X relative to the kernel κ ,denoted c κ ( Q ), is defined by(2.1) c κ ( Q ) := (cid:2) inf µ ∈E + κ ( Q, X ) κ ( µ, µ ) (cid:3) − (see e.g. [13, 21]). Then 0 c κ ( Q ) ∞ . (As usual, here and in the sequel the infimumover the empty set is taken to be + ∞ . We also set 1 (cid:14) (+ ∞ ) = 0 and 1 (cid:14) ∞ .) Because ONDENSERS WITH TOUCHING PLATES 5 of the strict positive definiteness of the kernel κ ,(2.2) c κ ( K ) < ∞ for every compact K ⊂ X. Furthermore, by [13, p. 153, Eq. 2],(2.3) c κ ( Q ) = sup c κ ( K ) ( K ⊂ Q, K compact) . We shall often use the fact that c κ ( Q ) = 0 if and only if µ ∗ ( Q ) = 0 for every µ ∈ E + κ ( X ), µ ∗ ( · ) being the inner measure of a set [13, Lemma 2.3.1].As in [19, p. 134], we call a measure µ ∈ M ( X ) c κ -absolutely continuous if µ ( K ) = 0for every compact set K ⊂ X with c κ ( K ) = 0. It follows from (2.3) that for such a µ , | µ | ∗ ( Q ) = 0 for every Q ⊂ X with c κ ( Q ) = 0. Hence every µ ∈ E κ ( X ) is c κ -absolutelycontinuous; but not conversely [19, pp. 134–135]. Definition 2.2.
Following [13], we call a (strictly positive definite) kernel κ perfect if everystrong Cauchy sequence in E + κ ( X ) converges strongly to any of its vague cluster points . Remark 2.3. On X = R n , n >
3, the α -Riesz kernel κ α ( x, y ) = | x − y | α − n , α ∈ (0 , n ),is strictly positive definite and perfect [8, 9]; thus so is the Newtonian kernel κ ( x, y ) = | x − y | − n [6]. Recently it has been shown by the present authors that if X = D where D is an arbitrary open set in R n , n >
3, and g αD , α ∈ (0 , α -Green kernel on D [19,Chapter IV, Section 5], then κ = g αD is likewise strictly positive definite and perfect [16,Theorems 4.9 and 4.11]. Theorem 2.4 (see [13]) . If a kernel κ on a locally compact space X is perfect, then the cone E + κ ( X ) is strongly complete and the strong topology on E + κ ( X ) is finer than the ( induced ) vague topology on E + κ ( X ) . Remark 2.5.
In contrast to Theorem 2.4, for a perfect kernel κ the whole pre-Hilbertspace E κ ( X ) is in general strongly incomplete , and this is the case even for the α -Rieszkernel of order α ∈ (1 , n ) on R n , n > signed ) ν ∈ E κ α ( R n ) such that ν + and ν − are supported by closed nonintersecting setsin R n , n >
3. This result from [22] has been proved with the aid of Deny’s theorem [8]stating that E κ α ( R n ) can be completed by making use of tempered distributions on R n withfinite α -Riesz energy, defined in terms of its Fourier transform (compare with Remark 2.6). Remark 2.6.
The concept of perfect kernel is an efficient tool in minimum energy problemsover classes of positive scalar
Radon measures with finite energy. Indeed, if Q ⊂ X is closed, c κ ( Q ) ∈ (0 , + ∞ ), and κ is perfect, then the minimum energy problem (2.1) has a uniquesolution λ Q [13, Theorem 4.1]; we shall call such a λ Q the ( inner ) κ -capacitary measure on Q . Later the concept of perfectness has been shown to be efficient also in minimumenergy problems over classes of vector measures associated with a standard condenser [23]–[26] (see also Remarks 3.10 and 3.14 below for a short survey). In contrast to [22, It follows from Theorem 2.4 that for a perfect kernel such a vague cluster point exists and is unique.
ONDENSERS WITH TOUCHING PLATES 6
Theorem 1], the approach developed in [23]–[26] substantially used the assumption of theboundedness of the kernel on the product of the oppositely charged plates of a condenser,which made it possible to extend Cartan’s proof [6] of the strong completeness of the cone E + κ ( R n ) of all positive measures on R n with finite Newtonian energy to an arbitrary perfectkernel κ on a locally compact space X and suitable classes of ( signed ) measures µ ∈ E κ ( X ).3. Minimum energy problems for a generalized condenser in a locallycompact space
Vector measures associated with a generalized condenser.
A subset L of alocally compact space X is said to be locally closed if for every x ∈ L there is a neighborhood V of x in X such that V ∩ L is a closed subset of the subspace L [4, Chapter I, Section 3,Definition 2], or equivalently if L is the intersection of an open and a closed subset of X [4, Chapter I, Section 3, Proposition 5].Consider an ordered finite collection A = ( A i ) i ∈ I of nonempty, locally closed sets A i ⊂ X with the sign s i = sign A i = ± I + := { i ∈ I : s i = +1 } , I − := I \ I + and p := Card I , where p > I − is allowed to be empty. Definition 3.1.
We call A a generalized condenser in X if A + ∩ A − = ∅ , where A + := [ i ∈ I + A i and A − := [ j ∈ I − A j . The sets A i , i ∈ I + , and A j , j ∈ I − , are said to be the positive and negative plates of the(generalized) condenser A . To avoid trivialities, we shall always assume that(3.1) c κ ( A i ) > i ∈ I, the (strictly positive definite) kernel κ on X being given. Note that any two equally signedplates may intersect each other or even coincide. Also note that, though A i and A j aredisjoint for any i ∈ I + and j ∈ I − , their closures in X may intersect each other even ina set with c κ ( · ) >
0. The concept of generalized condenser thus defined generalizes thatintroduced recently in [11, Section 3].
Definition 3.2.
A generalized condenser A is said to be standard if all the (locally closed)sets A i , i ∈ I , are closed in X .Unless explicitly stated otherwise, in all that follows we assume A to be a generalized condenser in X . Let M + ( A ; X ) consist of all positive vector measures µ = ( µ i ) i ∈ I whereeach µ i , i ∈ I , is a positive scalar Radon measure on X that is concentrated on A i , i.e. M + ( A ; X ) := Y i ∈ I M + ( A i ; X ) . ONDENSERS WITH TOUCHING PLATES 7
Elements of M + ( A ; X ) are said to be ( vector ) measures associated with A . If a measure µ ∈ M + ( A ; X ) and a vector-valued function u = ( u i ) i ∈ I with µ i -measurable components u i : X → [ −∞ , ∞ ] are given, then we write(3.2) h u , µ i := X i ∈ I h u i , µ i i = X i ∈ I Z u i dµ i . Being the intersection of an open and a closed subset of X , each A i , i ∈ I , is universallymeasurable, and hence M + ( A i ; X ) consists of all the restrictions µ | A i where µ ranges over M + ( X ). On the other hand, according to [4, Chapter I, Section 9, Proposition 13], A i itselfcan be thought of as a locally compact subspace of X . Thus M + ( A i ; X ) consists, in fact,of all those ν ∈ M + ( A i ) for each of which there exists b ν ∈ M + ( X ) with the property(3.3) b ν ( ϕ ) = h A i ϕ, ν i for every ϕ ∈ C ( X ) . We say that such b ν extends ν ∈ M + ( A i ) by 0 off A i to all of X . A sufficient condition for(3.3) to hold is that ν be bounded.Since A + ∩ A − = ∅ , there corresponds to each µ ∈ M + ( A ; X ) a ( signed ) scalar Radonmeasure R A µ := P i ∈ I s i µ i ∈ M ( X ), the ’resultant’ of µ , whose positive and negativeparts in the Hahn–Jordan decomposition are given by(3.4) R A µ + = X i ∈ I + µ i and R A µ − = X j ∈ I − µ j . For the sake of brevity we shall use the short notation R instead of R A if this will notcause any misunderstanding.The mapping M + ( A ; X ) ∋ µ R µ ∈ M ( X ) is in general non-injective. We shall call µ , µ ∈ M + ( A ; X ) R -equivalent if R µ = R µ . Note that the relation of R -equivalence on M + ( A ; X ) is that of identity if and only if all the A i , i ∈ I , are mutually disjoint. Alsoobserve that µ ∈ M + ( A ; X ) is R -equivalent to (if and) only if µ = .3.2. A (semi)metric structure on classes of vector measures.
For a given (strictlypositive definite) kernel κ on X and a given (generalized) condenser A , let E + κ ( A ; X ) consistof all µ ∈ M + ( A ; X ) such that κ ( µ i , µ i ) < ∞ for all i ∈ I ; in other words, E + κ ( A ; X ) := Y i ∈ I E + κ ( A i ; X ) . In view of [13, Lemma 2.3.1], we see from (3.1) that E + κ ( A ; X ) = { } .In accordance with an electrostatic interpretation of a condenser, we say that the interac-tion between the components µ i , i ∈ I , of µ ∈ E + κ ( A ; X ) is characterized by the matrix( s i s j ) i,j ∈ I . Given µ , µ ∈ E + κ ( A ; X ), we define the mutual energy (3.5) κ ( µ , µ ) := X i,j ∈ I s i s j κ ( µ i , µ j ) ONDENSERS WITH TOUCHING PLATES 8 and the vector potential κ µ = ( κ µ ,i ) i ∈ I where(3.6) κ µ ,i ( x ) := X j ∈ I s i s j κ ( x, µ j ) , x ∈ X. An assertion U ( x ) involving a variable point x ∈ X is said to hold c κ -n.e. on Q ⊂ X if c κ ( N ) = 0 where N consists of all x ∈ Q for which U ( x ) fails to hold. Lemma 3.3.
For any µ ∈ E + κ ( A ; X ) all the κ µ ,i , i ∈ I , are well defined and finite c κ -n.e.on X . Moreover, (3.7) κ µ ,i ( · ) = s i κ ( · , R µ ) c κ -n.e. on X. Proof.
Since µ i ∈ E + κ ( X ) for every i ∈ I , κ ( · , µ i ) is finite c κ -n.e. on X [13, p. 164]. Further-more, the set of all x ∈ X with κ ( x, µ i ) = ∞ is universally measurable, for κ ( · , µ i ) is l.s.c.on X . Combined with the fact that the inner capacity c κ ( · ) is subadditive on universallymeasurable sets [13, Lemma 2.3.5], this implies that κ µ ,i is well defined and finite c κ -n.e.on X . Finally, (3.7) is obtained directly from (3.4) and (3.6). (cid:3) Lemma 3.4.
For any µ , µ ∈ E + κ ( A ; X ) we have (3.8) κ ( µ , µ ) = κ ( R µ , R µ ) ∈ ( −∞ , ∞ ) . Proof.
This is obtained directly from (3.4) and (3.5). (cid:3)
For µ = µ ∈ E + κ ( A ; X ) the mutual energy κ ( µ , µ ) becomes the energy κ ( µ , µ ) of µ .Because of the strict positive definiteness of the kernel κ , we have from (3.8)(3.9) κ ( µ , µ ) = κ ( R µ , R µ ) ∈ [0 , ∞ ) for all µ ∈ E + κ ( A ; X ) , where κ ( µ , µ ) = 0 if and only if µ = .In order to introduce a (semi)metric structure on E + κ ( A ; X ), we define(3.10) k µ − µ k E + κ ( A ; X ) := k R µ − R µ k κ . Based on (3.8), we see by straightforward calculation that, in fact,(3.11) k µ − µ k E + κ ( A ; X ) = X i,j ∈ I s i s j κ ( µ i − µ i , µ j − µ j ) . Theorem 3.5. E + κ ( A ; X ) is a semimetric space with the semimetric defined by either ofthe ( equivalent ) relations (3.10) or (3.11) , and this space is isometric to its R -image in E κ ( X ) . This semimetric is a metric if and only if all the A i , i ∈ I , are mutually essentiallydisjoint, i.e. with c κ ( A i ∩ A j ) = 0 for all i = j .Proof. The former assertion is obvious by (3.10). Since a nonzero positive scalar measure offinite energy does not charge any set of zero capacity [13, Lemma 2.3.1], the sufficiency partof the latter assertion lemma holds. To prove the necessity part, assume on the contrarythat there are two equally signed plates A k and A ℓ , k = ℓ , with c κ ( A k ∩ A ℓ ) >
0. By
ONDENSERS WITH TOUCHING PLATES 9 [13, Lemma 2.3.1], there is a nonzero measure τ ∈ E + κ ( A k ∩ A ℓ ; X ). Choose µ = ( µ i ) i ∈ I ∈E + κ ( A ; X ) such that µ k | A k ∩ A ℓ − τ >
0, and define µ m = ( µ im ) i ∈ I ∈ E + κ ( A ; X ), m = 1 , µ k = µ k − τ and µ i = µ i for all i = k , while µ ℓ = µ ℓ + τ and µ i = µ i for all i = ℓ .Then R µ = R µ , hence k µ − µ k E + κ ( A ; X ) = 0, though µ = µ . (cid:3) Similarly to the terminology for the pre-Hilbert space E κ ( X ), the topology of the semi-metric space E + κ ( A ; X ) is termed strong . We say that µ , µ ∈ E + κ ( A ; X ) are equivalent in E + κ ( A ; X ) if k µ − µ k E + κ ( A ; X ) = 0, or equivalently if R µ = R µ .3.3. The vague topology on M + ( A ; X ) . In Section 3.3 we consider a standard condenser A (see Definition 3.2). The set of all (vector) measures associated with A can be endowedwith the vague topology defined as follows. Definition 3.6.
The vague topology on M + ( A ; X ) is the topology of the product space Q i ∈ I M + ( A i ; X ) where each of the M + ( A i ; X ) is endowed with the vague topology inducedfrom M ( X ). Namely, a sequence { µ k } k ∈ N ⊂ M + ( A ; X ) converges to µ ∈ M + ( A ; X ) vaguely if for every i ∈ I , µ ik → µ i vaguely in M ( X ) as k → ∞ .Since all the A i , i ∈ I , are closed in X , M + ( A ; X ) is vaguely closed in M ( X ) p . Besides,since every M + ( A i ; X ) is Hausdorff in the vague topology, so is M + ( A ; X ) [4, Chapter I,Section 8, Proposition 7]. Hence, a vague limit of any sequence in M + ( A ; X ) belongsto M + ( A ; X ) and is unique (whenever it exists). We call a set F ⊂ M + ( A ; X ) vaguelybounded if for every ϕ ∈ C ( X ),sup µ ∈ F | µ i ( ϕ ) | < ∞ for all i ∈ I. Lemma 3.7.
A vaguely bounded set F ⊂ M + ( A ; X ) is vaguely relatively compact.Proof. It is clear from the above definition that for every i ∈ I the set F i := (cid:8) µ i ∈ M + ( A i ; X ) : µ = ( µ j ) j ∈ I ∈ F (cid:9) is vaguely bounded in M + ( X ); hence, by [2, Chapitre III, Section 2, Proposition 9], F i isvaguely relatively compact in M ( X ). As F ⊂ Q i ∈ I F i , the lemma follows from Tychonoff’stheorem on the product of compact spaces [4, Chapter I, Section 9, Theorem 3]. (cid:3) An unconstrained weighted minimum energy problem for vector measures.
Let a (generalized) condenser A = ( A i ) i ∈ I and a (strictly positive definite) kernel κ on X be given. Fix a vector-valued function f = ( f i ) i ∈ I , where each f i : X → [ −∞ , ∞ ] is µ -measurable for every µ ∈ E + κ ( A i ; X ) and f i is treated as an external field acting on thecharges (measures) from E + κ ( A i ; X ). The f -weighted vector potential and the f -weightedenergy of µ ∈ E + κ ( A ; X ) are given by W µ κ, f := κ µ + f , (3.12) G κ, f ( µ ) := κ ( µ , µ ) + 2 h f , µ i , (3.13) ONDENSERS WITH TOUCHING PLATES 10 respectively. Thus W µ κ, f = (cid:0) W µ ,iκ, f (cid:1) i ∈ I , where W µ ,iκ, f := κ µ ,i + f i (see (3.6)). Let E + κ, f ( A ; X )consist of all µ ∈ E + κ ( A ; X ) with finite G κ, f ( µ ), or equivalently with finite h f , µ i . Lemma 3.8.
Suppose that a set E ⊂ E + κ, f ( A ; X ) is convex. Then there exists λ ∈ E with (3.14) G κ, f ( λ ) = min µ ∈ E G κ, f ( µ ) if and only if (3.15) X i ∈ I (cid:10) W λ ,iκ, f , µ i − λ i (cid:11) > for all µ ∈ E . Proof.
By direct calculation, for any µ , ν ∈ E and any h ∈ (0 ,
1] we get G κ, f (cid:0) h µ + (1 − h ) ν (cid:1) − G κ, f ( ν ) = 2 h X i ∈ I (cid:10) W ν ,iκ, f , µ i − ν i (cid:11) + h k µ − ν k E + κ ( A ; X ) . If ν = λ satisfies (3.14), then the left (hence, also the right) side of this display is >
0, whichleads to (3.15) by letting h →
0. Conversely, if (3.15) holds, then the preceding formulawith ν = λ and h = 1 gives G κ, f ( µ ) > G κ, f ( λ ) for all µ ∈ E , and (3.14) follows. (cid:3) Fix a numerical vector a = ( a i ) i ∈ I with 0 < a i < ∞ , i ∈ I , and write M + ( A , a ; X ) := (cid:8) µ ∈ M + ( A ; X ) : µ i ( A i ) = a i for all i ∈ I (cid:9) , E + κ ( A , a ; X ) := E + κ ( A ; X ) ∩ M + ( A , a ; X ) , E + κ, f ( A , a ; X ) := E + κ, f ( A ; X ) ∩ M + ( A , a ; X ) . If the class E + κ, f ( A , a ; X ) is nonempty, or equivalently if G κ, f ( A , a ; X ) := inf µ ∈E + κ, f ( A , a ; X ) G κ, f ( µ ) < ∞ , then the following unconstrained weighted minimum energy problem makes sense. Problem 3.9.
Given X , κ , A , a and f , does there exist λ A ∈ E + κ, f ( A , a ; X ) with G κ, f ( λ A ) = G κ, f ( A , a ; X )?If I = I + = { } , A is closed, a = 1 and f = 0, then Problem 3.9 reduces to theproblem (2.1) solved in [13, Theorem 4.1] (see Remark 2.6 above). Remark 3.10.
Let A be a standard condenser in X and let(3.16) sup x ∈ A + , y ∈ A − κ ( x, y ) < ∞ . Under these assumptions, in [25, 26] an approach has been worked out based on both thevague and the strong topologies on E + κ ( A ; X ) which made it possible to provide a fairly G κ, f ( · ) is also known as the Gauss functional (see e.g. [21]; compare with [17]). Note that when defining G κ, f ( · ), we have used the notation (3.2). ONDENSERS WITH TOUCHING PLATES 11 complete analysis of Problem 3.9. In more detail, it was shown that if the kernel κ isperfect and if for all i ∈ I either f i ∈ Ψ( X ) or f i = s i κ ( · , ζ ) for some (signed) ζ ∈ E κ ( X ),then the requirement(3.17) c κ ( A + ∪ A − ) < ∞ is sufficient for Problem 3.9 to be solvable for every vector a [25, Theorem 8.1]. However,if (3.17) does not hold then in general there exists a vector a ′ such that the problem has no solution (see [25]). Therefore, it was interesting to give a description of the set ofall vectors a for which Problem 3.9 is nevertheless solvable. Such a characterization hasbeen established in [26] (see also footnote 7 below). On the other hand, if assumption(3.16) is omitted, then the approach developed in [25, 26] breaks down, and (3.17) doesnot guarantee anymore the existence of a solution to Problem 3.9. This has been illustratedby [11, Theorem 4.6] pertaining to the Newtonian kernel.3.5. A constrained weighted minimum energy problem for vector measures.
A measure σ i ∈ M + ( A i ; X ) is said to be a constraint for elements of M + ( A i , a i ; X ) if σ i ( A i ) > a i . Let C ( A i ; X ) consist of all these σ i , and let C ( A ; X ) := Y i ∈ I C ( A i ; X ) . Consider ξ = ( ξ i ) i ∈ I such that for each i ∈ I either ξ i = σ i ∈ C ( A i ; X ) or ξ i = ∞ , where theformal notation ξ i = ∞ means that no upper constraint on the elements of M + ( A i , a i ; X )is imposed, and define M ξ ( A ; X ) := (cid:8) µ ∈ M + ( A ; X ) : µ i ξ i for all i ∈ I (cid:9) . (If ξ i = σ i ∈ C ( A i ; X ), then µ i ξ i means that ξ i − µ i >
0, while we make the obviousconvention that any positive scalar Radon measure is ∞ .) Also write M ξ ( A , a ; X ) := M + ( A , a ; X ) ∩ M ξ ( A ; X ) , E ξ κ ( A , a ; X ) := E + κ ( A , a ; X ) ∩ M ξ ( A ; X ) , E ξ κ, f ( A , a ; X ) := E + κ, f ( A , a ; X ) ∩ M ξ ( A ; X ) . If the class E ξ κ, f ( A , a ; X ) is nonempty, or equivalently if G ξ κ, f ( A , a ; X ) := inf µ ∈E ξ κ, f ( A , a ; X ) G κ, f ( µ ) < ∞ , then the following constrained weighted minimum energy problem makes sense. Problem 3.11.
Given X , κ , A , a , f and ξ , does there exist λ ξ A ∈ E ξ κ, f ( A , a ; X ) such that G κ, f ( λ ξ A ) = G ξ κ, f ( A , a ; X )? Let S ξ κ, f ( A , a ; X ) consist of all these λ ξ A . In the case of the α -Riesz kernels of order 1 < α R , some of the (theoretical) results on thesolvability or unsolvability of Problem 3.9 mentioned in [25] have been illustrated in [18, 20] by means ofnumerical experiments. ONDENSERS WITH TOUCHING PLATES 12
Lemma 3.12.
Any two solutions λ , ˘ λ ∈ S ξ κ, f ( A , a ; X ) are R -equivalent.Proof. This can be established by standard methods based on the convexity of the class E ξ κ, f ( A , a ; X ), the isometry between this class and its R -image in E κ ( X ), and the pre-Hilbertstructure on the space E κ ( X ). Indeed, in view of the convexity of E ξ κ, f ( A , a ; X ), relations(3.9) and (3.13) imply4 G ξ κ, f ( A , a ; X ) G κ, f (cid:16) λ + ˘ λ (cid:17) = k R λ + R ˘ λ k κ + 4 h f , λ + ˘ λ i . On the other hand, applying the parallelogram identity in E κ ( X ) to R λ and R ˘ λ and thenadding and subtracting 4 h f , λ + ˘ λ i we get k R λ − R ˘ λ k κ = −k R λ + R ˘ λ k κ − h f , λ + ˘ λ i + 2 G κ, f ( λ ) + 2 G κ, f (˘ λ ) . When combined with the preceding relation, this yields0 k R λ − R ˘ λ k κ − G ξ κ, f ( A , a ; X ) + 2 G κ, f ( λ ) + 2 G κ, f (˘ λ ) = 0 , which establishes the lemma because of the strict positive definiteness of κ . (cid:3) Corollary 3.13.
If the class S ξ κ, f ( A , a ; X ) is nonempty, then it reduces to a single elementwhenever all the A i , i ∈ I , are mutually essentially disjoint.Proof. This follows directly from Lemma 3.12 and Theorem 3.5. (cid:3)
Remark 3.14.
Assume for a moment that (3.16) holds, the condenser A is standard, thekernel κ is perfect and the external field f is as described in Remark 3.10. It has been shownin [24, Theorem 6.2] that then condition (3.17) guarantees the existence of a solution toProblem 3.11 for any ξ and any vector a . α -Riesz balayage and α -Green kernel In all that follows fix n > α ∈ (0 ,
2] and a domain D ⊂ R n with c κ α ( D c ) >
0, where D c := R n \ D , and assume that either κ ( x, y ) = κ α ( x, y ) := | x − y | α − n is the α -Riesz kernel on X = R n , or κ ( x, y ) = g αD ( x, y ) is the α -Green kernel on X = D . For the definition of g αD , see [19, Chapter IV, Section 5] or see below.For given x ∈ R n and r ∈ (0 , ∞ ) write B ( x, r ) := { y ∈ R n : | y − x | < r } , S ( x, r ) := { y ∈ R n : | y − x | = r } and B ( x, r ) := B ( x, r ) ∪ S ( x, r ). Let ∂Q denote the boundary of a set Q ⊂ R n in the topology of R n . Actually, this result and those described in Remark 3.10 have been obtained in [24]–[26] even for infinite dimensional vector measures.
ONDENSERS WITH TOUCHING PLATES 13
We shall simply write α instead of κ α if κ α serves as an index. When speaking of a positivescalar Radon measure µ ∈ M + ( R n ), we always assume κ α ( · , µ ) + ∞ . This impliesthat(4.1) Z | y | > dµ ( y ) | y | n − α < ∞ (see [19, Eq. 1.3.10]), and consequently that κ α ( · , µ ) is finite c α -n.e. on R n [19, Chapter III,Section 1]; these two implications can actually be reversed.We shall usually use the short form ’n.e.’ instead of ’ c α -n.e.’ if this will not cause anymisunderstanding. Definition 4.1. ν ∈ M ( D ) is called extendible if there exist c ν + and c ν − extending ν + and ν − , respectively, by 0 off D to R n (see (3.3)), and if these c ν + and c ν − satisfy (4.1). Weidentify such a ν ∈ M ( D ) with its extension b ν := c ν + − c ν − , and we therefore write b ν = ν .Every bounded measure ν ∈ M ( D ) is extendible. The converse holds if D is bounded, butnot in general (e.g. not if D c is compact). The set of all extendible measures consists of allthe restrictions µ | D where µ ranges over M ( R n ).The α -Green kernel g = g αD on D is defined by g αD ( x, y ) = κ α ( x, ε y ) − κ α ( x, ε D c y ) for all x, y ∈ D, where ε y denotes the unit Dirac measure at a point y and ε D c y its α -Riesz balayage ontothe (closed) set D c , uniquely determined in the frame of the classical approach by [16,Theorem 3.6] pertaining to positive Radon measures on R n . See also the book by Bliedtnerand Hansen [1] where balayage is studied in the setting of balayage spaces.We shall simply write µ ′ instead of µ D c when speaking of the α -Riesz balayage of µ ∈ M + ( D ; R n ) onto D c . According to [16, Corollaries 3.19 and 3.20], for any µ ∈ M + ( D ; R n ) the balayage µ ′ is c α -absolutely continuous and it is determined uniquely by relation (4.2) κ α ( · , µ ′ ) = κ α ( · , µ ) n.e. on D c among the c α -absolutely continuous measures supported by D c . Furthermore, there holdsthe integral representation (4.3) µ ′ = Z ε ′ y dµ ( y )(see [16, Theorem 3.17]). If moreover µ ∈ E + α ( D ; R n ), then the balayage µ ′ is in fact theorthogonal projection of µ onto the convex cone E + α ( D c ; R n ) (see [15, Theorem 4.12] or [16, In the literature the integral representation (4.3) seems to have been more or less taken for granted,though it has been pointed out in [3, p. 18, Remarque] that it requires that the family ( ε ′ y ) y ∈ D is µ -adequate in the sense of [3, Section 3, D´efinition 1] (see also counterexamples (without µ -adequacy) in Exercises 1and 2 at the end of that section). We therefore have brought in [16, Lemma 3.16] a proof of this adequacy. ONDENSERS WITH TOUCHING PLATES 14
Theorem 3.1]), i.e. µ ′ ∈ E + α ( D c ; R n ) and(4.4) k µ − θ k α > k µ − µ ′ k α for all θ ∈ E + α ( D c ; R n ) , θ = µ ′ . If now ν ∈ M ( D ) is an extendible (signed) measure, then ν ′ := ν D c := ( ν + ) D c − ( ν − ) D c is said to be a balayage of ν onto D c . It follows from [19, Chapter III, Section 1, n ◦ ν ′ is determined uniquely by (4.2) with ν in place of µ amongthe c α -absolutely continuous measures supported by D c .The following definition goes back to Brelot (see [5, Theorem VII.13]). Definition 4.2.
A closed set F ⊂ R n is said to be α -thin at infinity if either F is compact,or the inverse of F relative to S (0 ,
1) has x = 0 as an α -irregular boundary point (cf. [19,Theorem 5.10]). Theorem 4.3 (see [16, Theorem 3.22]) . The set D c is not α -thin at infinity if and onlyif for every bounded measure µ ∈ M + ( D ) we have (4.5) µ ′ ( R n ) = µ ( R n ) . As noted in Remark 2.3 above, the α -Riesz kernel κ α on R n and the α -Green kernel g αD on D are both strictly positive definite and moreover perfect. Furthermore, the α -Rieszkernel κ α (with α ∈ (0 , α -Green kernel g , the following assertion holds. Theorem 4.4 (see [16, Theorem 4.6]) . Let µ, ν ∈ M + ( D ) be extendible, g ( µ, µ ) < ∞ , andlet w be a positive α -superharmonic function on R n [19, Chapter I, Section n ◦ . Ifmoreover g ( · , µ ) g ( · , ν ) + w ( · ) µ -a.e. on D , then the same inequality holds on all of D . The following three lemmas establish relations between potentials and energies relative tothe kernels κ α and g = g αD , respectively. Lemma 4.5.
For any extendible measure µ ∈ M ( D ) , the α -Green potential g ( · , µ ) is welldefined and finite ( c α - ) n.e. on D and given by (4.6) g ( · , µ ) = κ α ( · , µ − µ ′ ) n.e. on D. Proof.
It is seen from Definition 4.1 that κ α ( · , µ ) is well defined and finite n.e. on R n , andhence so is κ α ( · , µ ′ ). Applying (4.3) to µ ± , we get by [3, Section 3, Th´eor`eme 1] g ( · , µ ) = Z (cid:2) κ α ( · , ε y ) − κ α ( · , ε ′ y ) (cid:3) dµ ( y ) = κ α ( · , µ ) − κ α ( · , µ ′ )n.e. on D , and the lemma follows. (cid:3) If Q is a given subset of D , then any assertion involving a variable point holds n.e. on Q if and only ifit holds c g -n.e. on Q [10, Lemma 2.6]. ONDENSERS WITH TOUCHING PLATES 15
Lemma 4.6.
Suppose that µ ∈ M ( D ) is extendible and the extension belongs to E α ( R n ) .Then µ ∈ E g ( D ) , (4.7) µ − µ ′ ∈ E α ( R n ) , (4.8) k µ k g = k µ − µ ′ k α = k µ k α − k µ ′ k α . (4.9) Proof.
In view of the definition of a (signed) measure of finite energy (see Section 2), weobtain (4.7) from the inequality (4.10) g αD ( x, y ) < κ α ( x, y ) for all x, y ∈ D, while (4.8) from [16, Corollary 3.7] (or [16, Theorems 3.1 and 3.6]). According to Lemma 4.5and footnote 9, g ( · , µ ) is finite c g -n.e. on D and given by (4.6), while by (4.7) the same holds | µ | -a.e. on D (see [13, Lemma 2.3.1]). Integrating (4.6) with respect to µ ± , we thereforeobtain by subtraction(4.11) ∞ > g ( µ, µ ) = κ α ( µ − µ ′ , µ ) . Since κ α ( · , µ − µ ′ ) = 0 n.e. on D c by (4.2) and since µ ′ is c α -absolutely continuous, we alsohave(4.12) κ α ( µ − µ ′ , µ ′ ) = 0 , which results in the former equality in (4.9) when combined with (4.11). Due to (4.8),relation (4.12) takes the form k µ ′ k α = κ α ( µ, µ ′ ), and the former equality in (4.9) impliesthe latter. (cid:3) Lemma 4.7.
Assume that µ ∈ M ( D ) has compact support S µD . Then µ ∈ E g ( D ) if andonly if its extension belongs to E α ( R n ) .Proof. According to Lemma 4.6, it is enough to establish the ’only if’ part of the lemma.We may clearly assume that µ is positive. Since κ α ( · , µ ′ ) is continuous on D and hencebounded on the compact set S µD , we have(4.13) κ α ( µ, µ ′ ) < ∞ . But g ( µ, µ ) is finite by assumption, and therefore likewise as in the preceding proof relation(4.11) holds. Combining (4.11) with (4.13) yields µ ∈ E α ( R n ). (cid:3) Minimum α -Riesz energy problems for generalized condensers An unconstrained weighted minimum α -Riesz energy problem. Consider ageneralized condenser A = ( A i ) i ∈ I in R n with p := Card I > I + := { , . . . , p − } and I − := { p } (see Section 3.1). Also require that the negative plate A p is closed in R n ,while all the positive plates A j , j ∈ I + , are relatively closed subsets of the (open) set The strict inequality in (4.10) is caused by our convention that c α ( D c ) > ONDENSERS WITH TOUCHING PLATES 16 D := A cp = R n \ A p . For the sake of simplicity, in all that follows assume that D is adomain. Recall that, by convention (3.1), c α ( A i ) > i ∈ I .When speaking of an external field f = ( f i ) i ∈ I acting on the vector measures of the class E + α ( A ; R n ), we shall always tacitly assume that either Case I or Case II holds, whereI. f i ∈ Ψ( R n ) for every i ∈ I and moreover (5.1) f p = 0 n.e. on A p . II. f i = s i κ α ( · , ζ − ζ ′ ) for every i ∈ I , where ζ is an extendible ( signed ) Radon measureon D with κ α ( ζ, ζ ) < ∞ .Observe that (5.1) holds also in Case II (see (4.2)). Since a nonzero positive scalar measureof finite energy does not charge any set of zero capacity [13, Lemma 2.3.1], we thus seethat under the stated assumptions no external field acts on the measures from E + α ( A p ; R n ).Furthermore, D c is ν -negligible for any ν ∈ M + ( A j ; R n ), j ∈ I + (see Section 2). We arethus led to the following conclusion. Lemma 5.1.
For any µ ∈ E + α ( A ; R n ) , G α, f ( µ ) can (equivalently ) be defined as (5.2) G α, f ( µ ) = κ α ( µ , µ ) + 2 h f , µ i = κ α ( µ , µ ) + 2 h f + , µ + i , where f + := ( f j | D ) j ∈ I + and µ + := ( µ j ) j ∈ I + . If Case II holds, then for every µ ∈ E + α ( A ; R n ) we get from (3.4) and (3.9) G α, f ( µ ) = k R µ k α + 2 X i ∈ I s i κ α ( ζ − ζ ′ , µ i ) = k R µ k α + 2 κ α ( ζ − ζ ′ , R µ ) , hence(5.3) ∞ > G α, f ( µ ) = k R µ + ζ − ζ ′ k α − k ζ − ζ ′ k α > −k ζ − ζ ′ k α > −∞ . Thus in either Case I or Case II G α, f ( µ ) > − M > −∞ for all µ ∈ E + α ( A ; R n ) , which is clear from (3.9) and (5.2) if Case I holds, or from (5.3) otherwise.Fix a numerical vector a = ( a i ) i ∈ I with a i > i ∈ I . Using the notations of Section 3.4with X = R n and κ = κ α , we obtain from the preceding display(5.4) G α, f ( A , a ; R n ) := inf µ ∈E + α, f ( A , a ; R n ) G α, f ( µ ) > −∞ . By [4, Chapter I, Section 3, Proposition 5], this is in agreement with our general requirement that thesets A i , i ∈ I be locally closed in R n (see the beginning of Section 3.1). Cf. (3.13) with X = R n and κ = κ α . ONDENSERS WITH TOUCHING PLATES 17 If E + α, f ( A , a ; R n ) is nonempty, or equivalently if G α, f ( A , a ; R n ) < ∞ , then we can consider(the unconstrained) Problem 3.9 on the existence of λ A ∈ E + α, f ( A , a ; R n ) with G α, f ( λ A ) = G α, f ( A , a ; R n ) . The following theorem shows that, in general, this problem has no solution. Theorem 5.2.
Suppose that D c is not α -thin at infinity, I + = { } , c g αD ( A ) = ∞ , andlet a = , f = . Then G α, f ( A , a ; R n ) = (cid:2) c g αD ( A ) (cid:3) − = 0; hence G α, f ( A , a ; R n ) cannot be an actual minimum because / ∈ E + α, f ( A , a ; R n ) .Proof. Since G α, f ( µ ) = κ α ( µ , µ ) because of f = , Problem 3.9 reduces to the problem ofminimizing κ α ( µ , µ ) over E + α ( A , a ; R n ). Thus by (3.9)(5.5) G α, f ( A , a ; R n ) > . Consider compact sets K ℓ ⊂ A , ℓ ∈ N , such that K ℓ ↑ A as ℓ → ∞ . By (2.3),(5.6) c g ( K ℓ ) ↑ c g ( A ) = ∞ as ℓ → ∞ , and hence there is no loss of generality in assuming that c g ( K ℓ ) > ℓ ∈ N .Furthermore, since the α -Green kernel g is strictly positive definite and moreover perfect(Remark 2.3), we see from (2.2) that c g ( K ℓ ) < ∞ and, by Remark 2.6, there exists a(unique) g -capacitary measure λ ℓ on K ℓ , i.e. λ ℓ ∈ E + g ( K ℓ , D ) with k λ ℓ k g = 1 /c g ( K ℓ ) < ∞ . According to Lemma 4.7 with λ ℓ in place of µ , κ α ( λ ℓ , λ ℓ ) is finite along with g ( λ ℓ , λ ℓ ).Hence, by Lemma 4.6, k λ ℓ k g = k λ ℓ − λ ′ ℓ k α . Applying Theorem 4.3, we get µ ℓ := ( λ ℓ , λ ′ ℓ ) ∈ E + α, f ( A , a ; R n ), which together with the twopreceding displays and (3.9) and (5.5) gives1 /c g ( K ℓ ) = k λ ℓ − λ ′ ℓ k α = κ α ( µ ℓ , µ ℓ ) > G α, f ( A , a ; R n ) > . Letting here ℓ → ∞ , we obtain the theorem from (5.6). (cid:3) Using the electrostatic interpretation, which is possible for the Coulomb kernel | x − y | − on R , we say that under the hypotheses of Theorem 5.2 a short-circuit occurs between theoppositely signed plates of the generalized condenser A . It is therefore meaningful to askwhat kinds of additional requirements on the objects in question will prevent this blow-upeffect, and secure that a solution to the corresponding minimum α -Riesz energy problemdoes exist. To this end we have succeeded in working out a substantive theory by imposinga proper upper constraint on the vector measures under consideration. ONDENSERS WITH TOUCHING PLATES 18
A constrained weighted minimum α -Riesz energy problem. Let A , a and f be as at the beginning of Section 5.1. In the rest of the paper we assume additionally that A p ( = D c ) is not α -thin at infinity and (5.7) a p = X j ∈ I + a j . Using notation of Section 3.5, fix ξ = ( ξ i ) i ∈ I with(5.8) ξ j ∈ C ( A j ; R n ) ∩ E + α ( A j ; R n ) for all j ∈ I + , and ξ p = ∞ . Unless explicitly stated otherwise, for these A , a , f , and ξ we shall always require that (5.9) G ξ α, f ( A , a ; R n ) < ∞ . The main purpose of this paper is to analyze Problem on the existence of λ ξ A ∈E ξ α, f ( A , a ; R n ) with G α, f ( λ ξ A ) = G ξ α, f ( A , a ; R n ). Recall that S ξ α, f ( A , a ; R n ) denotes theclass of all solutions to this problem (provided these exist). By Lemma 3.12, any twosolutions are R -equivalent, while by Corollary 3.13, S ξ α, f ( A , a ; R n ) reduces to a singleelement whenever the A j , j ∈ I + , are mutually essentially disjoint.Conditions on A , f and ξ which guarantee that (5.9) holds are given in the followingLemma 5.3. Write(5.10) A ◦ j := (cid:8) x ∈ A j : | f j ( x ) | < ∞ (cid:9) , j ∈ I + . Lemma 5.3.
Relation (5.9) holds if either Case II takes place, or ( in the presence ofCase I) if (5.11) ξ j ( A j \ A ◦ j ) = 0 for all j ∈ I + . Proof.
Assume first that (5.11) holds. Then for every j ∈ I + we have ξ j ( A ◦ j ) > a j , and bythe universal measurability of A ◦ j there is a compact set K j ⊂ A ◦ j such that ξ j ( K j ) > a j and | f j | M j < ∞ on K j for some constant M j (see (5.10)). Define µ := ( µ i ) i ∈ I , where µ j := a j ξ j | K j /ξ j ( K j ) for all j ∈ I + and µ p is any measure from E + α ( A p , a p ; R n ) (such µ p exists since c α ( A p ) > ξ j | K j ∈ E + α ( K j ; R n ) for all j ∈ I + by (5.8), we get µ ∈ E ξ α, f ( A , a ; R n ) which yields (5.9). To complete the proof, it is left to observe that(5.11) holds automatically if Case II takes place, because then κ α ( · , ζ − ζ ′ ) is finite n.e. on R n , hence ξ j -a.e. for all j ∈ I + by [13, Lemma 2.3.1]. (cid:3) The theory developed in the present study includes sufficient and/or necessary conditionsfor the existence of solutions λ ξ A = ( λ i A ) i ∈ I to Problem 3.11 with A , a , f and ξ chosenabove (see Theorems 7.1 and 7.2). We also provide descriptions of the f -weighted α -Riesz G ξ α, f ( A , a ; R n ) is then actually finite, for G ξ α, f ( A , a ; R n ) > −∞ by E ξ α, f ( A , a ; R n ) ⊂ E + α, f ( A , a ; R n )and (5.4). ONDENSERS WITH TOUCHING PLATES 19 vector potentials of the solutions λ ξ A , single out their characteristic properties, and analyzethe supports of the λ i A , i ∈ I (see Theorems 7.3, 7.5 and 7.6). These results are illustratedin Examples 10.1 and 10.2. See also Section 7.2 for an extension of the theory to the casewhere ξ p = ∞ . The proofs of Theorems 7.1–7.6 are given in Sections 8 and 9; they aresubstantially based on Theorem 6.1 which is a subject of the next section.6. Relations between minimum α -Riesz and α -Green energy problems Throughout this section, A , a , f and ξ are as indicated at the beginning of Section 5.2,except for (5.9) which is temporarily not required. The aim of Theorem 6.1 below is toestablish a relationship between, on the one hand, the solvability (or the non-solvability)of Problem 3.11 for R n , κ α , A , a , f , ξ and, on the other hand, that for D , g = g αD , A + , a + , f + and ξ + , where A + := ( A j ) j ∈ I + , a + := ( a j ) j ∈ I + , f + := ( f j | D ) j ∈ I + , ξ + := ( ξ j ) j ∈ I + . (Note that A + is a standard condenser in X = D consisting of only positive plates.)Observe that since for every given j ∈ I + we have(6.1) M + ( A j ; R n ) ⊂ M + ( A j ; D ) , the measure ξ j can certainly be thought of as an element of C ( A j ; D ).For any µ = ( µ i ) i ∈ I ∈ M + ( A ; R n ) write µ + := ( µ j ) j ∈ I + ; then µ + belongs to M + ( A + ; D )by (6.1). If moreover κ α ( µ , µ ) < ∞ , then µ + belongs to E + α ( A + ; R n ), as well as to E + g ( A + ; D ), the latter being clear from (4.7). Theorem 6.1.
Under the just mentioned assumptions on A , a , f and ξ , (6.2) G ξ α, f ( A , a ; R n ) = G ξ + g, f + ( A + , a + ; D ) . If moreover these ( equal ) extremal values are finite, then S ξ α, f ( A , a ; R n ) is nonempty ifand only if so is S ξ + g, f + ( A + , a + ; D ) , and in the affirmative case the following two assertionsare equivalent for any λ A = ( λ i A ) i ∈ I ∈ M + ( A ; R n ):(i) λ A ∈ S ξ α, f ( A , a ; R n ) . (ii) λ + A = ( λ j A ) j ∈ I + ∈ S ξ + g, f + ( A + , a + ; D ) and, in addition, (6.3) λ p A = (cid:16) X j ∈ I + λ j A (cid:17) ′ . Proof.
We begin by establishing the inequality(6.4) G ξ + g, f + ( A + , a + ; D ) > G ξ α, f ( A , a ; R n ) . ONDENSERS WITH TOUCHING PLATES 20
Assuming G ξ + g, f + ( A + , a + ; D ) < ∞ , choose µ = ( µ j ) j ∈ I + ∈ E ξ + g, f + ( A + , a + ; D ). Then, accord-ing to (3.9) and (3.13) with X = D and κ = g , G g, f + ( µ ) = g ( µ , µ ) + 2 h f + , µ i = k R A + µ k g + 2 h f + , µ i . Being bounded, each of the µ j , j ∈ I + , is extendible (see Section 4). Furthermore, theextension in question has finite α -Riesz energy, for so does the extension of the constraint ξ j by (5.8). Applying (4.9) to R A + µ ∈ E + α ( A + ; R n ) in place of µ , we thus get G g, f + ( µ ) = k R A + µ − ( R A + µ ) ′ k α + 2 h f + , µ i . Since A p ( = D c ) is not α -thin at infinity, we conclude from (4.5) and (5.7) that (cid:0) R A + µ (cid:1) ′ ∈E + α ( A p , a p ; R n ), and therefore ˜ µ = (˜ µ i ) i ∈ I ∈ E ξ α ( A , a ; R n ) where(6.5) ˜ µ + = µ and ˜ µ p = (cid:0) R A + µ (cid:1) ′ = (cid:16) X j ∈ I + µ j (cid:17) ′ . Here we have used the (permanent) assumption that ξ p = ∞ . Furthermore, h f , ˜ µ i = h f + , µ i < ∞ , the equality being valid because f p = 0 n.e. on A p (see Section 5.1), hence ˜ µ p -a.e. by[13, Lemma 2.3.1], and also because D c is µ j -negligible for every j ∈ I + . Thus ˜ µ ∈E ξ α, f ( A , a ; R n ) and G α, f ( ˜ µ ) = G g, f + ( µ ), where the latter relation follows from the precedingthree displays. This yields G g, f + ( µ ) = G α, f ( ˜ µ ) > G ξ α, f ( A , a ; R n ) , which in view of the arbitrary choice of µ ∈ E ξ + g, f + ( A + , a + ; D ) establishes (6.4).On the other hand, in view of (4.7) and (5.2) for any ν ∈ E ξ α, f ( A , a ; R n ) we have ν + ∈E ξ + g, f + ( A + , a + ; D ). Thus, by (3.9), (4.4), (4.9) and (5.2), G α, f ( ν ) = κ α ( ν , ν ) + 2 h f + , ν + i = k R A ν + − ν p k α + 2 h f + , ν + i > k R A ν + − ( R A ν + ) ′ k α + 2 h f + , ν + i = k R A ν + k g + 2 h f + , ν + i = g ( ν + , ν + ) + 2 h f + , ν + i = G g, f + ( ν + ) > G ξ + g, f + ( A + , a + ; D ) . (6.6)Since ν ∈ E ξ α, f ( A , a ; R n ) has been chosen arbitrarily, this together with (6.4) proves (6.2).Now suppose that there exists µ = ( µ j ) j ∈ I + ∈ S ξ + g, f + ( A + , a + ; D ). Define ˜ µ = (˜ µ i ) i ∈ I as in(6.5). Then the same arguments as those applied in the first paragraph of this proof enableus to see that ˜ µ ∈ E ξ α, f ( A , a ; R n ) and also that G α, f ( ˜ µ ) = G g, f + ( µ ). The latter yields G α, f ( ˜ µ ) = G ξ + g, f + ( A + , a + ; D ) . Substituting (6.2) into the last display shows that, actually, ˜ µ ∈ S ξ α, f ( A , a ; R n ), which inview of the latter relation in (6.5) proves that, indeed, (ii) implies (i). ONDENSERS WITH TOUCHING PLATES 21
To establish the converse implication, assume that there is ν = ( ν i ) i ∈ I ∈ S ξ α, f ( A , a ; R n ).Then ν + ∈ E ξ + g, f + ( A + , a + ; D ) (see the second paragraph of the proof) and, in addition,(6.6) holds. Since for this ν the first term in (6.6) equals G ξ α, f ( A , a ; R n ), we concludefrom (6.2) that all the inequalities in (6.6) are in fact equalities. This implies that ν + ∈ S ξ + g, f + ( A + , a + ; D ) and also that ν p = ( R ν + ) ′ , the latter being clear from (4.4). (cid:3) Main results
Throughout Section 7 we keep all the assumptions on A , a , f and ξ imposed at thebeginning of Section 5.2, except for (5.9). Formulations of the main results.Theorem 7.1.
Assume now that (5.9) is fulfilled and, moreover, (7.1) ξ j ( A j ) < ∞ for all j ∈ I + . Then the class S ξ α, f ( A , a ; R n ) of the solutions to Problem is nonempty, and for anyone of its elements λ ξ A = ( λ i A ) i ∈ I we have λ p A = (cid:0)P j ∈ I + λ j A (cid:1) ′ . Theorem 7.1 is sharp in the sense that it no longer holds if requirement (7.1) is omittedfrom its hypotheses (see the following Theorem 7.2).
Theorem 7.2.
Condition (7.1) is in general also necessary for the solvability of Prob-lem . More precisely, suppose that I + = { } , c α ( A ) = ∞ and that Case II holds with ζ > . Then there is a constraint ξ ∈ C ( A ; R n ) ∩ E + α ( A ; R n ) with ξ ( A ) = ∞ such that (7.2) G ξ α, f ( A , a ; R n ) = G ξ g,f | D ( A , a ; D ) = 0; hence G ξ α, f ( A , a ; R n ) cannot be an actual minimum because / ∈ E ξ α, f ( A , a ; R n ) . The following three assertions provide descriptions of the f -weighted α -Riesz potentials W λ ξ A α, f , cf. (3.12), of the solutions λ ξ A = ( λ i A ) i ∈ I ∈ S ξ α, f ( A , a ; R n ) (whenever these exist),single out their characteristic properties, and analyze the supports of the λ i A , i ∈ I . Theorem 7.3.
Let (5.11) hold, and let each f j , j ∈ I + , be lower bounded on A j . Fix λ A ∈E ξ α, f ( A , a ; R n ) ( which exists according to footnote . Then the following two assertionsare equivalent: (i) λ A ∈ S ξ α, f ( A , a ; R n ) . Under the hypotheses of any of Theorems 7.2–7.6, (5.9) holds in consequence of Lemma 5.3.
ONDENSERS WITH TOUCHING PLATES 22 (ii)
There exists a vector ( c j ) j ∈ I + ∈ R p − such that for all j ∈ I + W λ A ,jα, f > c j ( ξ j − λ j A ) -a.e. , (7.3) W λ A ,jα, f c j λ j A -a.e. , (7.4) and in addition we have (7.5) W λ A ,pα, f = 0 n.e. on A p . If moreover Case II holds, then relation (7.5) actually holds for every W λ A ,iα, f , i ∈ I ,and it takes now the form (7.6) W λ A ,iα, f = 0 on A p \ I α,A p , i ∈ I, where I α,A p denotes the set of all α -irregular ( boundary ) points of A p . Remark 7.4.
The lower boundedness of f j , j ∈ I + , assumed in Theorem 7.3, holdsautomatically provided that Case I takes place. Furthermore, in Case I relation (7.4) isequivalent to the following apparently stronger assertion: W λ A ,jα, f c j on S λ j A D . Let ˘ Q denote the c α -reduced kernel of Q ⊂ R n [19, p. 164], which is the set of all x ∈ Q such that for any r > c α (cid:0) B ( x, r ) ∩ Q (cid:1) > α = 2 the domain D is simply connected. Theorem 7.5.
If a solution λ ξ A = ( λ i A ) i ∈ I ∈ S ξ α, f ( A , a ; R n ) exists, then (7.7) S λ p A R n = (cid:26) ˘ A p if α < ,∂D if α = 2 . Assume now that I + = { } , a = , f = and that there is a solution λ ξ A = ( λ A , λ A ) ∈ S ξ α, f ( A , ; R n ). Then, equivalently, λ := R A λ ξ A = λ A − λ A is a solution to the minimum α -Riesz energy problem(7.8) inf κ α ( µ, µ ) , where µ ranges over all ( signed scalar Radon) measures with µ + ∈ E ξ α ( A , R n ) and µ − ∈ E + α ( A , R n ). Since f = , we also see from (3.7) and (3.12) that(7.9) κ α ( · , λ ) = s i κ λ ξ A ,iα ( · ) = s i W λ ξ A ,iα, f ( · ) n.e. on R n , i = 1 , . Theorem 7.6.
With these assumptions and notations, we have (7.10) κ α ( · , λ ) = (cid:26) g αD ( · , λ + ) n.e. on D, on D c \ I α,D c . ONDENSERS WITH TOUCHING PLATES 23
Furthermore, assertion (ii) of Theorem holds, and (7.3) and (7.4) now take the form κ α ( · , λ ) = c ( ξ − λ + ) -a.e. , (7.11) κ α ( · , λ ) c on R n , (7.12) respectively, where < c < ∞ . In addition, (7.11) and (7.12) together with κ α ( · , λ ) = 0 n.e. on D c ( cf. (7.10)) determine uniquely the solution to the problem (7.8) among theadmissible measures µ . If moreover κ α ( · , ξ ) is ( finitely ) continuous on D , then also (7.13) κ α ( · , λ ) = c on S ξ − λ + D , (7.14) c g αD (cid:0) S ξ − λ + D (cid:1) < ∞ . Omitting now the requirement of the continuity of κ α ( · , ξ ) , assume further that α < and m n ( D c ) > where m n is the n -dimensional Lebesgue measure. Then (7.15) S λ + D = S ξ D , (7.16) κ α ( · , λ ) < c on D \ S ξ D (cid:0) = D \ S λ + D (cid:1) . An extension of the theory.
Parallel with Problem 3.11 for a constraint ξ givenby (5.8) and acting only on measures concentrated on the positive plates A j , j ∈ I + ,of the generalized condenser A = ( A i ) i ∈ I , consider also Problem 3.11 for σ = ( σ i ) i ∈ I ∈ M + ( A ; R n ) (in place of ξ ) defined as follows:(7.17) σ j = ξ j for all j ∈ I + , σ p > (cid:16) X j ∈ I + σ j (cid:17) ′ h = (cid:16) X j ∈ I + ξ j (cid:17) ′ i . Since in consequence of (4.5), (5.7) and (7.17) we have σ p ( A p ) > a p , the measure σ thusdefined can be thought of as an element of C ( A ; R n ). In contrast to ξ , the constraint σ isacting on all the components of µ ∈ E + α ( A , a ; R n ). Also note that σ p ( A p ) and κ α ( σ p , σ p )may both be infinite. Theorem 7.7.
The following identity holds: (7.18) G σ α, f ( A , a ; R n ) = G ξ α, f ( A , a ; R n ) . If moreover these ( equal ) extremal values are finite, then Problem for A , a , f and ξ is solvable if and only if so is that for A , a , f and σ , and in the affirmative case (7.19) S σ α, f ( A , a ; R n ) = S ξ α, f ( A , a ; R n ) . Proof.
Indeed, G σ α, f ( A , a ; R n ) > G ξ α, f ( A , a ; R n ) follows from the relation(7.20) E σ α, f ( A , a ; R n ) ⊂ E ξ α, f ( A , a ; R n ) . ONDENSERS WITH TOUCHING PLATES 24
To establish the converse inequality, assume that G ξ α, f ( A , a ; R n ) < ∞ and fix a minimizer ν ∈ E ξ α, f ( A , a ; R n ). Define ˜ ν = (˜ ν i ) i ∈ I ∈ M + ( A ; R n ) by the equalities(7.21) ˜ ν + = ν + and ˜ ν p = ( R A ν + ) ′ . Clearly ˜ ν ∈ E + α ( A ; R n ), and moreover ˜ ν ∈ E + α ( A , a ; R n ) which follows from (4.5), (5.7) and(7.21) since A p is not α -thin at infinity. By the linearity of balayage and (7.17) we actuallyhave ˜ ν ∈ E σ α ( A , a ; R n ), and finally ˜ ν ∈ E σ α, f ( A , a ; R n ) by (5.2). In consequence of (4.4),(5.2) and (7.21) we therefore obtain G α, f ( ν ) = κ α ( ν , ν ) + 2 h f + , ν + i = k R A ν + − ν p k α + 2 h f + , ν + i > k R A ν + − ( R A ν + ) ′ k α + 2 h f + , ν + i = G α, f (˜ ν ) > G σ α, f ( A , a ; R n ) , which establishes (7.18) in view of the arbitrary choice of ν ∈ E ξ α, f ( A , a ; R n ).Assuming now that (5.9) holds, we proceed to prove (7.19). The inclusion S σ α, f ( A , a ; R n ) ⊂ S ξ α, f ( A , a ; R n ) is obvious because of (7.18) and (7.20). To establish the converse inclusion,fix λ = ( λ i ) i ∈ I ∈ S ξ α, f ( A , a ; R n ). Then by (6.3) we have λ p = ( R A λ + ) ′ , and in the samemanner as in the preceding paragraph we obtain λ ∈ E σ α, f ( A , a ; R n ). Hence λ also belongsto S σ α, f ( A , a ; R n ), for G α, f ( λ ) = G ξ α, f ( A , a ; R n ) = G σ α, f ( A , a ; R n ) by (7.18). (cid:3) Thus, the theory of minimum α -Riesz energy problems with a constraint ξ given by (5.8) and acting only on measures concentrated on the A j , j ∈ I + , developed in Section ,remains valid in its full generality for the constraint σ , defined by (7.17) and acting on allthe components of µ ∈ E + α ( A , a ; R n ).8. Proofs of Theorems 7.1 and 7.2
Observe that, if Case II takes place, then(8.1) ζ ∈ E g ( D ) , (8.2) f j = κ α ( · , ζ − ζ ′ ) = g ( · , ζ ) c g -n.e. on D for all j ∈ I + . Indeed, (8.1) is obvious by (4.7), and (8.2) holds by Lemma 4.5 and footnote 9. By (8.1)and (8.2), in Case II for every ν ∈ E + g ( A + ; D ) we get G g, f + ( ν ) = k R A + ν k g + 2 X j ∈ I + g ( ζ, ν j )= k R A + ν k g + 2 g ( ζ, R A + ν ) = k R A + ν + ζ k g − k ζ k g . (8.3) ONDENSERS WITH TOUCHING PLATES 25
Proof of Theorem 7.1.
By Theorem 6.1, Theorem 7.1 will be proved once we haveestablished the following assertion.
Theorem 8.1.
Under the assumptions of Theorem , Problem for D , g , A + , a + , f + and ξ + is solvable, i.e. there is µ ∈ E ξ + g, f + ( A + , a + ; D ) with G g, f + ( µ ) = G ξ + g, f + ( A + , a + ; D ) . Proof.
Note that Problem 3.11 for D , g , A + , a + , f + and ξ + makes sense since by assump-tion (5.9) and identity (6.2) we have(8.4) G ξ + g, f + ( A + , a + ; D ) < ∞ . Actually, G ξ + g, f + ( A + , a + ; D ) is finite, which is clear from (6.2) and footnote 13. In view of(8.4), there is a sequence { µ k } k ∈ N ⊂ E ξ + g, f + ( A + , a + ; D ) such that(8.5) lim k →∞ G g, f + ( µ k ) = G ξ + g, f + ( A + , a + ; D ) . Since the α -Green kernel g satisfies the energy principle [16, Theorem 4.9], E g ( D ) forms apre-Hilbert space with the inner product g ( ν, ν ) and the energy norm k ν k g = p g ( ν, ν ).Furthermore, E ξ + g, f + ( A + , a + ; D ) is a convex cone and R A + is an isometric mapping betweenthe semimetric space E + g ( A + ; D ) and its R A + -image into E g ( D ) (see Theorem 3.5). We aretherefore able to apply to the set { µ k : k ∈ N } arguments similar to those in the proof ofLemma 3.12, and we get0 k R A + µ k − R A + µ ℓ k g − G ξ + g, f + ( A + , a + ; D ) + 2 G g, f + ( µ k ) + 2 G g, f + ( µ ℓ ) . Letting here k, ℓ → ∞ and combining the relation thus obtained with (8.5), we see in viewof the finiteness of G ξ + g, f + ( A + , a + ; D ) that { R A + µ k } k ∈ N forms a strong Cauchy sequence inthe metric space E + g ( D ). In particular, this implies(8.6) sup k ∈ N k R A + µ k k g < ∞ . Since the sets A j , j ∈ I + , are (relatively) closed in D , the cones M ξ j ( A j ; D ), j ∈ I + ,are vaguely closed in M ( D ), and therefore M ξ + ( A + ; D ) is vaguely closed in M ( D ) p − (cf. Definition 3.6). Furthermore, M ξ + ( A + , a + ; D ) is vaguely bounded, hence vaguelyrelatively compact by Lemma 3.7. Thus, there is a vague cluster point µ of the sequence { µ k } k ∈ N chosen above, which belongs to M ξ + ( A + ; D ). Passing to a subsequence andchanging notations, we assume that(8.7) µ k → µ vaguely as k → ∞ . We assert that the µ is a solution to Problem 3.11 for D , g , A + , a + , f + and ξ + . ONDENSERS WITH TOUCHING PLATES 26
Fix j ∈ I + . Applying Lemma 2.1 to 1 D ∈ Ψ( D ), we obtain from (8.7) µ j ( D ) lim k →∞ µ jk ( D ) = a j . We proceed by showing that the inequality here is in fact an equality, and hence(8.8) µ ∈ M ξ + ( A + , a + ; D ) . Consider an exhaustion of A j by an increasing sequence of compact sets K ℓ ⊂ A j , ℓ → ∞ .Since each − K ℓ ∈ Ψ( D ), we get from Lemma 2.1 a j > h D , µ j i = lim ℓ →∞ h K ℓ , µ j i > lim ℓ →∞ lim sup k →∞ h K ℓ , µ jk i = a j − lim ℓ →∞ lim inf k →∞ h A j \ K ℓ , µ jk i . Thus (8.8) will follow once we show that(8.9) lim ℓ →∞ lim inf k →∞ h A j \ K ℓ , µ jk i = 0 . By (7.1), ∞ > ξ j ( D ) = lim ℓ →∞ h K ℓ , ξ j i and therefore lim ℓ →∞ h A j \ K ℓ , ξ j i = 0 . Combined with h A j \ K ℓ , µ jk i h A j \ K ℓ , ξ j i for all k ∈ N , this implies (8.9), and hence (8.8).Furthermore, as R A + µ k → R A + µ vaguely in M + ( D ), [2, Chapitre III, Section 5, Exer-cice 5] implies that R A + µ k ⊗ R A + µ k → R A + µ ⊗ R A + µ vaguely in M + ( D × D ). Lemma 2.1with X = D × D and ψ = g therefore yields g ( R A + µ , R A + µ ) lim inf k →∞ k R A + µ k k g < ∞ , the latter holds by (8.6). Together with (3.9) and (8.8) this gives µ ∈ E ξ + g ( A + , a + ; D ).Since G g, f + ( µ ) > −∞ , µ ∈ S ξ + g, f + ( A + , a + ; D ) will be established once we have shown that(8.10) G g, f + ( µ ) lim k →∞ G g, f + ( µ k ) . Since the kernel g is perfect [16, Theorem 4.11], the sequence { R A + µ k } k ∈ N , being strongCauchy in E + g ( D ) and vaguely convergent to R A + µ , converges to the same limit stronglyin E + g ( D ), i.e. lim k →∞ k R A + µ k − R A + µ k g = 0 , which in view of (3.10) is equivalent to the relation(8.11) lim k →∞ k µ k − µ k E + g ( A + ; D ) = 0 . ONDENSERS WITH TOUCHING PLATES 27
Also note that the mapping ν G g, f + ( ν ) is vaguely l.s.c., resp. strongly continuous, on E + g, f + ( A + ; D ) if Case I, resp. Case II, takes place. In fact, since g ( ν , ν ) is vaguely l.s.c. on E + g ( A + ; D ), the former assertion follows from Lemma 2.1. As for the latter assertion, it isobvious by (8.3). In view of this observation, (8.7) and (8.11) result in (8.10). (cid:3) Corollary 8.2.
Suppose that the assumptions of Theorem are fulfilled. Then the ( nonempty ) class S ξ + g, f + ( A + , a + ; D ) of all solutions to Problem for D , g , A + , a + , f + and ξ + is vaguely compact in M ( D ) p − .Proof. According to Lemma 3.12, any solutions µ k ∈ S ξ + g, f + ( A + , a + ; D ), k ∈ N , forma strong Cauchy sequence in E + g ( A + ; D ). Furthermore, the set { µ k : k ∈ N } is vaguelyclosed and vaguely relatively compact in M ( D ) p − (see Section 3.3 with X = D ). Thereforein the same manner as in the proof of Theorem 8.1 one can see that any vague cluster pointof { µ k } k ∈ N belongs to S ξ + g, f + ( A + , a + ; D ). (cid:3) Proof of Theorem 7.2.
Assume that the requirements of the latter part of thetheorem are fulfilled. By Theorem 6.1, the former equality in (7.2) holds. Furthermore,since Case II with ζ > G g,f | D ( ν ) = k ν k g + 2 g ( ζ, ν ) ∈ [0 , ∞ ) for all ν ∈ E + g ( A ; D ) . Consider numbers r ℓ > ℓ ∈ N , such that r ℓ ↑ ∞ as ℓ → ∞ , and write B r ℓ := B (0 , r ℓ ), A ,r ℓ := A ∩ B r ℓ . Since c α ( A ) = ∞ by assumption and since c α ( B r ℓ ) < ∞ for every ℓ ∈ N ,we infer from the subadditivity of c α ( · ) on universally measurable sets [13, Lemma 2.3.5]that c α ( A \ B r ℓ ) = ∞ . Hence for every ℓ ∈ N there is ξ ℓ ∈ E + α ( A \ B r ℓ , a ; R n ) of compactsupport S ξ ℓ D such that(8.13) k ξ ℓ k α ℓ − . Clearly, the r ℓ can be chosen successively so that A ,r ℓ ∪ S ξ ℓ D ⊂ A ,r ℓ +1 . Any compactset K ⊂ R n is contained in a ball B r ℓ with ℓ large enough, and hence K has points incommon with only finitely many S ξ ℓ D . Therefore, ξ defined by ξ ( ϕ ) := X ℓ ∈ N ξ ℓ ( ϕ ) for any ϕ ∈ C ( R n )is a positive Radon measure on R n carried by A . Furthermore, ξ ( A ) = ∞ and ξ ∈E + α ( R n ). To prove the latter, note that η k := ξ + · · · + ξ k ∈ E + α ( R n ) in view of (8.13) andthe triangle inequality in E α ( R n ). Also observe that η k → ξ vaguely because for any ϕ ∈ C ( R n ) there is k such that ξ ( ϕ ) = η k ( ϕ ) for all k > k . As k η k k α L := P ℓ ∈ N ℓ − < ∞ for all k ∈ N , Lemma 2.1 with X = A × A and ψ = κ α | A × A yields k ξ k α L .Each ξ ℓ belongs to E + g ( A , a ; D ) and moreover, by (4.10) and (8.13),(8.14) k ξ ℓ k g k ξ ℓ k α ℓ − . ONDENSERS WITH TOUCHING PLATES 28
As Case II with ζ > ξ ℓ ∈ E ξ g,f | D ( A , a ; D ) for all ℓ ∈ N by (8.12). Therefore,by the Cauchy–Schwarz (Bunyakovski) inequality in E g ( D ),0 G ξ g,f | D ( A , a ; D ) lim ℓ →∞ (cid:2) k ξ ℓ k g + 2 g ( ζ, ξ ℓ ) (cid:3) k ζ k g lim ℓ →∞ k ξ ℓ k g = 0 , where the first and the second inequalities hold by (8.12), and the third inequality and theequality are valid by (8.14). Hence G ξ g,f | D ( A , a ; D ) = 0, and the theorem follows.9. Proofs of Theorems 7.3, 7.5 and 7.6
Throughout this section we maintain all the requirements on A , a , f , and ξ imposed at thebeginning of Section 5.2, except for (5.9) which follows automatically from the hypothesesof the assertions under proving in view of Lemma 5.3.9.1. Proof of Theorem 7.3.
Fix λ A ∈ E + α, f ( A , a ; R n ). Then each λ i A , i ∈ I , has finite α -Riesz energy, and hence it is c α -absolutely continuous. Note that, since f p = 0 n.e. on A p , (7.5) can alternatively be written as κ λ A ,pα = 0 n.e. on A p , which by (3.7) (with X = R n and κ = κ α ) is equivalent to the relation κ α ( · , R A λ + A − λ p ) = 0 n.e. on A p . In view of the characteristic property (4.2) of the swept measures, this shows that for thegiven λ A , (7.5) and (6.3) are equivalent. On account of Theorem 6.1, we thus see that when proving the equivalence of assertions (i) and (ii) of Theorem , there is no loss ofgenerality in assuming λ A to satisfy (6.3).Substituting (6.3) into (3.7), we therefore get for every i ∈ I (9.1) κ λ A ,iα ( · ) = s i κ α (cid:0) · , R A λ + A − ( R A λ + A ) ′ (cid:1) n.e. on R n . In particular, for every j ∈ I + we have(9.2) κ λ A ,jα ( · ) = g ( · , R A + λ + A ) = g λ + A ,j ( · ) n.e. on D and hence, by (3.12), W λ A ,jα, f = W λ + A ,jg, f + n.e. on D. (Note that (9.2) has been obtained from (9.1) with the aid of (4.6), applied to R A + λ + A inplace of µ , and (3.7), the latter now with X = D and κ = g .)If Case II holds, then for every i ∈ I we also get from (9.1) and (3.12) W λ A ,iα, f ( · ) = s i κ α (cid:0) · , ( R A λ + A + ζ ) − ( R A λ + A + ζ ) ′ (cid:1) n.e. on R n . By [16, Corollary 3.14], the function on the right (hence, also that on the left) in thisrelation takes the value 0 at every α -regular point of A p , which gives (7.6).By Theorem 6.1, what has been shown just above yields that Theorem 7.3 will be provedonce the following theorem has been established. ONDENSERS WITH TOUCHING PLATES 29
Theorem 9.1.
Under the hypotheses of Theorem the following two assertions areequivalent for any λ ∈ E ξ + g, f + ( A + , a + ; D ):(i ′ ) λ ∈ S ξ + g, f + ( A + , a + ; D ) . (ii ′ ) There is a vector ( c j ) j ∈ I + ∈ R p − such that for all j ∈ I + W λ ,jg, f + > c j ( ξ j − λ j ) -a.e. , (9.3) W λ ,jg, f + c j λ j -a.e. (9.4) Proof.
Suppose first that (i ′ ) holds. To verify (ii ′ ), fix j ∈ I + . For every µ = ( µ ℓ ) ℓ ∈ I + ∈E ξ + g, f + ( A + , a + ; D ) write µ j := ( µ ℓj ) ℓ ∈ I + where µ ℓj := µ ℓ for all ℓ = j and µ jj = 0; then µ j ∈ E + g, f + ( A + ). Also define ˜ f j := f j | D + g λ j ,j . By substituting (3.6) with κ = g we thenobtain(9.5) ˜ f j = f j | D + X ℓ ∈ I + , ℓ = j g ( · , λ ℓ ) . Since g ( · , λ ℓ ) > D for all ℓ ∈ I + according to [16, Lemma 4.1] and since f j is lowerbounded on A j by assumption, the function(9.6) W λ j g, ˜ f j := g ( · , λ j ) + ˜ f j , j ∈ I + , is likewise lower bounded on A j . Furthermore, both ˜ f j and W λ j g, ˜ f j are finite n.e. on the set A ◦ j , which is clear from (5.10) and Lemma 3.3.Applying (3.8) and (3.13) we get for any µ ∈ E ξ + g, f + ( A + , a + ; D ) with the additional propertythat µ j = λ j (in particular for µ = λ ) G g, f + ( µ ) = G g, f + ( λ j ) + G g, ˜ f j ( µ j ) . Combined with G g, f + ( µ ) > G g, f + ( λ ), this yields G g, ˜ f j ( µ j ) > G g, ˜ f j ( λ j ). Hence, λ j mini-mizes G g, ˜ f j ( ν ) where ν ranges over E ξ j g, ˜ f j ( A j , a j ; D ). This enables us to show that there is c j ∈ R such that W λ j g, ˜ f j > c j ( ξ j − λ j )-a.e. , (9.7) W λ j g, ˜ f j c j λ j -a.e.(9.8)In doing this we shall use permanently the fact that both ξ j and λ j have finite α -Rieszenergy are hence they are c α -absolutely continuous.Indeed, (9.7) holds with c j := L j := sup (cid:8) t ∈ R : W λ j g, ˜ f j > t ( ξ j − λ j )-a.e. (cid:9) . ONDENSERS WITH TOUCHING PLATES 30
In turn, (9.7) with c j = L j implies L j < ∞ , for W λ j g, ˜ f j < ∞ n.e. on A ◦ j , and hence ( ξ j − λ j )-a.e. on A j by (5.11). Also L j > −∞ , because W λ j g, ˜ f j is lower bounded on A j (see above).We next establish (9.8) with c j = L j . To this end, write for any w ∈ R A + j ( w ) := (cid:8) x ∈ A j : W λ j g, ˜ f j ( x ) > w (cid:9) , A − j ( w ) := (cid:8) x ∈ A j : W λ j g, ˜ f j ( x ) < w (cid:9) . On the contrary, let (9.8) with c j = L j do not hold, i.e. λ j (cid:0) A + j ( L j ) (cid:1) >
0. Since W λ j g, ˜ f j is λ j -measurable, one can choose w j ∈ ( L j , ∞ ) so that λ j (cid:0) A + j ( w j ) (cid:1) >
0. At the same time, as w j > L j , (9.7) with c j = L j yields ( ξ j − λ j ) (cid:0) A − j ( w j ) (cid:1) >
0. Therefore, there exist compactsets K ⊂ A + j ( w j ) and K ⊂ A − j ( w j ) such that(9.9) 0 < λ j ( K ) < ( ξ j − λ j )( K ) . Write τ j := ( ξ j − λ j ) | K ; then g ( τ j , τ j ) < κ α ( τ j , τ j ) < ∞ , where the former inequalityholds by (4.10). As (cid:10) W λ j g, ˜ f j , τ j (cid:11) w j τ j ( K ) < ∞ , we therefore get h ˜ f j , τ j i < ∞ . Define θ j := λ j − λ j | K + b j τ j , where b j := λ j ( K ) /τ j ( K ) ∈ (0 ,
1) by (9.9). Straightforwardverification then shows that θ j ( A j ) = a i and θ j ξ j , and hence θ j ∈ E ξ j g, ˜ f j ( A j , a j ; D ). Onthe other hand, (cid:10) W λ j g, ˜ f j , θ j − λ j (cid:11) = (cid:10) W λ j g, ˜ f j − w j , θ j − λ j (cid:11) = − (cid:10) W λ j g, ˜ f j − w j , λ j | K (cid:11) + b j (cid:10) W λ j g, ˜ f j − w j , τ j (cid:11) < , which is impossible by Lemma 3.8 with the (convex) set E = E ξ j g, ˜ f j ( A j , a j ; D ). This contra-diction establishes (9.8).Substituting (9.5) into (9.6) and then comparing the result obtained with (3.6) and (3.12),we see that(9.10) W λ j g, ˜ f j = W λ ,jg, f + . Combined with (9.7) and (9.8), this establishes (9.3) and (9.4), thus completing the proofthat (i ′ ) implies (ii ′ ).Conversely, let (ii ′ ) hold. On account of (9.10), for every j ∈ I + relations (9.7) and (9.8)are then fulfilled with ˜ f j defined by (9.5). This yields λ j (cid:0) A + j ( c j ) (cid:1) = 0 and (cid:0) ξ j − λ j (cid:1)(cid:0) A − j ( c j ) (cid:1) = 0 . For any ν ∈ E ξ + g, f + ( A + , a + ; D ) we therefore get (cid:10) W λ ,jg, f + , ν j − λ j (cid:11) = (cid:10) W λ j g, ˜ f j − c j , ν j − λ j (cid:11) = (cid:10) W λ j g, ˜ f j − c j , ν j | A + j ( c j ) (cid:11) + (cid:10) W λ j g, ˜ f j − c j , ( ν j − ξ j ) | A − j ( c j ) (cid:11) > . ONDENSERS WITH TOUCHING PLATES 31
Summing up these inequalities over all j ∈ I + , we conclude from Lemma 3.8 with the(convex) set E = E ξ + g, f + ( A + , a + ; D ) that λ satisfies (i ′ ). (cid:3) Proof of Theorem 7.5.
For any x ∈ D consider the inverse K x of Cℓ R n A p relative to S ( x, R n being the one-point compactification of R n . Since K x is compact, there existsthe (unique) κ α -equilibrium measure γ x ∈ E + α ( K x ; R n ) on K x possessing the properties k γ x k α = γ x ( K x ) = c α ( K x ),(9.11) κ α ( · , γ x ) = 1 n.e. on K x , and κ α ( · , γ x ) R n . Note that γ x = 0, for c α ( K x ) > c α ( A p ) > ◦ S γ x R n = (cid:26) ˘ K x if α < ,∂ R n K x if α = 2 . The latter equality in (9.12) follows from [19, Chapter II, Section 3, n ◦ we first note that S γ x R n ⊂ ˘ K x by the c α -absolute continuity of γ x .As for the converse inclusion, assume on the contrary that there is x ∈ ˘ K x such that x / ∈ S γ x R n . Choose r > B ( x , r ) ∩ S γ x R n = ∅ . But c α (cid:0) B ( x , r ) ∩ ˘ K x (cid:1) > y ∈ B ( x , r ) such that κ α ( y, γ x ) = 1. The function κ α ( · , γ x ) is α -harmonic on B ( x , r ) [19, Chapter I, Section 5, n ◦ B ( x , r ), and takesat y ∈ B ( x , r ) its maximum value 1. Applying [19, Theorem 1.28] we obtain κ α ( · , γ x ) = 1 m n -a.e. on R n , hence everywhere on ˘ K cx by the continuity of κ α ( · , γ x ) on (cid:0) S γ x R n (cid:1) c (cid:2) ⊃ ( ˘ K x ) c (cid:3) ,and altogether n.e. on R n by (9.11). This means that γ x serves as the α -Riesz equilibriummeasure on the whole of R n , which is impossible.Based on (6.3) and on the integral representation (4.3), we then arrive at the claimedrelation (7.7) in view of the fact that, for every x ∈ D , ε ′ x is the Kelvin transform of theequilibrium measure γ x [16, Section 3.3].9.3. Proof of Theorem 7.6.
Combining (7.6), (7.9) and (9.2) gives (7.10). Substitutingthe first relation from (7.10) into (7.4) shows that under the stated assumptions the number c from Theorem 7.3 is >
0, while (7.3) now takes the (equivalent) form(9.13) κ α ( · , λ ) > c > ξ − λ + )-a.e.Having rewritten (7.4) as κ α ( · , λ + ) κ α ( · , λ − ) + c λ + -a.e. , we infer from [19, Theorems 1.27, 1.29, 1.30] that the same inequality holds on all of R n ,which amounts to (7.12). In turn, (7.12) yields (7.11) when combined with (9.13). It followsdirectly from Theorem 7.3 that (7.11) and (7.12) together with the relation κ α ( · , λ ) = 0 We have brought here this proof, since we did not find a reference for this possibly known assertion.
ONDENSERS WITH TOUCHING PLATES 32 n.e. on D c determine uniquely the solution λ to the problem (7.8) among the admissiblemeasures.Assume now that κ α ( · , ξ ) is continuous on D . Then so is κ α ( · , λ + ). Indeed, since κ α ( · , λ + )is l.s.c. and since κ α ( · , λ + ) = κ α ( · , ξ ) − κ α ( · , ξ − λ + ) with κ α ( · , ξ ) continuous and κ α ( · , ξ − λ + ) l.s.c., it follows that κ α ( · , λ + ) is also upper semicontinuous, hence continuous.Therefore, by the continuity of κ α ( · , λ + ) on D , (7.11) implies (7.13). Thus, by (7.10) and(7.13), g ( · , λ + ) = c on S ξ − λ + D , which implies (7.14) in view of [13, Lemma 3.2.2] (with X = D and κ = g ).Omitting now the requirement of continuity of κ α ( · , ξ ), assume further that α < m n ( D c ) >
0. If on the contrary (7.15) is not fulfilled, then there is x ∈ S ξ D such that x / ∈ S λ + D . Hence one can choose r > B ( x , r ) ⊂ D and B ( x , r ) ∩ S λ + D = ∅ . Then ( ξ − λ + ) (cid:0) B ( x , r ) (cid:1) >
0, and hence there exists y ∈ B ( x , r ) with the property κ α ( y, λ ) = c (cf. (7.11)), or equivalently(9.15) κ α ( y, λ + ) = κ α ( y, λ − ) + c . Since κ α ( · , λ + ) is α -harmonic on B ( x , r ) and continuous on B ( x , r ) and since κ α ( · , λ − ) + c is α -superharmonic on R n , we conclude from (7.12) and (9.15) with the aid of [19,Theorem 1.28] that(9.16) κ α ( · , λ ) = κ α (cid:0) · , λ − ) + c m n -a.e. on R n . This implies c = 0, because by (4.2) and (6.3) κ α ( · , λ + ) = κ α (cid:0) · , ( λ + ) ′ (cid:1) = κ α ( · , λ − ) n.e. on D c , hence m n -a.e. on D c . A contradiction.Similar arguments enable us to establish (7.16). Indeed, if (7.16) were not fulfilled at some x ∈ D \ S λ + D , then (9.15) would hold with x in place of y (cf. (7.12)) and, furthermore, onecould choose r > x in place of x . Therefore, since κ α ( · , λ + ) is α -harmonic on B ( x , r ) and continuous on B ( x , r ) and since κ α ( · , λ − ) + c is α -superharmonic on R n , we would arrive again at (9.16) and hence at c = 0, which isimpossible. 10. Examples
The purpose of the examples below is to illustrate the assertions from Section 7.1. Notethat in either Example 10.1 or Example 10.2 the set A = D c is not α -thin at infinity. ONDENSERS WITH TOUCHING PLATES 33
Example 10.1.
Let n > α < D = B r := B (0 , r ), where r ∈ (0 , ∞ ), and let I + = { } , A = D , a = , f = . Define ξ := qλ r , where q ∈ (1 , ∞ ) and λ r is the κ α -capacitarymeasure on B r := B (0 , r ) (Remark 2.6). As follows from [19, Chapter II, Section 3,n ◦ ξ ∈ E + α ( A , q ; R n ), S ξ D = D and κ α ( · , ξ ) is continuous on R n . Since f = ,Problem 3.11 reduces to the problem (7.8) of minimizing κ α ( µ, µ ) over all µ ∈ E α ( R n ) suchthat µ + ∈ E ξ α ( A , R n ) and µ − ∈ E + α ( A , R n ), which by Theorem 6.1 is equivalent tothe problem of minimizing g αD ( ν, ν ) over E ξ g αD ( A , D ). According to Theorems 7.1, 6.1and Corollary 3.13, these two constrained minimum energy problems are uniquely solvable(no short-circuit occurs between D and D c ), and their solutions, denoted respectively by λ α, A = λ + − λ − and λ g,A , are related to each other as follows: λ α, A = λ g,A − λ ′ g,A . Furthermore, by (7.7), (7.14) and (7.15), S λ + D = S λ g,A D = S ξ D = D, S λ − R n = D c , (10.1) c g αD (cid:0) S ξ − λ + D (cid:1) < ∞ , while by (7.10), (7.12) and (7.13),(10.2) κ α ( · , λ α, A ) = (cid:26) c on S ξ − λ + D , D c , (10.3) κ α ( · , λ α, A ) c on D \ S ξ − λ + D , where c >
0. (In (10.2) we have used the fact that for the given α and D , I α,D c = ∅ .)Moreover, according to Theorem 7.3, relations (10.2) and (10.3) determine uniquely thesolution λ α, A among the admissible measures. Example 10.2.
Let n = 3, α = 2, f = , a = . Define D := { x = ( x , x , x ) ∈ R : x > } and A := P k ∈ N K k , where K k := (cid:8) ( x , x , x ) ∈ D : x = k − , x + x k (cid:9) , k ∈ N . Let λ k be the κ -capacitary measure on K k (Remark 2.6); hence λ k ( K k ) = 1 and k λ k k = π / (2 k ) by [19, Chapter II, Section 3, n ◦ ξ := X k ∈ N k − λ k . In the same manner as in the proof of Theorem 7.2 one can see that ξ is a positiveRadon measure carried by A with κ ( ξ , ξ ) < ∞ and ξ ( A ) ∈ (1 , ∞ ). By Theorem 7.1,Problem 3.11 for the constraint ξ = ( ξ , ∞ ) and the condenser A = ( A , D c ) has thereforea (unique) solution λ α, A = λ + − λ − (no short-circuit occurs between A and D c ), although D c ∩ Cℓ R A = ∂D and hence c (cid:0) D c ∩ Cℓ R A (cid:1) = ∞ . ONDENSERS WITH TOUCHING PLATES 34
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