Constrained energy problems with external fields for infinite dimensional vector measures
aa r X i v : . [ m a t h . C A ] O c t Constrained energy problems with external fieldsfor infinite dimensional vector measures
Natalia Zorii
Abstract.
We consider a constrained minimal energy problem with an ex-ternal field over noncompact classes of infinite dimensional vector measures( µ i ) i ∈ I on a locally compact space. The components µ i are positive measureswith the properties R g i dµ i = a i and σ i − µ i > a i , g i , and σ i aregiven) and supported by closed sets A i with the sign +1 or − A i ∩ A j = ∅ whenever sign A i = sign A j , and the law of interactionof µ i , i ∈ I , is determined by the matrix (cid:0) sign A i sign A j (cid:1) i,j ∈ I . For all posi-tive definite kernels satisfying Fuglede’s condition of consistency between thevague (= weak ∗ ) and strong topologies, sufficient conditions for the existenceof minimizers are established and their uniqueness and vague compactness arestudied. Examples illustrating the sharpness of the sufficient conditions areprovided. We also analyze continuity properties of minimizers in the vagueand strong topologies when A i and σ i are varied simultaneously. The resultsare new even for classical kernels in R n , which is important in applications.
1. Introduction
In all that follows, X denotes a locally compact Hausdorff space and M = M (X)the linear space of all real-valued scalar Radon measures ν on X equipped withthe vague (= weak ∗ ) topology, i.e., the topology of pointwise convergence on theclass C (X) of all real-valued continuous functions ϕ on X with compact support.A kernel κ on X is meant to be an element from Φ(X × X), where Φ(Y)consists of all lower semicontinuous functions ψ : Y → ( −∞ , ∞ ] such that ψ > ν, ν ∈ M , the mutual energy and the potential relativeto the kernel κ are defined by κ ( ν, ν ) := Z κ ( x, y ) d ( ν ⊗ ν )( x, y ) and κ ( · , ν ) := Z κ ( · , y ) dν ( y ) , respectively. (When introducing notation, we always tacitly assume the corre-sponding object on the right to be well defined — as a finite number or ±∞ .) N. ZoriiFor ν = ν the mutual energy κ ( ν, ν ) defines the energy κ ( ν, ν ) of ν . Wedenote by E = E κ (X) the set of all ν ∈ M with −∞ < κ ( ν, ν ) < ∞ .We shall mainly be concerned with a positive definite kernel κ , which meansthat it is symmetric (i.e., κ ( x, y ) = κ ( y, x ) for all x, y ∈ X) and the energy κ ( ν, ν ), ν ∈ M , is nonnegative whenever defined. Then E forms a pre-Hilbert space with thescalar product κ ( ν, ν ) and the seminorm k ν k E := k ν k κ := p κ ( ν, ν ) (see [12]); thetopology on E , determined by this seminorm, is called strong . A positive definitekernel κ is strictly positive definite if the seminorm k · k E is a norm.Given a closed set F ⊂ X, we denote by M + ( F ) the convex cone of allnonnegative ν ∈ M supported by F , and let E + ( F ) := M + ( F ) ∩ E . Also write M + := M + (X) and E + := E + (X).We consider a countable, locally finite collection A = ( A i ) i ∈ I of fixed closedsets A i ⊂ X with the sign +1 or − M + ( A ) stand for the Cartesian product Q i ∈ I M + ( A i );then an element µ of M + ( A ) is a (nonnegative) vector measure ( µ i ) i ∈ I with thecomponents µ i ∈ M + ( A i ). The topology of the product space Q i ∈ I M + ( A i ),where every M + ( A i ) is endowed with the vague topology, is likewise called vague .If a vector measure µ ∈ M + ( A ) and a vector-valued function u = ( u i ) i ∈ I with µ i -measurable components u i : A i → [ −∞ , ∞ ] are given, then for brevity wewrite h u , µ i := X i ∈ I Z u i dµ i . Let a kernel κ be fixed. In accordance with an electrostatic interpretationof a condenser, we assume that the interaction between the charges lying on theconductors A i , i ∈ I , is characterized by the matrix ( α i α j ) i,j ∈ I , where α i :=sign A i . Then the energy of µ ∈ M + ( A ) is defined by the formula κ ( µ , µ ) := X i,j ∈ I α i α j κ ( µ i , µ j ) . We denote by E + ( A ) the set of all µ ∈ M + ( A ) with −∞ < κ ( µ , µ ) < ∞ .Also fix a vector-valued function f = ( f i ) i ∈ I , treated as an external field , andassume it to satisfy one of the following two cases: Case I. f i ∈ Φ(X) for all i ∈ I ; Case
II. f i = α i κ ( · , ζ ) for all i ∈ I , where ζ ∈ E is a signed measure .Furthermore, suppose each f i to affect the charges on A i only; then the f - weightedenergy of µ ∈ M + ( A ) is given by the expression G f ( µ ) := κ ( µ , µ ) + 2 h f , µ i . (1.1)Let E + f ( A ) consist of all µ ∈ E + ( A ) with −∞ < G f ( µ ) < ∞ . Here and in the sequel, an expression P i ∈ I c i is meant to be well defined provided so is everysummand c i and the sum does not depend on the order of summation — though might be ±∞ .Then, by Riemann series theorem, the sum is finite if and only if the series converges absolutely. onstrained energy problems with external fields 3Also fix a vector measure σ ∈ M + ( A ), serving as a constraint , a numer-ical vector a = ( a i ) i ∈ I with a i > i ∈ I , and a vector-valued function g = ( g i ) i ∈ I , where all the g i : A i → (0 , ∞ ) are continuous. In the study, we areinterested in the problem of minimizing G f ( µ ) over the class of all µ ∈ E + f ( A )with the properties that h g i , µ i i = a i and σ i − µ i > i ∈ I .Along with its electrostatic interpretation, such a problem has found variousimportant applications in approximation theory (see, e.g., [8, 9, 21]).The main question is whether minimizers λ σ A in the constrained minimal f -weighted energy problem exist. If A is finite, all the A i are compact, κ ( x, y )is continuous on A ℓ × A j whenever α ℓ = α j , and Case I takes place, then theexistence of those λ σ A can easily be established by exploiting the vague topologyonly, since then the class of admissible vector measures is vaguely compact, while G f ( µ ) is vaguely lower semicontinuous (cf. [14, 19, 20, 22]).However, these arguments break down if any of the above-mentioned fourassumptions is dropped, and then the problem on the existence of minimizersbecomes rather nontrivial. In particular, the class of admissible vector measuresis no longer vaguely compact if any of the A i is noncompact. Another difficulty isthat G f ( µ ) might not be vaguely lower semicontinuous when Case II holds.To solve the problem on the existence of minimizers λ σ A in the general case,we restrict ourselves to a positive definite kernel κ and develop an approach basedon the following crucial arguments.The set E + ( A ) is shown to be a semimetric space with the semimetric k µ − µ k E + ( A ) := h X i,j ∈ I α i α j κ ( µ i − µ i , µ j − µ j ) i / , (1.2)and one can define an inclusion R of E + ( A ) into the pre-Hilbert space E such that E + ( A ) becomes isometric to its R -image, the latter being regarded as a semimetricsubspace of E (see Theorem 3.11). Similar to the terminology in E , we thereforecall the topology of the semimetric space E + ( A ) strong .Another crucial fact is that, for rather general κ , g , and a , the topologicalsubspace of E + ( A ) consisting of all µ such that h g i , µ i i a i and σ i − µ i > i ∈ I turns out to be strongly complete (see Theorem 7.4).Using these arguments, we obtain sufficient conditions for the existence ofminimizers λ σ A and establish statements on their uniqueness and vague compact-ness (see Lemma 4.1 and Theorem 6.2). Examples illustrating the sharpness of thesufficient conditions are provided (see Sec. 12). We also analyze continuity prop-erties of λ σ A relative to the vague and strong topologies when both σ and A arevaried (see Theorems 6.7, 6.9 and Corollaries 6.8, 6.10).The results obtained hold true, e.g., for the Newtonian, Green or Riesz kernelsin R n , n >
2, as well as for the restriction of the logarithmic kernel in R to theopen unit disk, which is important in applications. N. Zorii
2. Preliminaries: topologies, consistent and perfect kernels
In all that follows, we suppose the kernel κ to be positive definite. In addition tothe strong topology on E , determined by the seminorm k ν k := k ν k E := k ν k κ := p κ ( ν, ν ), it is often useful to consider the weak topology on E , defined by meansof the seminorms ν
7→ | κ ( ν, µ ) | , µ ∈ E (see [12]). The Cauchy–Schwarz inequality | κ ( ν, µ ) | k ν k k µ k , where ν, µ ∈ E , implies immediately that the strong topology on E is finer than the weak one.In [12, 13], B. Fuglede introduced the following two equivalent properties ofconsistency between the induced strong, weak, and vague topologies on E + :(C ) Every strong Cauchy net in E + converges strongly to any of its vague clusterpoints; (C ) Every strongly bounded and vaguely convergent net in E + converges weaklyto the vague limit. Definition 2.1.
Following Fuglede [12], we call a kernel κ consistent if it satisfieseither of the properties (C ) and (C ), and perfect if, in addition, it is strictlypositive definite. Remark . One has to consider nets or filters in M + instead of sequences, sincethe vague topology in general does not satisfy the first axiom of countability. Wefollow Moore’s and Smith’s theory of convergence, based on the concept of nets(see [18]; cf. also [11, Chap. 0] and [16, Chap. 2]). However, if X is metrizable andcountable at infinity, then M + satisfies the first axiom of countability (see [12,Lemma 1.2.1]) and the use of nets may be avoided. Theorem 2.3 (
Fuglede [12]) . A kernel κ is perfect if and only if E + is stronglycomplete and the strong topology on E + is finer than the vague one.Remark . In R n , n >
3, the Newtonian kernel | x − y | − n is perfect [5]. So arethe Riesz kernels | x − y | α − n , 0 < α < n , in R n , n > − log | x − y | in R to the open unit disk [17]. Furthermore,if D is an open set in R n , n >
2, and its generalized Green function g D exists (see,e.g., [15, Th. 5.24]), then the kernel g D is perfect as well [10]. Remark . As is seen from the above definitions and Theorem 2.3, the conceptof consistent or perfect kernels is an efficient tool in minimal energy problemsover classes of nonnegative scalar
Radon measures with finite energy. Indeed, thetheory of capacities of sets has been developed in [12] for exactly those kernels. Weshall show below that this concept is efficient, as well, in minimal energy problemsover classes of vector measures of finite or infinite dimensions. This is guaranteedby a theorem on the strong completeness of proper subspaces of the semimetricspace E + ( A ), to be stated in Sec. 7.2.onstrained energy problems with external fields 5
3. Condensers. Vector measures and their energies
Throughout the article, let I + and I − be fixed countable, disjoint sets of indices,where the latter is allowed to be empty, and let I denote their union. Assume thatto every i ∈ I there corresponds a (unique) nonempty, closed set A i ⊂ X. Definition 3.1.
A collection A = ( A i ) i ∈ I is called an ( I + , I − )- condenser (or simplya condenser ) in X if every compact subset of X intersects with at most finitelymany A i and A i ∩ A j = ∅ for all i ∈ I + , j ∈ I − . (3.1)A condenser A is called compact if so are all A i , i ∈ I , and finite if I is finite.The sets A i , i ∈ I + , and A j , j ∈ I − , are called the positive and, respectively, the negative plates of A . (Note that any two equally signed plates can intersect eachother or even coincide.) In the sequel, also the following notation will be used: A + := [ i ∈ I + A i , A − := [ i ∈ I − A i . Observe that A + and A − might both be noncompact even for a compact A .Given a condenser A , let M + ( A ) consist of all nonnegative vector measures µ = ( µ i ) i ∈ I , where µ i ∈ M + ( A i ) for all i ∈ I ; that is, M + ( A ) := Q i ∈ I M + ( A i ).The product topology on M + ( A ), where every M + ( A i ) is equipped with the vaguetopology, is likewise called vague . Since the space M (X) is Hausdorff, so is M + ( A )(cf. [16, Chap. 3, Th. 5]).A set F ⊂ M + ( A ) is vaguely bounded if, for every ϕ ∈ C (X) and every i ∈ I ,sup µ ∈ F | µ i ( ϕ ) | < ∞ . Lemma 3.2. If F ⊂ M + ( A ) is vaguely bounded, then it is vaguely relatively com-pact.Proof. Since by [2, Chap. III, §
2, Prop. 9] any vaguely bounded part of M isvaguely relatively compact, the lemma follows from Tychonoff’s theorem on theproduct of compact spaces (see, e.g., [16, Chap. 5, Th. 13]). (cid:3) R : M + ( A ) → M . Relation of R -equivalency on M + ( A )Since each compact subset of X intersects with at most finitely many A i , for every ϕ ∈ C (X) only a finite number of µ i ( ϕ ) (where µ ∈ M + ( A ) is given) are nonzero.This yields that to every vector measure µ ∈ M + ( A ) there corresponds a uniquescalar Radon measure R µ ∈ M such that R µ ( ϕ ) = X i ∈ I α i µ i ( ϕ ) for all ϕ ∈ C (X) , where α i := (cid:26) +1 if i ∈ I + , − i ∈ I − . N. ZoriiThen, because of (3.1), the positive and negative parts in the Hahn–Jordan de-composition of R µ can respectively be written in the form R µ + = X i ∈ I + µ i , R µ − = X i ∈ I − µ i . Of course, the inclusion R of M + ( A ) into M , thus defined, is in generalnon-injective, i.e., one may choose µ , µ ∈ M + ( A ) so that µ = µ , though R µ = R µ . We call µ , µ ∈ M + ( A ) R -equivalent if R µ = R µ — or, which isequivalent, whenever P i ∈ I µ i = P i ∈ I µ i .Observe that the relation of R -equivalency implies that of identity (and,hence, these two relations on M + ( A ) are actually equivalent) if and only if all A i , i ∈ I , are mutually disjoint. Lemma 3.3.
The vague convergence of ( µ s ) s ∈ S ⊂ M + ( A ) to µ ∈ M + ( A ) impliesthe vague convergence of ( R µ s ) s ∈ S to R µ .Proof. This is obvious in view of the fact that the support of any ϕ ∈ C (X) canhave points in common with only finitely many A i . (cid:3) Remark . Lemma 3.3 in general can not be inverted. However, if all the A i are mutually disjoint, then the vague convergence of ( R µ s ) s ∈ S to R µ implies thevague convergence of ( µ s ) s ∈ S to µ , which is seen by using the Tietze–Urysohnextension theorem [11, Th. 0.2.13]. κ ( µ , µ ) and κ ( R µ , R µ ) are related to each other? In accordance with an electrostatic interpretation of a condenser A , we assumethat the law of interaction between the charges lying on the plates A i , i ∈ I , isdetermined by the matrix ( α i α j ) i,j ∈ I . Then the mutual energy of µ , µ ∈ M + ( A )is given by the expression κ ( µ , µ ) := X i,j ∈ I α i α j κ ( µ i , µ j ) . (3.2)For µ = µ the mutual energy defines the energy κ ( µ , µ ) of µ . Let E + ( A ) consistof all µ ∈ M + ( A ) with −∞ < κ ( µ , µ ) < ∞ . Lemma 3.5.
For µ ∈ M + ( A ) to have finite energy, it is necessary and sufficientthat µ i ∈ E for all i ∈ I and P i ∈ I k µ i k < ∞ .Proof. This follows immediately from the above definitions due to the inequality2 κ ( ν , ν ) k ν k + k ν k for ν , ν ∈ E . (cid:3) In view of the convexity of M + ( A ), Lemma 3.5 yields that also E + ( A ) formsa convex cone .In order to establish relations between the mutual energies of vector measuresand those of their (scalar) R -images, we need the following two lemmas, the formerbeing well known (see, e.g., [12]). In both, Y is a locally compact Hausdorff space. It will be shown below (see Corollary 3.10) that the mutual energy is well defined and finite(hence, the series in (3.2) converges absolutely) at least for all measures from E + ( A ). onstrained energy problems with external fields 7 Lemma 3.6. If ψ ∈ Φ(Y) is given, then the map ν
7→ h ψ, ν i is vaguely lowersemicontinuous on M + (Y) . In particular, this implies that the potential κ ( · , ν ) of any ν ∈ M + (X) belongsto Φ(X). Lemma 3.7.
Consider an ( L + , L − ) -condenser B = ( B ℓ ) ℓ ∈ L in Y , a vector measure ω = ( ω ℓ ) ℓ ∈ L ∈ M + ( B ) , and a function ψ ∈ Φ(Y) . For h ψ, R ω i to be finite, it isnecessary and sufficient that P ℓ ∈ L α ℓ h ψ, ω ℓ i converge absolutely, and then h ψ, R ω i = X ℓ ∈ L α ℓ h ψ, µ ℓ i . Proof.
We can assume ψ to be nonnegative, for if not, we replace ψ by a func-tion ψ ′ > ψ a suitable constant c >
0, which is alwayspossible since a lower semicontinuous function is bounded from below on a compactspace. Hence, h ψ, R ω + i > X ℓ ∈ L + , ℓ N h ψ, ω ℓ i for all N ∈ L + . On the other hand, the sum of ω ℓ over all ℓ ∈ L + that do not exceed N ap-proaches R ω + vaguely as N → ∞ ; consequently, by Lemma 3.6, h ψ, R ω + i lim N →∞ X ℓ ∈ L + , ℓ N h ψ, ω ℓ i . Combining the last two inequalities and then letting N → ∞ , we get h ψ, R ω + i = X ℓ ∈ L + h ψ, ω ℓ i . Since the same holds true for R ω − and L − instead of R ω + and L + , the lemmafollows. (cid:3) To apply Lemma 3.7 to the condenser A × A := ( A i × A j ) ( i,j ) ∈ I × I in X × Xwith α ( i,j ) := α i α j , we observe that any ω ∈ M + ( A × A ) can be written as µ ⊗ µ := ( µ i ⊗ µ j ) ( i,j ) ∈ I × I , where µ , µ ∈ M + ( A ). Therefore, R ( µ ⊗ µ ) = X i,j ∈ I α i α j µ i ⊗ µ j = R µ ⊗ R µ . If, moreover, ψ = κ ∈ Φ(X × X), then we arrive at the following assertion.
Corollary 3.8.
Given µ , µ ∈ M + ( A ) , we κ ( µ , µ ) = κ ( R µ , R µ ) , the identitybeing understood in the sense that each of its sides is finite whenever so is theother and then they coincide. Hence, µ ∈ M + ( A ) belongs to E + ( A ) if and only if R µ ∈ E and, furthermore, κ ( µ , µ ) = κ ( R µ , R µ ) for all µ ∈ E + ( A ) . (3.3)In view of the positive definiteness of the kernel, this yields the following propertyof positivity of the energy κ ( µ , µ ), which was not obvious a priori. N. Zorii Corollary 3.9.
For all µ ∈ E + ( A ) , it is true that κ ( µ , µ ) > . Corollary 3.10.
For any µ , µ ∈ E + ( A ) , we have κ ( µ , µ ) = κ ( R µ , R µ ) = X i,j ∈ I α i α j κ ( µ i , µ j ) , (3.4) and the series here converges absolutely.Proof. For any µ , µ ∈ E + ( A ), we get R µ , R µ ∈ E ; hence, κ ( R µ , R µ ) is finite.Therefore, repeated application of Corollary 3.8 gives the desired conclusion. (cid:3) E + ( A ) forms a semimetric space with the semimetric k · k E + ( A ) ,defined by (1.2) , and this space is isometric to its R -image. The semimetric k ·k E + ( A ) is a metric if and only if the kernel κ is strictly positive definite while all A i , i ∈ I , are mutually disjoint.Proof. Fix µ , µ ∈ E + ( A ). Applying Corollary 3.10 to κ ( R µ k , R µ t ), k, t = 1 , k R µ − R µ k E = X i,j ∈ I α i α j κ ( µ i − µ i , µ j − µ j ) , where the series converges absolutely. Compared with (1.2), this relation yields k µ − µ k E + ( A ) = k R µ − R µ k E . (3.5)Since k · k E is a seminorm on E , the theorem follows. (cid:3) From now on, E + ( A ) will always be treated as a semimetric space with thesemimetric k · k := k · k E + ( A ) . Since E + ( A ) and its R -image are isometric, similarto the terminology in E we shall call the topology on E + ( A ) strong .Two elements of E + ( A ), µ and µ , are said to be equivalent in E + ( A ) if k µ − µ k = 0. Observe that the equivalence in E + ( A ) implies R -equivalence (i.e.,then R µ = R µ ) provided the kernel κ is strictly positive definite, and it impliesthe identity (i.e., then µ = µ ) if, moreover, all A i , i ∈ I , are mutually disjoint.
4. Constrained minimal f -weighted energy problem Consider an external field f = ( f i ) i ∈ I satisfying Case I or Case II (see the Introduc-tion), and assume each f i to affect the charges on A i only. The f - weighted energy G f ( µ ) of µ ∈ M + ( A ) is defined by (1.1), and let E + f ( A ) consist of all µ ∈ E + ( A )with −∞ < G f ( µ ) < ∞ .Also fix a nonnegative vector measure σ ∈ M + ( A ), called a constraint as-sociated with A , a numerical vector a = ( a i ) i ∈ I with a i >
0, and a vector-valuedfunction g = ( g i ) i ∈ I , where all the g i : X → (0 , ∞ ) are continuous. We define M + σ ( A ) := (cid:8) µ ∈ M + ( A ) : µ σ (cid:9) , onstrained energy problems with external fields 9where µ σ means that σ i − µ i > i ∈ I , and M + σ ( A , a , g ) := (cid:8) µ ∈ M + σ ( A ) : h g i , µ i i = a i for all i ∈ I (cid:9) , E + σ ( A , a , g ) := M + σ ( A , a , g ) ∩ E + ( A ) , E + σ , f ( A , a , g ) := M + σ ( A , a , g ) ∩ E + f ( A )and then we introduce the extremal value G σ f ( A , a , g ) := inf µ ∈E + σ , f ( A , a , g ) G f ( µ ) . (4.1)In (4.1), as usual, the infimum over the empty set is taken to be + ∞ .If E + σ , f ( A , a , g ) is nonempty or, which is equivalent, if it is true that G σ f ( A , a , g ) < ∞ , (4.2)then the following problem makes sense. Problem.
Does there exist λ σ A ∈ E + σ , f ( A , a , g ) with G f ( λ σ A ) = G σ f ( A , a , g )?Along with its electrostatic interpretation, such a problem has found variousimportant applications in approximation theory (see, e.g., [8, 9, 21]). The problemis called solvable if the class S σ f ( A , a , g ) of all the minimizers λ = λ σ A is nonempty. If λ and b λ belong to S σ f ( A , a , g ) , then k λ − b λ k E + ( A ) = 0 . (4.3) Proof.
It follows from the convexity of E + ( A ) (see Sec. 3.3) that so is E + σ , f ( A , a , g ),which makes it possible to conclude from (1.1), (3.3), and (4.1) that4 G σ f ( A , a , g ) G f (cid:16) λ + b λ (cid:17) = k R λ + R b λ k + 4 h f , λ + b λ i . On the other hand, applying the parallelogram identity in the pre-Hilbert space E to R λ and R b λ and then adding and subtracting 4 h f , λ + b λ i , we get k R λ − R b λ k = −k R λ + R b λ k − h f , λ + b λ i + 2 G f ( λ ) + 2 G f ( b λ ) . When combined with the preceding relation, this yields0 k R λ − R b λ k − G σ f ( A , a , g ) + 2 G f ( λ ) + 2 G f ( b λ ) = 0 , which establishes (4.3) because of (3.5). (cid:3) Thus, any two minimizers (if exist) are equivalent in E + ( A ). Consequently,they are R -equivalent if the kernel κ is strictly positive definite, and they are equalif, moreover, all A i , i ∈ I , are mutually disjoint. See Lemma 5.5 below for necessary and (or) sufficient conditions for (4.2) to hold. Then, actually, G σ f ( A , a , g ) has to be finite (see Corollary 5.4).
5. Elementary properties of G σ f ( A , a , g ) Before analyzing the existence of minimizers and their continuity, we provide someauxiliary results, to be needed in the sequel. Write g i, inf := inf x ∈ A i g i ( x ) , g i, sup := sup x ∈ A i g i ( x ) . G σ f ( A , a , g )On the collection of all ( I + , I − )-condensers in X, it is natural to introduce anordering relation by declaring A ′ A to mean that A ′ i ⊂ A i for all i ∈ I . Here, A ′ = ( A ′ i ) i ∈ I . If now σ is a constraint associated with A and σ ′ is that associatedwith A ′ , then we write ( A ′ , σ ′ ) ( A , σ ) provided A ′ A and σ ′ σ . Then G σ f ( A , a , g ) is a nonincreasing function of ( A , σ ), namely G σ f ( A , a , g ) G σ ′ f ( A ′ , a , g ) whenever ( A ′ , σ ′ ) ( A , σ ) . (5.1)We shall employ the technique of exhaustion of A by compact K . In doingso, we shall need the following notation and elementary lemma.Given A , let { K } A stand for the increasing family of all compact condensers K = ( K i ) i ∈ I such that K A . For any µ ∈ M + ( A ) and K ∈ { K } A , let µ i K denote the trace of µ i upon K i , i.e. µ i K := µ iK i , and let µ K := ( µ i K ) i ∈ I . Observethat, if σ is a constraint associated with A , then σ K = ( σ i K ) i ∈ I is that associatedwith K . We further write b µ K := (ˆ µ i K ) i ∈ I , whereˆ µ i K := a i h g i , µ i K i µ i K . (5.2) Lemma 5.1.
Fix µ ∈ E + σ , f ( A , a , g ) . For every ε > , there exists K ∈ { K } A suchthat, for all K ∈ { K } A that follow K , b µ K ∈ E +(1+ ε ) σ K , f ( K , a , g ) . (5.3) Proof.
Application of [12, Lemma 1.2.2] yields h g i , µ i i = lim K ↑ A h g i , µ i K i , i ∈ I, (5.4) h f i , µ i i = lim K ↑ A h f i , µ i K i , i ∈ I, (5.5) κ ( µ i , µ j ) = lim K ↑ A κ ( µ i K , µ j K ) , i, j ∈ I. (5.6)Fix ε >
0. By (5.4)–(5.6), for every i ∈ I one can choose a compact set K i ⊂ A i so that, for all compact sets K i with the property K i ⊂ K i ⊂ A i , a i h g i , µ iK i i < ε i − , (5.7) (cid:12)(cid:12) h f i , µ i i − h f i , µ iK i i (cid:12)(cid:12) < ε i − , (5.8) (cid:12)(cid:12) k µ i k − k µ iK i k (cid:12)(cid:12) < ε i − . (5.9)onstrained energy problems with external fields 11Having denoted K := ( K i ) i ∈ I , for every K ∈ { K } A that follows K we get b µ K ∈ E +(1+ ε ) σ K ( K , a , g ) , the finiteness of the energy being obtained from (5.2), (5.7), and (5.9) with helpof Lemma 3.5. Furthermore, since P i ∈ I h f i , µ i i converges absolutely, we concludefrom (5.7) and (5.8) that so does P i ∈ I h f i , ˆ µ i K i . This means (5.3) as claimed. (cid:3) G σ f ( A , a , g ) > −∞ To prove the estimate, announced in the title, we need the following two lemmas.
Lemma 5.2.
Let Case II take place, i.e., let f i = α i κ ( · , ζ ) for all i ∈ I , where ζ ∈ E is given. Then the classes E + ( A ) and E + f ( A ) coincide and, furthermore, G f ( µ ) = k R µ + ζ k − k ζ k for all µ ∈ E + ( A ) . (5.10) Proof.
Applying Lemma 3.7 to µ ∈ E + ( A ) and each of κ ( · , ζ + ) and κ ( · , ζ − ), weget h f , µ i = X i ∈ I α i Z κ ( x, ζ ) dµ i ( x ) = κ ( ζ, R µ ) , (5.11)where the series converges absolutely. Hence, µ ∈ E + f ( A ). Now, substituting (3.3)and (5.11) into (1.1) gives (5.10) as required. (cid:3) Lemma 5.3.
Consider a condenser B = ( B ℓ ) ℓ ∈ L in a locally compact space Y , u = ( u ℓ ) ℓ ∈ L with u ℓ ∈ Φ(Y) , and F ⊂ M + ( B ) with the property that sup ω ∈ F ω ℓ (Y) < ∞ for all ℓ ∈ L (5.12) unless Y is noncompact. Then h u , ω i is well defined for all ω ∈ F , and −∞ < inf ω ∈ F h u , ω i ∞ . Proof.
We can assume Y to be compact, for if not, then u ℓ > ℓ ∈ L andthe lemma is obvious. But then B is to be finite while every u ℓ , being lower semi-continuous, is bounded from below by − c ℓ , where 0 < c ℓ < ∞ . Hence, by (5.12), −∞ < − c ℓ sup ω ∈ F ω ℓ (Y) h u ℓ , ω ℓ i ∞ , which in view of the finiteness of L yields the lemma. (cid:3) Corollary 5.4. G σ f ( A , a , g ) > −∞ .Proof. We can consider Case I, since otherwise the corollary follows from (5.10).Then f i ∈ Φ(X) for all i ∈ I . Furthermore, if X is compact, then g i, inf > µ ∈ M + σ ( A , a , g ) µ i (X) a i g − i, inf < ∞ . By Lemma 5.3, − ∞ < M h f , µ i ∞ for all µ ∈ M + σ ( A , a , g ) , (5.13)which together with Corollary 3.9 completes the proof. (cid:3) G σ f ( A , a , g ) < ∞ hold? Let C ( E ) = C κ ( E ) denote the interior capacity of a set E ⊂ X relative to thekernel κ (see [12]).The following assertion provides necessary and (or) sufficient conditions forrelation (4.2) to hold (or, which is equivalent, for E + σ , f ( A , a , g ) to be nonempty). Lemma 5.5. If (4.2) is true, then necessarily C (cid:0) { x ∈ A i : | f i ( x ) | < ∞} (cid:1) > for all i ∈ I. In the case where X i ∈ I a i g − i, inf < ∞ , (5.14) for (4.2) to hold, it is sufficient that the following conditions be both satisfied: (a) for every K ∈ { K } A , σ K has finite energy; (b) there exists M ∈ (0 , ∞ ) not depending on i and such that h g i , σ iA Mi i > a i ,where A Mi := { x ∈ A i : | f i ( x ) | M } , i ∈ I .Proof. To prove the necessity part of the lemma, fix µ ∈ E + σ , f ( A , a , g ); then, byLemma 5.1, b µ K has finite f -weighted energy provided K ∈ { K } A is sufficientlylarge. Suppose, contrary to our claim, that C (cid:0) { x ∈ A i : | f i ( x ) | < ∞} (cid:1) = 0for some i ∈ I . Since ˆ µ i K has finite energy and is compactly supported in A i ,[12, Lemma 2.3.1] shows that | f i ( x ) | = ∞ holds ˆ µ i K -almost everywhere (ˆ µ i K -a.e.)in X. This is impossible, for ˆ µ i K is nonzero while h f , b µ K i is finite.To establish the sufficient part, suppose (5.14), (a), and (b) to be satisfied.Then for every i ∈ I one can choose a compact set K i ⊂ A Mi so that h g i , σ iK i i > a i , (5.15)which is seen from (b) due to [12, Lemma 1.2.2]. Having denoted K := ( K i ) i ∈ I , weconsider the vector measure b σ K with the components ˆ σ i K , defined by (5.2) with σ i K in place of µ i K . It follows from (5.15) that b σ K ∈ E + σ K ( K , a , g ), the finiteness ofthe energy being obtained from (a) in view of Lemma 3.5. Furthermore, since X i ∈ I h| f i | , ˆ σ i K i M X i ∈ I a i σ i K (X) h g i , σ i K i M X i ∈ I a i g − i, inf , we actually have b σ K ∈ E + σ K , f ( K , a , g ) by (5.14), and so G σ K f ( K , a , g ) < ∞ . Since( K , σ K ) ( A , σ ), this together with (5.1) yields (4.2) as was to be proved. (cid:3) Remark . If A is finite, then Lemma 5.5 remains true with (b) replaced by thefollowing assumption: for every i ∈ I , h g i , σ i i > a i while | f i | 6 = ∞ locally σ i -a.e.(see [25, Lemma 4]).onstrained energy problems with external fields 13
6. Main results
From now on, (4.2) is always assumed to hold. Observe that, according to Corol-lary 5.4, G σ f ( A , a , g ) is then actually finite.Suppose for a moment that the condenser A is compact. Then the class M + σ ( A , a , g ) is vaguely bounded and closed and hence, by Lemma 3.2, it is vaguelycompact. If, moreover, A is finite, κ is continuous on A + × A − , and Case I holds,then G f ( µ ) is vaguely lower semicontinuous on E + f ( A ) and, therefore, the existenceof minimizers λ σ A immediately follows (cf. [14, 19, 20, 22]).However, these arguments break down if any of the above-mentioned fourassumptions is dropped, and then the problem on the existence of minimizers λ σ A becomes rather nontrivial. In particular, M + σ ( A , a , g ) is no longer vaguely compactif any of the A i is noncompact. Another difficulty is that G f ( µ ) might not bevaguely lower semicontinuous on E + f ( A ) when Case II takes place.To solve the problem on the existence of minimizers λ σ A in the general case,we develop an approach based on both the vague and strong topologies in thesemimetric space E + ( A ), introduced for vector measures of finite dimensions in [25,26, 28]. For I = { } , see also [27] (compare with [8, 9, 21]). In addition to (4.2), in all that follows it is always required that the kernel κ isconsistent and either I − = ∅ , or there hold (5.14) and the following condition:sup x ∈ A + , y ∈ A − κ ( x, y ) < ∞ . (6.1) Remark . Note that these assumptions on a kernel are not too restrictive. Inparticular, they all are satisfied by the Newtonian, Riesz, or Green kernels in R n , n >
2, provided the Euclidean distance between A + and A − is nonzero, as well asby the restriction of the logarithmic kernel in R to the open unit disk. A proposition u ( x ) involving a variable point x ∈ X is said to subsist nearlyeverywhere (n.e.) in E , where E ⊂ X, if the set of all x ∈ E for which u fails tohold is of interior capacity zero. Theorem 6.2.
Under the standing assumptions, suppose, moreover, for every i ∈ I the following (a)–(c) to hold: (a) Either g i, inf > or A i can be written as a countable union of compact sets; (b) Either g i, sup < ∞ or there exist r i ∈ (1 , ∞ ) and τ i ∈ E with the property g r i i ( x ) κ ( x, τ i ) n.e. in A i ; (6.2)(c) A i either is compact or has finite interior capacity.Then, for any σ , f , and a , S σ f ( A , a , g ) is nonempty and vaguely compact. Remark . If I − is nonempty, then condition (a) follows immediately from (5.14)and, hence, it can be omitted. It also holds automatically if the space X is countableat infinity (e.g., for X = R n ). Remark . Regarding condition (c), note that a compact set K ⊂ X mightbe of infinite capacity; C ( K ) is necessarily finite provided the kernel is strictlypositive definite [12]. On the other hand, even for the Newtonian kernel, sets offinite capacity might be noncompact [17]. Remark . Condition (c) is essential for the validity of Theorem 6.2. See Sec. 12for some examples, illustrating its sharpness.
Corollary 6.6. If A = K is compact, then, for any σ , f , g , and a , S σ f ( A , a , g ) isnonempty and vaguely compact.Proof. This is an immediate consequence of Theorem 6.2, since g i is boundedon K i . (cid:3) G σ f ( A , a , g ) and λ σ A with respect to ( A , σ )We write A s ↓ A , where A s = ( A si ) i ∈ I , s ∈ S , is a net of condensers, if A s A s whenever s s and \ s ∈ S A si = A i for all i ∈ I. Theorem 6.7.
Let A s ↓ A , and let for some s ∈ S all the assumptions of The-orem with A s instead of A be satisfied. Let σ s be a constraint associatedwith A s , and let ( σ s ) s ∈ S decrease and converge vaguely to σ . Then G σ f ( A , a , g ) = lim s ∈ S G σ s f ( A s , a , g ) . Fix arbitrary λ σ s A s ∈ S σ s f ( A s , a , g ) , where s > s , and λ σ A ∈ S σ f ( A , a , g ) — suchminimizers exist. Then every vague cluster point of the net ( λ σ s A s ) s ∈ S is an elementof S σ f ( A , a , g ) . Furthermore, λ σ s A s → λ σ A strongly, i.e. lim s ∈ S k λ σ s A s − λ σ A k E + ( A s ) = 0 . Corollary 6.8.
Under the assumptions of Theorem , if, moreover, the kernel κ isstrictly positive definite ( hence, perfect ) and all A s i , i ∈ I , are mutually disjoint,then the ( unique ) minimizer λ σ s A s ∈ S σ s f ( A s , a , g ) , where s > s , approaches the ( unique ) minimizer λ σ A ∈ S σ f ( A , a , g ) both vaguely and strongly. In the rest of Sec. 6.3, let A and g satisfy all the conditions of Theorem 6.2.We proceed by analyzing continuity properties of G σ f ( A , a , g ) and λ σ A under ex-haustion of A by compact K . Including the standing ones. onstrained energy problems with external fields 15
Theorem 6.9.
There exists a net ( β ∗ K ) K ∈{ K } A ⊂ (1 , ∞ ) decreasing to and suchthat, for any β K ∈ [1 , β ∗ K ] , G σ f ( A , a , g ) = lim K ↑ A G β K σ K f ( K , a , g ) , (6.3) where σ K := ( σ iK i ) i ∈ I . Fix arbitrary λ β K σ K K ∈ S β K σ K f ( K , a , g ) , where K ∈ { K } A is sufficiently large, and λ σ A ∈ S σ f ( A , a , g ) — such minimizers exist. Then ev-ery vague cluster point of the net ( λ β K σ K K ) K ∈{ K } A is an element of S σ f ( A , a , g ) .Furthermore, λ β K σ K K → λ σ A strongly, i.e. lim K ↑ A k λ β K σ K K − λ σ A k E + ( A ) = 0 . Corollary 6.10.
With the notation of Theorem , if the kernel κ is strictly positivedefinite ( hence, perfect ) and all A i , i ∈ I , are mutually disjoint, then the ( unique ) minimizer λ β K σ K K ∈ S β K σ K f ( K , a , g ) , where K ∈ { K } A , approaches the ( unique ) minimizer λ σ A ∈ S σ f ( A , a , g ) both vaguely and strongly.Remark . The value G σ f ( A , a , g ) remains unchanged if E + σ , f ( A , a , g ) in itsdefinition is replaced by the class of all µ ∈ E + σ , f ( A , a , g ) such that supp µ i , i ∈ I ,are compact . Indeed, this is concluded from (6.3) with β K = 1 for all K ∈ { K } A .The proofs of Theorems 6.2, 6.7 and 6.9, to be given in Sections 9, 10 and 11below (see also Sec. 8 for some crucial auxiliary notions and results), are basedon a theorem on the strong completeness of proper subspaces of the semimetricspace E + ( A ), which is a subject of Sec. 7.
7. Strong completeness of classes of vector measures
Recall that we are working under the standing assumptions, stated in Sec. 6.1.Write M + ( A , a , g ) := (cid:8) µ ∈ M + ( A ) : h g i , µ i i a i for all i ∈ I (cid:9) , M + σ ( A , a , g ) := M + ( A , a , g ) ∩ M + σ ( A ) , E + ( A , a , g ) := M + ( A , a , g ) ∩ E + ( A ) , and E + σ ( A , a , g ) := E + ( A , a , g ) ∩ M + σ ( A ) . Our next purpose is to show that E + ( A , a , g ) and E + σ ( A , a , g ), treated astopological subspaces of the semimetric space E + ( A ), are strongly complete .6 N. Zorii M + ( A , a , g ) and M + σ ( A , a , g ) are vaguely bounded and, hence,they are vaguely compact.Proof. Fix i ∈ I , and let a compact set K i ⊂ A i be given. Since g i is positive andcontinuous, the relation a i > h g i , µ i i > µ i ( K i ) min x ∈ K i g i ( x ) , where µ ∈ M + ( A , a , g ) , yields sup µ ∈ M + ( A , a , g ) µ i ( K i ) < ∞ . This implies that M + ( A , a , g ) is vaguely bounded; hence, by Lemma 3.2, it isvaguely relatively compact. In fact, it is vaguely compact, since it is vaguely closedin consequence of Lemma 3.6 with Y = A i and ψ = g i . Having observed that also M + σ ( A ) is vaguely closed, we then conclude that M + σ ( A , a , g ) is vaguely compactas well, which completes the proof. (cid:3) Lemma 7.2.
If a net ( µ s ) s ∈ S ⊂ E + ( A , a , g ) is strongly bounded, then every itsvague cluster point µ has finite energy.Proof. Note that, by (3.3), the net of scalar measures ( R µ s ) s ∈ S ⊂ E is stronglybounded as well. We proceed by showing that so are ( R µ + s ) s ∈ S and ( R µ − s ) s ∈ S ,i.e., sup s ∈ S k R µ ± s k < ∞ . (7.1)Of course, this needs to be verified only when I − = ∅ ; then, according to thestanding assumptions, (5.14) and (6.1) hold. Since h g i , µ is i a i , we conclude that µ is (X) a i g − i, inf for all i ∈ I and s ∈ S . Therefore, by (5.14),sup s ∈ S R µ ± s (X) X i ∈ I a i g − i, inf < ∞ . Because of (6.1), this implies that κ ( R µ + s , R µ − s ) remains bounded from aboveon S ; hence, so do k R µ + s k and k R µ − s k .If ( µ d ) d ∈ D is a subnet of ( µ s ) s ∈ S that converges vaguely to µ , then ( R µ + d ) d ∈ D and ( R µ − d ) d ∈ D converge vaguely to R µ + and R µ − , respectively. Using the factthat the map ( ν , ν ) ν ⊗ ν from M + (X) × M + (X) into M + (X × X) is vaguelycontinuous (see [2, Chap. 3, §
5, exerc. 5]) and applying Lemma 3.6 to Y = X × Xand ψ = κ , we conclude from (7.1) that R µ + and R µ − are both of finite energy.By Corollary 3.8, this means µ ∈ E + ( A ), as was to be proved. (cid:3) Corollary 7.3.
If a net ( µ s ) s ∈ S ⊂ E + ( A , a , g ) is strongly bounded, then sup s ∈ S k µ is k < ∞ , i ∈ I. (7.2)onstrained energy problems with external fields 17 Proof.
As an application of Lemma 5.3, we obtaininf s ∈ S X i = j, i,j ∈ I ± h κ, µ is ⊗ µ js i > −∞ . When combined with (3.4) and (7.1), this yields the corollary. (cid:3) E + ( A , a , g ) and E + σ ( A , a , g ) Theorem 7.4.
The following assertions hold: (i)
The semimetric space E + ( A , a , g ) is complete. In more detail, if ( µ s ) s ∈ S isa strong Cauchy net in E + ( A , a , g ) and µ is one of its vague cluster points ( such a µ exists ) , then µ ∈ E + ( A , a , g ) and µ s → µ strongly, i.e. lim s ∈ S k µ s − µ k E + ( A ) = 0 . (7.3)(ii) If the kernel κ is strictly positive definite while all A i , i ∈ I , are mutuallydisjoint, then the strong topology on E + ( A , a , g ) is finer than the vague one.In more detail, if ( µ s ) s ∈ S ⊂ E + ( A , a , g ) converges strongly to µ ∈ E + ( A ) ,then actually µ ∈ E + ( A , a , g ) and µ s → µ vaguely. (iii) Both the assertions (i) and (ii) remain valid if E + ( A , a , g ) is replacedeverywhere by E + σ ( A , a , g ) .Proof. To verify (i), fix a strong Cauchy net ( µ s ) s ∈ S ⊂ E + ( A , a , g ). Since such anet converges strongly to any of its strong cluster points, ( µ s ) s ∈ S can be assumedto be strongly bounded. Then, by Lemmas 7.1 and 7.2, a vague cluster point µ of ( µ s ) s ∈ S exists and, moreover, µ ∈ E + ( A , a , g ) . (7.4)We next proceed by proving (7.3). Of course, there is no loss of generality inassuming ( µ s ) s ∈ S to converge vaguely to µ . Then, by Lemma 3.3, ( R µ + s ) s ∈ S and( R µ − s ) s ∈ S converge vaguely to R µ + and R µ − , respectively. Since, by (7.1), thesenets are strongly bounded in E + , the property (C ) (see Sec. 2) shows that theyapproach R µ + and R µ − , respectively, in the weak topology as well, and so R µ s → R µ weakly. This gives, by (3.5), k µ s − µ k = k R µ s − R µ k = lim ℓ ∈ S κ ( R µ s − R µ , R µ s − R µ ℓ )and hence, by the Cauchy–Schwarz inequality, k µ s − µ k k µ s − µ k lim inf ℓ ∈ S k µ s − µ ℓ k , which proves (7.3) as required, because k µ s − µ ℓ k becomes arbitrarily small when s, ℓ ∈ S are sufficiently large. The proof of (i) is complete.To establish (ii), suppose now that the kernel κ is strictly positive defi-nite, while all A i , i ∈ I , are mutually disjoint, and let the net ( µ s ) s ∈ S convergestrongly to some µ ∈ E + ( A ). Given a vague limit point µ of ( µ s ) s ∈ S , we concludefrom (7.3) that k µ − µ k = 0, hence R µ = R µ since κ is strictly positive definite,and finally µ = µ because A i , i ∈ I , are mutually disjoint. In view of (7.4), this8 N. Zoriimeans that µ ∈ E + ( A , a , g ), which is a part of the desired conclusion. More-over, µ has thus been shown to be identical to any vague cluster point of ( µ s ) s ∈ S .Since the vague topology is Hausdorff, this implies that µ is actually the vaguelimit of ( µ s ) s ∈ S (cf. [1, Chap. I, §
9, n ◦
1, cor.]), as claimed.Finally, we observe that all the arguments applied above remain valid if E + ( A , a , g ) is replaced everywhere by E + σ ( A , a , g ). This yields (iii). (cid:3) Remark . Since the semimetric spaces E + ( A , a , g ) and E + σ ( A , a , g ) areisometric to their R -images, Theorem 7.4 has thus singled out strongly complete topological subspaces of the pre-Hilbert space E , whose elements are signed mea-sures. This is of a independent interest because, according to a well-known coun-terexample by H. Cartan [5], the whole space E is strongly incomplete even for theNewtonian kernel | x − y | − n in R n , n >
8. Extremal measures in the constrained energy problem
To apply Theorem 7.4 to the constrained energy problem, we next proceed byintroducing the concept of extremal measure defined as a strong and, simultane-ously, a vague limit point of a minimizing net. See below for strict definitions andrelated auxiliary results.
We call a net ( µ s ) s ∈ S minimizing if ( µ s ) s ∈ S ⊂ E + σ , f ( A , a , g ) andlim s ∈ S G f ( µ s ) = G σ f ( A , a , g ) . (8.1)Let M σ f ( A , a , g ) consist of all minimizing nets, and let M σ f ( A , a , g ) be theunion of the vague cluster sets of ( µ s ) s ∈ S , where ( µ s ) s ∈ S ranges over M σ f ( A , a , g ). Definition 8.2.
We call γ ∈ E + ( A ) extremal if there exists ( µ s ) s ∈ S ∈ M σ f ( A , a , g )that converges to γ both strongly and vaguely; such a net ( µ s ) s ∈ S is said to generate γ . The class of all extremal measures will be denoted by E σ f ( A , a , g ). Lemma 8.3.
The following assertions hold true: (i)
From every minimizing net one can select a subnet generating an extremalmeasure; hence, E σ f ( A , a , g ) is nonempty. Furthermore, E σ f ( A , a , g ) ⊂ E + σ ( A , a , g ) (8.2) and E σ f ( A , a , g ) = M σ f ( A , a , g ) . (8.3)(ii) Every minimizing net converges strongly to every extremal measure; hence, E σ f ( A , a , g ) is contained in an equivalence class in E + ( A ) . (iii) The class E σ f ( A , a , g ) is vaguely compact. onstrained energy problems with external fields 19 Proof.
Fix ( µ s ) s ∈ S and ( ν t ) t ∈ T in M σ f ( A , a , g ). Thenlim ( s,t ) ∈ S × T k µ s − ν t k = 0 , (8.4)where S × T is the directed product (see, e.g., [16, Chap. 2, § S and T . Indeed, in the same manner as in the proof of Lemma 4.1 we get0 k R µ s − R ν t k − G σ f ( A , a , g ) + 2 G f ( µ s ) + 2 G f ( ν t ) , which yields (8.4) when combined with (8.1).Relation (8.4) implies that ( µ s ) s ∈ S is strongly fundamental. Therefore, byTheorem 7.4, (iii), for every vague cluster point µ of ( µ s ) s ∈ S (such a µ exists) wehave µ ∈ E + σ ( A , a , g ) and µ s → µ strongly. Thus, µ is an extremal measure;actually, M σ f ( A , a , g ) ⊂ E σ f ( A , a , g ) . Since the inverse inclusion is obvious, relations (8.2) and (8.3) follow.Having thus proved (i), we proceed by verifying (ii). Fix arbitrary ( µ s ) s ∈ S ∈ M σ f ( A , a , g ) and γ ∈ E σ f ( A , a , g ). Then, according to Definition 8.2, there exists anet in M σ f ( A , a , g ), say ( ν t ) t ∈ T , that converges to γ strongly. Repeated applicationof (8.4) shows that also ( µ s ) s ∈ S converges to γ strongly, as claimed.To establish (iii), it is enough to prove that M σ f ( A , a , g ) is vaguely compact.Fix ( γ s ) s ∈ S ⊂ M σ f ( A , a , g ). It follows from (8.2) and Lemma 7.1 that there exists avague cluster point γ of ( γ s ) s ∈ S ; let ( γ t ) t ∈ T be a subnet of ( γ s ) s ∈ S that convergesvaguely to γ . Then for every t ∈ T one can choose ( µ s t ) s t ∈ S t ∈ M σ f ( A , a , g )converging vaguely to γ t . Consider the Cartesian product Q { S t : t ∈ T } — that is,the collection of all functions β on T with β ( t ) ∈ S t , and let D denote the directedproduct T × Q { S t : t ∈ T } . Given ( t, β ) ∈ D , write µ ( t,β ) := µ β ( t ) . Then thetheorem on iterated limits from [16, Chap. 2, §
4] yields that the net ( µ ( t,β ) ) ( t,β ) ∈ D belongs to M σ f ( A , a , g ) and converges vaguely to γ . Thus, γ ∈ M σ f ( A , a , g ). (cid:3) Corollary 8.4.
If Case II takes place, then G f ( γ ) = G σ f ( A , a , g ) for all γ ∈ E σ f ( A , a , g ) . (8.5) Proof.
Applying (5.10) to µ s , s ∈ S , and γ , where ( µ s ) s ∈ S ∈ M σ f ( A , a , g ) and γ ∈ E σ f ( A , a , g ) are arbitrarily given, and using the fact that, in accordance withLemma 8.3, µ s → γ strongly, we get G f ( γ ) = k R γ + ζ k − k ζ k = lim s ∈ S (cid:2) k R µ s + ζ k − k ζ k (cid:3) = lim s ∈ S G f ( µ s ) . Substituting (8.1) into the last relation gives (8.5), as desired. (cid:3) g i -masses of the i -componentsLemma 8.5. Let κ , A , a , and g satisfy all the assumptions (a)–(c) of Theorem ,and let a net ( µ s ) s ∈ S ⊂ E + ( A ) be strongly bounded and converge vaguely to µ .If, moreover, h g i , µ is i = a i for all s ∈ S and i ∈ I , then h g i , µ i i = a i for all i ∈ I. (8.6)0 N. Zorii Proof.
Fix i ∈ I . By Corollary 7.3, the net ( µ is ) s ∈ S is strongly bounded as well.Also note that A i can be written as a countable union of µ is -integrable sets,where s ∈ S is given. Indeed, this is obvious if A i is a countable union of compactsets; otherwise, due to condition (a) of Theorem 6.2, we have g i, inf > µ is ( A i ) a i g − i, inf < ∞ . Therefore, the concept of local µ is -negligibility and thatof µ is -negligibility coincide. Together with [12, Lemma 2.3.1], this yields that anyproposition holds µ is -a.e. in X provided it holds n.e. in A i .We proceed by establishing (8.6). Of course, this needs to be done only if theset A i is noncompact; then, by condition (c), its capacity has to be finite. Hence,by [12, Th. 4.1], for every E ⊂ A i there exists a measure θ E ∈ E + ( E ), called an interior equilibrium measure associated with E , which admits the properties θ E (X) = k θ E k = C ( E ) , (8.7) κ ( x, θ E ) > E. (8.8)Also observe that there is no loss of generality in assuming g i to satisfy (6.2)with some r i ∈ (1 , ∞ ) and τ i ∈ E . Indeed, otherwise, due to condition (b) ofTheorem 6.2, g i has to be bounded from above (say by M ), which combinedwith (8.8) again gives (6.2) for τ i := M r i θ A i , r i ∈ (1 , ∞ ) being arbitrary.We treat A i as a locally compact space with the topology induced from X.Given E ⊂ A i , let χ E denote its characteristic function and let E c := A i \ E .Further, let { K i } be the increasing family of all compact subsets K i of A i . Since g i χ K i is upper semicontinuous on A i while ( µ is ) s ∈ S converges to µ i vaguely, fromLemma 3.6 we get h g i χ K i , µ i i > lim sup s ∈ S h g i χ K i , µ is i for every K i ∈ { K i } . On the other hand, application of Lemma 1.2.2 from [12] yields h g i , µ i i = lim K i ∈{ K i } h g i χ K i , µ i i . Combining the last two relations, we obtain a i > h g i , µ i i > lim sup ( s,K i ) ∈ S ×{ K i } h g i χ K i , µ is i = a i − lim inf ( s,K i ) ∈ S ×{ K i } h g i χ K ci , µ is i ,S × { K i } being the directed product of the directed sets S and { K i } . Hence, if weprove lim inf ( s,K i ) ∈ S ×{ K i } h g i χ K ci , µ is i = 0 , (8.9)the desired relation (8.6) follows.Consider an interior equilibrium measure θ K ci , where K i ∈ { K i } is given.Then application of Lemma 4.1.1 and Theorem 4.1 from [12] shows that k θ K ci − θ ˜ K ci k k θ K ci k − k θ ˜ K ci k provided K i ⊂ ˜ K i . Furthermore, it is clear from (8.7) that the net k θ K ci k , K i ∈ { K i } , is bounded andnonincreasing, and hence fundamental in R . The preceding inequality thus yieldsthat the net ( θ K ci ) K i ∈{ K i } is strongly fundamental in E . Since, clearly, it convergesonstrained energy problems with external fields 21vaguely to zero, the property (C ) (see. Sec. 2) implies that zero is also one of itsstrong limits and, hence, lim K i ∈{ K i } k θ K ci k = 0 . (8.10)Write q i := r i ( r i − − , where r i ∈ (1 , ∞ ) is the number involved in condi-tion (6.2). Combining (6.2) with (8.8) shows that the inequality g i ( x ) χ K ci ( x ) κ ( x, τ i ) /r i κ ( x, θ K ci ) /q i subsists n.e. in A i and, hence, µ is -a.e. in X. Having integrated this relation withrespect to µ is , we then apply the H¨older and, subsequently, the Cauchy–Schwarzinequalities to the integrals on the right. This gives h g i χ K ci , µ is i hZ κ ( x, τ i ) dµ is ( x ) i /r i hZ κ ( x, θ K ci ) dµ is ( x ) i /q i k τ i k /r i k θ K ci k /q i k µ is k . Taking limits here along S × { K } and using (7.2) and (8.10), we obtain (8.9) asdesired. (cid:3) Corollary 8.6.
Under the assumptions of Theorem , we have E σ f ( A , a , g ) ⊂ E + σ ( A , a , g ) . (8.11) Proof.
Fix γ ∈ E σ f ( A , a , g ); then there exists a net ( µ s ) s ∈ S ⊂ E + σ , f ( A , a , g ) con-verging to γ strongly and vaguely. Taking a subnet if necessary, we assume ( µ s ) s ∈ S to be strongly bounded. Then, by Lemma 8.5, h g i , γ i i = a i for all i ∈ I , whichtogether with (8.2) gives (8.11). (cid:3)
9. Proof of Theorem 6.2
Fix an extremal measure γ ∈ E σ f ( A , a , g ) — it exists by Lemma 8.3, (i), and choosea net ( µ s ) s ∈ S ∈ M σ f ( A , a , g ) that converges to γ both strongly and vaguely. Weare going to show that γ is a minimizer, i.e. γ ∈ S σ f ( A , a , g ) . (9.1)According to Corollary 8.6, we have γ ∈ E + σ ( A , a , g ). Hence, (9.1) will beestablished once we prove that P i ∈ I h f i , γ i i converges absolutely, so that γ ∈ E + σ , f ( A , a , g ) , (9.2)and G f ( γ ) G σ f ( A , a , g ) . (9.3)To this end, assume Case I to hold, since otherwise (9.2) and (9.3) havealready been established by Lemma 5.2 and Corollary 8.4, respectively. Then, inconsequence of Lemma 5.3 (see (5.13) with µ = γ ), h f , γ i is well defined and h f , γ i > −∞ . (9.4)2 N. ZoriiBesides, from the strong and vague convergence of ( µ s ) s ∈ S to γ we obtain G σ f ( A , a , g ) = lim s ∈ S (cid:2) k µ s k + 2 h f , µ s i (cid:3) = k γ k + 2 lim s ∈ S h f , µ s i (9.5)(consequently, lim s ∈ S h f , µ s i exists and is finite) and h f , γ i = X i ∈ I h f i , γ i i X i ∈ I lim inf s ∈ S h f i , µ is i lim s ∈ S X i ∈ I h f i , µ is i = lim s ∈ S h f , µ s i < ∞ . (9.6)Combining (9.4) and (9.6) proves (9.2), while substituting (9.6) into (9.5) gives (9.3),and the required inclusion (9.1) follows.It has thus been proved that E σ f ( A , a , g ) ⊂ S σ f ( A , a , g ). This inclusion cancertainly be inverted, since any minimizer λ σ A can be thought as an extremalmeasure generated by the constant net ( λ σ A ). On account of (8.3), we get S σ f ( A , a , g ) = E σ f ( A , a , g ) = M σ f ( A , a , g ) . Therefore Lemma 8.3 shows that S σ f ( A , a , g ) is vaguely compact. The proof iscomplete. (cid:3)
10. Proof of Theorem 6.7
It is seen from (4.2), (5.1) and Corollary 5.4 that, under the assumptions of thetheorem, G σ s f ( A s , a , g ) increases as s ranges through S and −∞ < lim s ∈ S G σ s f ( A s , a , g ) G σ f ( A , a , g ) < ∞ . Besides, in accordance with Theorem 6.2, for every s > s there is a minimizer λ s := λ σ s A s ∈ S σ s f ( A s , a , g ). Therefore, lim s ∈ S G f ( λ s ) exists and − ∞ < lim s ∈ S G f ( λ s ) G σ f ( A , a , g ) < ∞ . (10.1)Also observe that, since A s and σ s decrease along S , it is true that λ s ∈ E + σ ℓ , f ( A ℓ , a , g ) for all s > ℓ > s . We proceed by showing that k λ s − λ s k G f ( λ s ) − G f ( λ s ) (10.2)whenever s s s . For every t ∈ (0 , µ := (1 − t ) λ s + t λ s belongs to theclass E + σ s , f ( A s , a , g ), and therefore G f ( µ ) > G f ( λ s ). Evaluating the left-handside of this inequality and then letting t →
0, we get −k λ s k + κ ( λ s , λ s ) − h f , λ s i + h f , λ s i > , and (10.2) follows.Due to (10.1), the net G f ( λ s ), s ∈ S , is fundamental in R . When com-bined with (10.2), this implies that λ s , s > ℓ > s , form a fundamental netonstrained energy problems with external fields 23in E + σ ℓ ( A ℓ , a , g ). Hence, by Theorem 7.4, there exists a vague cluster point λ of( λ s ) s ∈ S and the following two assertions hold:(i) λ ∈ E + σ s ( A s , a , g ) for all s > s ; (ii) λ s → λ strongly . However, Lemma 8.5 with A s instead of A shows that assertion (i) can bestrengthened as follows: λ ∈ E + σ s ( A s , a , g ) for all s > s . In turn, this implies that,actually, λ ∈ E + σ ( A , a , g ), since σ s → σ vaguely while A s ↓ A .What has already been established yields that the proof of the theorem willbe complete once we show that P i ∈ I h f i , λ i i converges absolutely, so that λ ∈ E + σ , f ( A , a , g ) , (10.3)and h f , λ i lim s ∈ S h f , λ s i . (10.4)Note that lim s ∈ S h f , λ s i exists and is finite, which is clear from (10.1) and (ii).We can suppose Case I to hold, since otherwise (10.3) is already known fromLemma 5.2 while (10.4) can be obtained directly from (5.11) and assertion (ii).Therefore, by (5.13) with µ = λ , h f , λ i is well defined and h f , λ i > −∞ . Taking asubnet if necessary, we can also assume that λ s → λ vaguely. Then, −∞ < h f , λ i = X i ∈ I h f i , λ i i X i ∈ I lim inf s ∈ S h f i , λ is i lim s ∈ S X i ∈ I h f i , λ is i < ∞ , and the required relations (10.3) and (10.4) follow. (cid:3)
11. Proof of Theorem 6.9
We begin by establishing the relation G σ f ( A , a , g ) = lim K ↑ A G σ K f ( K , a , g ) . (11.1)For every µ ∈ E + σ , f ( A , a , g ), consider b µ K = (ˆ µ i K ) i ∈ I defined by (5.2). Fix anarbitrary ε > K so that for all K that follow K inclusion (5.3)holds. This yields G f ( b µ K ) > G (1+ ε ) σ K f ( K , a , g ) . (11.2)We next proceed by showing that G f ( µ ) = lim K ↑ A G f ( b µ K ) . (11.3)To this end, it can be assumed that κ >
0; for if not, then A must be finite sinceX is compact, and (11.3) follows from (5.4)–(5.6). Therefore, for all K > K and i ∈ I we get k µ i K k k µ i k k R µ + + R µ − k , (11.4) k µ i − µ i K k < ε i − , (11.5)4 N. Zoriithe latter being clear from (5.9) because of κ ( µ i K , µ i − µ i K ) >
0. Also observe that (cid:12)(cid:12) k µ k − k b µ K k (cid:12)(cid:12) X i,j ∈ I (cid:12)(cid:12)(cid:12) κ ( µ i , µ j ) − a i h g i , µ i K i a j h g j , µ j K i κ ( µ i K , µ j K ) (cid:12)(cid:12)(cid:12) X i,j ∈ I h κ ( µ i − µ i K , µ j ) + κ ( µ i K , µ j − µ j K ) + (cid:16) a i h g i , µ i K i a j h g j , µ j K i − (cid:17) κ ( µ i K , µ j K ) i . When combined with (5.7), (5.8), (11.4), and (11.5), this yields (cid:12)(cid:12) G f ( µ ) − G f ( b µ K ) (cid:12)(cid:12) M ε for all K > K , where M is finite and independent of K , and the required relation (11.3) follows.Substituting (11.2) into (11.3), in view of (5.1) and the arbitrary choice of µ we get G σ f ( A , a , g ) > lim K ↑ A G (1+ ε ) σ K f ( K , a , g ) > G (1+ ε ) σ f ( A , a , g ) . Letting ε → A and K , we obtain G σ f ( A , a , g ) = lim ε → h lim K ↑ A G (1+ ε ) σ K f ( K , a , g ) i = lim K ↑ A G σ K f ( K , a , g ) , which proves (11.1) as desired.Fix λ σ K K ∈ S σ K f ( K , a , g ), where K ∈ { K } A . As follows from (11.1), the net( λ σ K K ) K ∈{ K } A belongs to M σ f ( A , a , g ) and, hence, it is strongly fundamental.Further, for every K ∈ { K } A choose β ∗ K ∈ (1 , ∞ ) such that ( β ∗ K ) K ∈{ K } A decreases to 1 and, for all β K ∈ [1 , β ∗ K ] and λ β K σ K K ∈ S β K σ K f ( K , a , g ),lim K ∈{ K } A k λ β K σ K K − λ σ K K k = 0 , (11.6)lim K ∈{ K } A (cid:2) G f ( λ β K σ K K ) − G f ( λ σ K K ) (cid:3) = 0 . (11.7)The existence of those β ∗ K follows from Theorem 6.7.Then, combining (11.1) and (11.7) gives (6.3) as required, while (11.6) to-gether with the property of strong fundamentality of ( λ σ K K ) K ∈{ K } A shows that( λ β K σ K K ) K ∈{ K } A is strongly fundamental as well. Hence, by Theorem 7.4 andLemma 8.5, the vague cluster set of ( λ β K σ K K ) K ∈{ K } A is nonempty and for everyits element λ the following assertions both hold: λ ∈ E + σ ( A , a , g ) and λ β K σ K K → λ strongly. To complete the proof, it is enough to show that λ ∈ S σ f ( A , a , g ), butthis can be done in the same way as at the end of the proof of Theorem 6.7. (cid:3)
12. On the sharpness of condition (c) in Theorem 6.2
Given a closed set F ⊂ R n , for brevity let M ( F ) denote the collection of allprobability measures supported by F . The examples below illustrate the sharpnessof condition (c) in Theorem 6.2.onstrained energy problems with external fields 25 Example . In R n , n >
2, consider the Riesz kernel κ α ( x, y ) := | x − y | α − n oforder α , α ∈ (0 , α < n , and a condenser A = ( A , A ), where I + = { } , I − = { } , A is compact, and C κ α ( A i ) > i = 1 ,
2. Also consider the α -Greenkernel g αA c of A c := R n \ A , defined by (see, e.g., [17, Chap. 4, § g αA c ( x, y ) := κ α ( x, ε y ) − κ α ( x, β αA ε y ) , where ε y is the (unit) Dirac measure at y and β αA is the operator of Riesz balayageonto A . Further, let K ⊂ ( A ∪ A ) c be a compact set with C κ α ( K ) >
0, and let θ denote the (unique) minimizer in the minimal α -Green energy probleminf ν ∈ M ( A ∪ K ) g αA c ( ν, ν );then it holds true that k θ k g αAc = k θ − β αA θ k κ α = (cid:2) C g αAc ( A ∪ K ) (cid:3) − . Assume, moreover, that f satisfies Case II with ζ = θ K , where θ K is the traceof θ upon K , and let a = θ ( A ), a = 1, g = g = 1. Also assume, for simplicity, A c to be connected. Proposition . Under the above notation and requirements, the following twoassertions are equivalent: (i)
One can choose a strictly increasing sequence of positive numbers R k , k ∈ N ,and measures ω k ∈ M (cid:0) A ∩ { R k | x | R k +1 } (cid:1) so that S σ f ( A , a , g ) = ∅ , (12.1) where σ = ( σ , σ ) is a constraint with the components σ := θ A and σ := β αA θ + (cid:2) − β αA θ ( A ) (cid:3) X k ∈ N ω k . (12.2)(ii) C κ α ( A ) = ∞ , though A is α -thin at ∞ R n . Recall that a closed set F ⊂ R n is α - thin at ∞ R n if the origin x = 0 is an α -irregular point for the inverse of F relative to the unit sphere (see [4]; cf. also [3,17]). See, e.g., [17, Chap. V] for the notion of α -regularity in case α ∈ (0 , Example . With the notation and the requirements of Example 1, let κ ( x, y ) bethe Newtonian kernel | x − y | − in R , and let A be a rotational body consistingof all x = ( x , x , x ) ∈ R such that q x < ∞ and 0 x + x ρ ( x ), where q ∈ R and ρ ( x ) approaches 0 as x → ∞ . Consider the following three cases: ρ ( r ) = r − s , where s ∈ [0 , ∞ ) , (12.3) ρ ( r ) = exp( − r s ) , where s ∈ (0 , , (12.4) ρ ( r ) = exp( − r s ) , where s > . (12.5)6 N. ZoriiAs has been shown in [23], A is not 2-thin at ∞ R in case (12.3), has finite (New-tonian) capacity in case (12.5), and it is 2-thin at ∞ R though has infinite (New-tonian) capacity in case (12.4). Consequently, assertion (i) from Proposition 12.1on the unsolvability of the corresponding constrained problem holds in case (12.4),and it fails to hold in both cases (12.3) and (12.5). Our arguments are based on Theorem 4 from [24], which asserts that, if F ⊂ R n is closed, ν > F c and if, for simplicity, F c is connected, then β αF ν ( R n ) = ν ( R n ) ⇐⇒ F is not α -thin at ∞ R n . (12.6)Fix an arbitrary µ = ( µ , µ ) ∈ E + ( A ). Then, by (5.10), G f ( µ ) = k R µ + θ K k κ α − k θ K k κ α = k µ + θ K − µ k κ α − k θ K k κ α . Since, by known facts from the Riesz and α -Green potential theory [17], k µ + θ K − µ k κ α > k µ + θ K − β αA ( µ + θ K ) k κ α = k µ + θ K k g αAc > k θ k g αAc , we get G f ( µ ) > k θ k g αAc − k θ K k κ α , (12.7)where the inequality is actually an equality if and only if µ + θ K = θ (hence, µ = θ A ) and µ = β αA θ. Assume (i) to hold; then necessarily C κ α ( A ) = ∞ , for if not, we would arriveat a contradiction with Theorem 6.2. To establish (ii), assume, on the contrary,that A is not α -thin at ∞ R n . Then, by (12.6), β αA θ ( A ) = 1 (12.8)and consequently, by (12.2), σ = θ A and σ = β αA θ. (12.9)It follows from (12.8) and (12.9) that σ ∈ E + σ ( A , a , g ) and inequality (12.7) for µ = σ is actually an equality. Thus, σ ∈ S σ f ( A , a , g ), which is impossible by (12.1).Now, assume (ii) to hold. Since A is α -thin at ∞ R n , from (12.6) we get c := 1 − β αA θ ( A ) > . (12.10)Choose a strictly increasing sequence ( R k ) k ∈ N with the property C κ α (cid:0) A ( k )2 (cid:1) > k ,where A ( k )2 := A ∩{ R k | x | R k +1 } , which is possible because of the assumption C ( A ) = ∞ , and let ω k minimize κ α ( ν, ν ) among all ν ∈ M (cid:0) A ( k )2 (cid:1) . Thenlim k →∞ k ω k k κ α = 0 = inf ν ∈ M ( A ) κ α ( ν, ν ) , which yields by standard arguments that ( ω k ) k ∈ N is a strong Cauchy sequencein E κ α ( R n ). Since ω k → κ α we thus get ω k → µ k = ( µ k , µ k ), k ∈ N , with µ k = θ A and µ k = β αA θ + cω k , where c is given by (12.10). Then µ k ∈ E + σ ( A , a , g ), where σ is defined by (12.2), and k µ k + θ K − µ k k κ α = k θ − β αA θ − cω k k κ α = k θ k g αAc + c k ω k k κ α − cκ α ( ω k , θ − β αA θ ) . Letting here k → ∞ , we then obtainlim k →∞ G ( µ k ) = k θ k g αAc − k θ K k κ α and so, by (12.7), ( µ k ) k ∈ N ∈ M σ f ( A , a , g ). Since µ k → γ := ( θ A , β αA θ ) stronglyand vaguely, we get γ ∈ E σ f ( A , a , g ). Moreover, in view of the strict positivedefiniteness of the Riesz kernel, γ is the only element of the class E σ f ( A , a , g ) (seeassertion (ii) of Lemma 8.3). However, because of (12.10), γ S σ f ( A , a , g ). Thisproves (12.1) and, hence, (i). (cid:3) Acknowledgments
The research was supported, in part, by the ”Scholar-in-Residence” program atIPFW, and the author acknowledges this institution for the support and theexcellent working conditions. The author also thanks Professors P. D. Dragnev,E. B. Saff, and W. L. Wendland for many valuable discussions about the contentof the paper.
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