Inner Riesz pseudo-balayage and its applications to minimum energy problems with external fields

For the Riesz kernel $\kappa_\alpha(x,y):=|x-y|^{\alpha-n}$, $0<\alpha<n$, on $\mathbb R^n$, $n\geqslant2$, we introduce the inner pseudo-balayage $\hat{\omega}^A$ of a (Radon) measure $\omega$ on $\mathbb R^n$ to a set $A\subset\mathbb R^n$ as the (unique) measure minimizing the Gauss functional \[\int\kappa_\alpha(x,y)\,d(\mu\otimes\mu)(x,y)-2\int\kappa_\alpha(x,y)\,d(\omega\otimes\mu)(x,y)\] over the class $\mathcal E^+(A)$ of all positive measures $\mu$ of finite energy, concentrated on $A$. For quite general signed $\omega$ (not necessarily of finite energy) and $A$ (not necessarily closed), such $\hat{\omega}^A$ does exist, and it maintains the basic features of inner balayage for positive measures (defined when $\alpha\leqslant2$), except for those implied by the domination principle. (To illustrate the latter, we point out that, in contrast to what occurs for the balayage, the inner pseudo-balayage of a positive measure may increase its total mass.) The inner pseudo-balayage $\hat{\omega}^A$ is further shown to be a powerful tool in the problem of minimizing the Gauss functional over all $\mu\in\mathcal E^+(A)$ with $\mu(\mathbb R^n)=1$, which enables us to improve substantially many recent results on this topic, by strengthening their formulations and/or by extending the areas of their applications. For instance, if $A$ is a quasiclosed set of nonzero inner capacity $c_*(A)$, and if $\omega$ is a signed measure, compactly supported in $\mathbb R^n\setminus{\rm Cl}_{\mathbb R^n}A$, then the problem in question is solvable if and only if either $c_*(A)<\infty$, or $\hat{\omega}^A(\mathbb R^n)\geqslant1$.


Inner pseudo-balayage: a motivation and a model case
This paper deals with the theory of potentials with respect to the α-Riesz kernels κ α (x, y) := |x − y| α−n of order 0 < α < n on R n , n 2, |x − y| being the Euclidean distance in R n .Our main goal is to proceed further with the study of minimum α-Riesz energy problems in the presence of external fields f , a point of interest for many researchers (see e.g. the monographs [2,20] and references therein, [14,18], [21]- [25], as well as [1,8,32,33], some of the most recent papers on this topic).
In the current work we improve substantially many recent results in this field, by strengthening their formulations and/or by extending the areas of their applications (see Section 6 for the results obtained).This has become possible due to the development of a new tool, the inner pseudo-balayage (see Section 3, cf.also the present section for a motivation of the proposed definition as well as for a model case).
It is well known that the α-Riesz balayage (sweeping out) serves as an efficient tool in the problems in question (see e.g.[8,23,25,33]).However, its application is only limited to the case of α ranging over (0, 2], and to external fields f of the form f (x) := −U ω (x) := − ˆκα (x, y) dω(y), (1.1) where ω is a suitable positive Radon measure.
To extend the area of application of such a tool to arbitrary α ∈ (0, n) and/or to external fields f given by (1.1), but now with signed ω involved, we generalize the standard concept of inner balayage of positive measures (defined for α ∈ (0, 2]) to the so-called inner pseudo-balayage of signed measures, by maintaining the basic features of the former concept -except for those implied by the domination principle.
Being crucial to our study of minimum energy problems with external fields, the concept of inner pseudo-balayage is also of independent interest, looking promising for further generalizations and other applications.Before introducing it, we first review some basic facts of the theory of α-Riesz potentials.
We denote by M the linear space of all (real-valued Radon) measures µ on R n , equipped with the vague topology of pointwise convergence on the class C 0 (R n ) of all continuous functions ϕ : R n → R of compact support, and by M + the cone of all positive µ ∈ M, where µ is positive if and only if µ(ϕ) 0 for all positive ϕ ∈ C 0 (R n ).Given µ, ν ∈ M, the potential U µ and the mutual energy I(µ, ν) are introduced by U µ (x) := ˆκα (x, y) dµ(y), x ∈ R n , I(µ, ν) := ˆκα (x, y) d(µ ⊗ ν)(x, y), respectively, provided that the integral on the right is well defined (as a finite number or ±∞).For µ = ν, I(µ, ν) defines the energy I(µ) := I(µ, µ) of µ ∈ M.
Another fact decisive to this paper is that the cone E + := E ∩ M + is strongly complete, and that the strong topology on E + is finer than the (induced) vague topology on E + (see J. Deny [6]; for α = 2, cf.also H. Cartan [4]).Thus any strong Cauchy sequence (net) (µ j ) ⊂ E + converges both strongly and vaguely to the same unique limit µ 0 ∈ E + , the strong topology on E as well as the vague topology on M being Hausdorff.(Following B. Fuglede [10], such a kernel is said to be perfect.) 1.1.A model case.As a model case for introducing the concept of inner α-Riesz pseudo-balayage, consider first a closed set F ⊂ R n and a (signed) measure ω ∈ M of finite energy.Since the class M + (F ) of all µ ∈ M + with the support S(µ) ⊂ F is vaguely closed [3, Section III.2, Proposition 6], the convex cone E + (F ) := M + (F ) ∩ E is strongly closed, and hence strongly complete, the α-Riesz kernel being perfect.By applying [9] (Theorem 1.12.3 and Proposition 1.12.4(2)),we therefore conclude that for the given ω ∈ E, there exists the unique P ω ∈ E + (F ) such that and the same P ω is uniquely characterized within E + (F ) by the two relations P ω − ω, P ω = 0. (1.4) This P ω is said to be the orthogonal projection of ω ∈ E onto E + (F ).
By a slight modification of [26, Proof of Theorem 3.1] we infer from the above that P ω is the only measure in E + (F ) having the two properties U P ω U ω n.e. on F , ( U P ω = U ω P ω-a.e., (1.6) where the abbreviation n.e.(nearly everywhere) means that the inequality holds true everywhere on F except for a subset N ⊂ F of inner capacity zero: c * (N) = 0.1 Assume for a moment that α 2, and that the above ω is positive, i.e. ω ∈ E + .By use of the complete maximum principle [17, Theorems 1.27, 1.29], we derive from (1.5) and (1.6) that P ω is then uniquely characterized within E + (F ) by the equality see [26,Theorem 3.1], and hence P ω is actually the balayage ω F of ω ∈ E + onto F : P ω = ω F .¶ However, if either α > 2, or if the above ω is signed, then relations (1.2)-(1.6)still hold, but they no longer result in (1.7).
Motivated by this observation, we introduce the following definition.
Definition 1.1.The pseudo-balayage ωF of ω ∈ E onto a closed set F ⊂ R n with respect to the α-Riesz kernel of arbitrary order α ∈ (0, n) is defined as the only measure in E + (F ) satisfying (1.2) (with ωF in place of P ω); or equivalently, as the unique measure in E + (F ) having properties (1.3)-(1.6)(with ωF in place of P ω).
Remark 1.2.Assume for a moment that ω is positive.It follows from the above that the pseudo-balayage ωF coincides with the balayage ω F whenever α 2; while otherwise, the former concept presents a natural extension of the latter, the problem of balayage for α > 2 being unsolvable. 2 But if now ω is signed, then for α 2, both ωF and ω F still exist and are unique, whereas, in general, ωF = ω F , the balayage of signed ω being defined by linearity (for more details see Remark 3.3).
Remark 1.3.In Section 3 below, we shall extend the above definition of the pseudo-balayage ωF , given in the model case of ω ∈ E and closed F ⊂ R n , to: • ω ∈ M that are not necessarily of finite energy.
• F ⊂ R n that are not necessarily closed.For the former goal, we observe that problem (1.2) is equivalent to that of minimizing the Gauss functional µ 2 − 2 ´Uω dµ, which makes sense not only for ω ∈ E.
We complete this section with some general conventions, used in what follows.From now on, when speaking of a (signed) measure µ ∈ M, we understand that its potential U µ is well defined and finite almost everywhere with respect to the Lebesgue measure on R n ; or equivalently (cf.[17,Section I.3.7]) that where |µ| := µ + + µ − , µ + and µ − being the positive and negative parts of µ in the Hahn-Jordan decomposition.Actually, then (and only then) U µ is finite q.e. on R n , cf. [17,Section III.1.1].This would necessarily hold if µ were required to be bounded (that is, with |µ|(R n ) < ∞), or of finite energy, cf.[10, Corollary to Lemma 3.
or equivalently if A is µ-measurable and µ = µ| A , µ| A being the restriction of µ to A. Denoting by M + (A) the cone of all µ ∈ M + concentrated on A, we further write E + (A) := M + (A) ∩ E, and let E ′ (A) stand for the closure of E + (A) in the strong topology on E + .We emphasize that E ′ (A) is strongly complete, being a strongly closed subcone of the strongly complete cone E + .
Given A ⊂ R n , denote by C A the upward directed set of all compact subsets K of A, where being a topological space, then we shall indicate this fact by writing

On the inner Riesz balayage
Before proceeding with an extension of the concept of pseudo-balayage announced in Remark 1.3, we first recall some basic facts of the theory of inner α-Riesz balayage.Such a theory, generalizing Cartan's pioneering work [5] on the inner Newtonian balayage (α = 2) to any α ∈ (0, 2], was initiated in the author's recent papers [26,27], and it was further developed in [28]- [30]. 3Throughout this section, 0 < α 2. Definition 2.1 ([26, Sections 3,4]).The inner balayage ω A of a measure ω ∈ M + to a set A ⊂ R n is defined as the measure of minimum potential in the class Γ A,ω , Γ A,ω := µ ∈ M + : U µ U ω n.e. on A .

That is, ω
Theorem 2.2 ([26, Sections 3,4]).Given arbitrary ω ∈ M + and A ⊂ R n , the inner balayage ω A , introduced by Definition 2.1, exists and is unique.Furthermore, 4 The same ω A can alternatively be characterized by means of either of the following (equivalent) assertions: (a) ω A is the unique measure in M + satisfying the symmetry relation where σ A denotes the only measure in E ′ (A) with U σ A = U σ n.e. on A. 5   3 See also [31] for an application of this theory to Deny's principle of positivity of mass. 4 As pointed out in [26, Remark 3.12], (2.2) no longer characterizes ω A uniquely (as it does for closed A and ω ∈ E + ).For more details see footnote 5, Corollary 2.4, Remark 2.5, and Theorem 2.6.
5 For any σ ∈ E + and any A ⊂ R n , the measure σ A ∈ E ′ (A) having the property U σ A = U σ n.e. on A, exists and is unique.It is, in fact, the orthogonal projection of σ in the pre-Hilbert space E onto the convex, strongly complete cone E ′ (A); that is (compare with (1.2)), The same σ A is uniquely characterized within M + by the extremal property (2.1) with µ := σ.
(b) ω A is the unique measure in M + satisfying either of the two limit relations where (ω j ) ⊂ E + is an arbitrary sequence having the property 6 U ω j ↑ U ω pointwise on R n as j → ∞, whereas ω A j denotes the only measure in E ′ (A) with U ω A j = U ω j n.e. on A.
Remark 2.3.For signed ω ∈ M, we define the inner balayage ω A by linearity: If moreover the mutual energy I(ω, σ) is well defined for all σ ∈ E, then this ω A is uniquely characterized by the symmetry relation which actually only needs to be verified for certain countably many σ ∈ E, independent of the choice of ω (cf.[30], Theorem 1.4 and Remark 1.4).
• In the rest of this paper, we shall always require A ⊂ R n to have the property 7 (P 1 ) E + (A) is strongly closed.
Then (and only then) E ′ (A) = E + (A), and hence Theorem 2.2 remains valid with E ′ (A) replaced throughout by E + (A).In particular, the following useful corollary holds true.
Corollary 2.4.For this A and for any ω ∈ E + , the inner balayage ω A is, in fact, the orthogonal projection of ω onto the (convex, strongly complete) cone E + (A): The same ω A is uniquely characterized within E + (A) by U ω A = U ω n.e. on A.

An extension of the concept of pseudo-balayage
In what follows, we assume that c * (A) > 0.
Being the difference between two lower semicontinuous (l.s.c.) functions, f is Borel measurable, and, due to (1.8), f is finite q.e. on R n . Let we have − ∞ ŵf (A) 0, (3.3) the upper estimate being caused by the fact that 0 Proof.This follows by standard methods based on the convexity of the class E + f (A) and the parallelogram identity in the pre-Hilbert space E, by use of the strict positive definiteness of the Riesz kernel.(See e.g.[21, Proof of Lemma 6].Note that this proof requires ŵf (A) to be finite, which however necessarily holds whenever ωA exists.)Remark 3.3.If ω is positive, then, as seen from Theorem 2.6, the concept of inner pseudo-balayage ωA extends that of inner balayage ω A (introduced for α 2) to arbitrary α ∈ (0, n).See the present section as well as Section 4 for details.
¶ But if ω is signed, then the inner balayage ω A , defined for α 2 by means of (2.3), may not coincide with the inner pseudo-balayage ωA .(Thus, in case α 2, the theory of inner pseudo-balayage may be thought of as an alternative theory of inner balayage for signed measures, which is not equivalent to the standard one.)Indeed, take , is obviously 0, and so ωA = ω A .Remark 3.4.It follows easily from Definition 3.1 that, if ωA exists, then so does ( cω) A for any c ∈ [0, ∞), and moreover However, this fails to hold if c < 0, cf.Remark 3.3.

3.1.
On the existence of the inner pseudo-balayage.Recall that we are working under the permanent requirements (P 1 ) and (3.1).In the rest of this paper, we also assume that ω ∈ M satisfies either of the following properties:11 (P 2 ) ω ∈ E.
Note that in case (P 2 ), the class E + f (A) actually coincides with the whole of E + (A), cf.(3.13), while in case (P 3 ), it necessarily contains all bounded µ ∈ E + (A).The same ωA can alternatively be characterized by either of the following (a) or (b): (a) ωA is the only measure in ) Proof.Fixing ν ∈ E + f (A), we shall first show that (3.9) and (3.10) hold true for ν in place of ωA if and only if so do (3.7) and (3.8).It is enough to verify only the "if" part of this claim, the opposite being obvious from the fact that any µ-measurable subset of A with c * (•) = 0 is µ-negligible for any µ ∈ E + (A).
The equivalence thereby verified enables us to prove the statement on the uniqueness in each of assertions (a) and (b).Indeed, suppose that (3.9) and (3.10) are fulfilled by some ν, ν ′ ∈ E + (A) in place of ωA .Noting from (3.10) that then necessarily ν, ν ′ ∈ E + f (A), we conclude by applying (3.7) and (3.8) to each of ν and ν ′ that Therefore, whence ν = ν ′ , by virtue of the strict positive definiteness of the α-Riesz kernel.
Case (P 2 ).Assume first that (P 2 ) holds; then the Gauss functional has the form whence Thus the problem on the existence of the inner pseudo-balayage ωA is reduced to that on the existence of the orthogonal projection of ω ∈ E onto E + (A), i.e.
ωA ∈ E + (A) and ω − ωA = min Since the convex cone E + (A) is strongly closed by (P 1 ), hence strongly complete, E + being strongly complete by the perfectness of the α-Riesz kernel, applying [9] (Theorem 1.12.3 and Proposition 1.12.4(2))shows that such an orthogonal projection does exist, and it is uniquely characterized within E + (A) by both (3.7) and (3.8).In view of the equivalence of (a) and (b) proved above, this implies the theorem.
Case (P 3 ).The remaining case (P 3 ) will be treated in four steps.
Step 1.The purpose of this step is to show that the inner pseudo-balayage ωA exists if and only if there exists the (unique) measure µ 0 ∈ E + f (A) satisfying (a) (equivalently, (b)), and then necessarily µ 0 = ωA .
Assume first that ωA exists.To verify (3.9), suppose to the contrary that there is a compact set K ⊂ A with c(K) > 0, such that U ωA < U ω on K.A straightforward verification then shows that for any τ ∈ E + (K), τ = 0, and any t ∈ (0, ∞), As τ < ∞, the value on the right in (3.14) (hence, also that on the left) is < 0 when t > 0 is small enough, which however contradicts Definition 3.1, for ωA + tτ ∈ E + f (A).Having thus established (3.9), we obtain Suppose now that (3.10) fails to hold; then there exists a compact set which again contradicts Definition 3.1 when t is small enough.(Note that, by (3.10), because U ω is upper semicontinuous (u.s.c.) on A by (P 3 ), while U ωA is l.s.c. on R n .)For the "if" part of the claim, assume that (a) holds true for some (unique) To show that then necessarily µ 0 = ωA , we only need to verify that But obviously and relations (3.7) and (3.8) (with µ 0 in place of ωA ) immediately lead to (3.17).
Step 3. Our next aim is to show that the constant L K , satisfying (3.20), can be defined to be independent of K ∈ C A large enough.(To be exact, here and in the sequel K ∈ C A is chosen to follow some K 0 with c(K 0 ) > 0.) By (3.9) and (3.16), U ωK = U ω n.e. on S := S(ω K ), hence γ S -a.e., where γ S denotes the capacitary measure on S (see [10,Theorem 2.5]).Since U γ S 1 holds true n.e. on S, hence ωK -a.e., we get, by Fubini's theorem, As U γ S 1 on S(γ S ), applying the Frostman maximum principle if α 2 (see [17, Theorem 1.10]), or [17, Theorem 1.5] otherwise, gives where We are thus led to the following conclusion, crucial to our proof.¶ We have M A ∈ [0, ∞) and L ∈ [0, ∞) being introduced by (3.5) and (3.21), respectively.The infimum ŵf,L (K) is an actual minimum with the minimizer ωK .
Step 4. To establish the existence of ωA for noncompact A, we first note that the net ŵf,L (K) K∈C A decreases, and moreover, by (3.23), For any compact K, K ′ ⊂ A such that K ⊂ K ′ and c(K) > 0, Applying the parallelogram identity to ωK , ωK ′ ∈ E + we therefore get Noting from (3.24) that the net The cone E + (A) being strongly closed (hence strongly complete) by (P 1 ), there exists , the mapping µ → µ(R n ) being vaguely l.s.c. on M + . 14 We claim that this ζ serves as the inner pseudo-balayage ωA .As shown above (Step 1), this will follow once we verify (3.9) and (3.10) for ζ in place of ωA .
To verify (3.9) (for ζ in place of ωA ), it is enough to do this for any given compact K * ⊂ A. The strong topology on E + being first-countable, in view of (3.26) there is a subsequence (ω K j ) j∈N of the net (ω K ) K∈C A such that K j ⊃ K * for all j, and Passing if necessary to a subsequence and changing the notations, we conclude from (3.27), by use of [10, p. 166, Remark], that Applying now (3.9) to each ωK j , and then letting j → ∞, on account of the countable subadditivity of inner capacity on Borel sets [17, p. 144] we infer from (3.28) that (3.9) (with ζ in place of ωA ) does indeed hold n.e. on K * , whence n.e. on A.
To establish (3.10) for ζ in place of ωA , we note from (3.16) applied to K j that (K j ) being the sequence chosen above.Since (ω K j ) converges to ζ vaguely, see (3.27), for every x ∈ S(ζ) there exist a subsequence (K j k ) of (K j ) and points x j k ∈ S(ω K j k ) such that x j k , k ∈ N, approach x as k → ∞.Thus, by (3.29), Letting here k → ∞, in view of the upper semicontinuity of U ω on A and the lower semicontinuity of the mapping thereby completing the proof of the whole theorem.
14 See [3, Section IV.1, Proposition 4] applied to the (positive, l.s.c.) function 1 on R n .It is useful to point out that in the case where the set in question is compact, the mapping µ → ´g dµ remains vaguely l.s.c. on M + (K) for any l.s.c.function g (not necessarily positive).This follows by replacing g by g ′ := g + c 0, where c ∈ (0, ∞), a l.s.c.function on a compact set being lower bounded, and then by making use of the vague continuity of the mapping µ → µ(K) on M + (K).
Remark 3.6.Assume for a moment that ω is positive, and that A is quasiclosed and Borel.If moreover either ω ∈ E + , or U ω | A is bounded while c * (A) < ∞, 15 then Theorem 3.5 can be deduced from Fuglede's result [12,Theorem 4.10] on the outer pseudo-balayage with respect to a perfect kernel on a locally compact (Hausdorff) space.The methods developed in the above proof are essentially different from those in [12], which enabled us to establish the existence of the inner Riesz pseudo-balayage ωA for pretty general ω and A, see (P 1 ) and (P 2 ), or (P 1 ) and (P 3 ).In this regard, it is also worth noting that the above proof seems to admit a generalization to suitable perfect kernels on locally compact spaces, which we plan to pursue in future work.
It is thus left to consider case (P 2 ).Then ω ∈ E, and therefore ωK , resp.ωA , is the orthogonal projection of ω onto the (convex, strongly complete) cone E + (K), resp.E + (A).A slight modification of the proof of (3.25) shows that Noting that the net ŵf (K) K∈C A is decreasing and, by (3.12), bounded: we conclude from the above that the net (ω K ) K∈C A ⊂ E + (A) is strong Cauchy, and hence converges strongly and vaguely to some (unique) µ 0 ∈ E + (A).This implies K↑A ŵf (K), the former equality being derived from the strong convergence of (ω K ) to µ 0 by use of (3.12).To verify that this µ 0 actually equals ωA , it thus remains to show that But for every µ ∈ E + (A), , where the equality follows by applying [10, Lemma 1.2.2] to each of the positive, l.s.c., µ-integrable functions κ α , U ω + , and U ω − , the set A being µ-measurable.Letting now µ range over E + (A) we get (3.31),thereby completing the proof of the theorem.Corollary 3.8.If U ω is u.s.c. on A (which holds in particular in case (P 3 )), then C n,α being introduced by (3.22).
Proof.In view of the upper semicontinuity of U ω on A, the proof of (3.21), provided in case (P 3 ), remains valid in case (P 2 ) as well.Hence, in both cases (P 2 ) and (P 3 ), which results in (3.32) since the net (ω K ) K∈C A converges vaguely to ωA (Theorem 3.7) while the mapping µ → µ(R n ) is vaguely l.s.c. on M + (cf.footnote 14).

The comparison of the concepts of pseudo-balayage and balayage
The aim of Examples 4.1-4.4below is to demonstrate that usual nice properties of the inner balayage may fail to hold when dealing with the inner pseudo-balayage, and this occurs even in the simplest case of the Dirac measure and a sphere.
¶ The first fact illustrating the difference between these two concepts is that the inner pseudo-balayage may increase the total mass of a positive measure (see e.g.(4.5), pertaining to α > 2).Recall that, if α ∈ (0, 2], then, by [26, Corollary 4.9], (4.1) ¶ The second one is that, for α > 2, there exist a set A ⊂ R n which is not inner α-thin at infinity 16 and a measure µ ∈ M + such that see e.g.(4.5).In contrast to that, not being inner α-thin at infinity is necessary and sufficient for equality to prevail in (4.1) for all µ ∈ M + , see [27,Corollary 5.3].
Example 4.1.Let D ⊂ R n be a bounded (connected, open) domain, ω := ε x 0 , where ε x 0 is the unit Dirac measure at x 0 ∈ D, and let A be the inverse of D \ {x 0 } with respect to the sphere S x 0 ,1 := {|x − x 0 | = 1}.For these A and ω, (P 1 ) and (P 3 ) are fulfilled, and hence the pseudo-balayage εA x 0 exists and is unique (Theorem 3.5).Assume first that α ∈ (0, 2].Since the balayage ε A x 0 of ε x 0 onto A is obviously of finite energy, Theorem 2.6 yields 2) Applying [17, Section IV.5.20] we therefore infer that the pseudo-balayage εA x 0 is actually the Kelvin transform γ * D of the capacitary measure γ D on D with respect to the sphere S x 0 ,1 .Hence, in particular, where ∂A := ∂ R n A. The set A not being α-thin at infinity, we also get, in consequence of (4.2) and [13, Theorem 3.22], Let now α ∈ (2, n).We aim to show that then, in contrast to (4.4), 17 the latter inequality being valid by virtue of (3.32).
x * being the inverse of x with respect to S x 0 ,1 .Combined with (4.6) and (4.
where γ B r is the capacitary measure on the ball B r , r := 1/R, and γ * Br is the Kelvin transform of γ Br with respect to the unit sphere S 1 .Therefore, by symmetry reasons applied to γ Br , εB c R 0 is uniformly distributed over the sphere S R , and such that

S(ε
Thus εS R 0 is uniformly distributed over the sphere S R , and such that Also note that εS R 0 is, in fact, the Kelvin transform of the capacitary measure γ Sr ( = γ Br ), where r := 1/R, with respect to the unit sphere S 1 .

The inner Gauss variational problem
As before, consider a set A ⊂ R n such that (3.1) and (P 1 ) are fulfilled, a (signed) measure ω ∈ M satisfying either (P 2 ) or (P 3 ), and the external field f given by According to Theorem 3.5, the problem of minimizing the Gauss functional I f (µ), is uniquely solvable, and its solution ωA , called the inner pseudo-balayage of ω onto A, is uniquely characterized within E + f (A) by both (3.7) and (3.8) -or, equivalently, by both (3.9) and (3.10).The rest of the paper is to show that the concept of inner pseudo-balayage serves as a powerful tool in the inner Gauss variational problem, which reads as follows.
Remark 5.2.If A = K is compact while f is l.s.c. on K, then the existence of the minimizer λ K,f can easily be verified, by use of the fact that the class Ȇ+ (K) is vaguely compact, cf.[3, Section III.1.9,Corollary 3], while the Gauss functional I f (•) is vaguely l.s.c. on M + (K), the latter being obvious from the principle of descent and the vague lower semicontinuity of the mapping µ → ´f dµ on M + (K) (footnote 14).However, such a proof, based on the vague topology only, is no longer applicable if either A is noncompact, or f is not l.s.c.
To investigate Problem 5.1 in the general case where A is noncompact and/or f is not l.s.c., we have recently developed an approach based on the systematic use of both the strong and vague topologies on the pre-Hilbert space E, which utilized essentially the perfectness of the Riesz kernels, see [33].However, if c * (A) = ∞, then the analysis performed in [33] was only limited to α 2 and ω 0, being mainly based on the theory of inner balayage for positive measures.
Motivated by this observation, we generalize the approach, suggested in [33], to arbitrary α ∈ (0, n) and signed ω, by use of the theory of inner pseudo-balayage, developed in Sections 3, 4 above.The results thereby established are formulated in Section 6, and are proved in Sections 7-9.It is worth emphasizing that those results improve substantially many recent ones from [8,33] (see Section 6.3 for some details).

Preliminary results.
To begin with, observe that under either of assumptions (P 2 ) or (P 3 ), the Gauss functional I f (µ) is finite for all µ ∈ Ȇ+ (A).Thus and therefore which indeed holds true by virtue of (3.1) and the fact that f is finite n.e. on R n .(Here the strengthened version of countable subadditivity for inner capacity has been utilized, see Lemma 2.7.)Finally, combining (5.2) and (3.6) gives w f (A) > −∞.
The solution λ A,f to Problem 5.1 is unique (if it exists), which can be proved by use of the convexity of the class Ȇ+ (A) and the parallelogram identity in the pre-Hilbert space E. Such λ A,f is said to be the inner f -weighted equilibrium measure.
The following theorem, providing characteristic properties of λ A,f , can be derived from the author's earlier paper [21] (see Theorems 1, 2 and Proposition 1 therein).
Theorem 5.4.For λ ∈ Ȇ+ (A) to be the (unique) solution λ A,f to Problem 5.1, it is necessary and sufficient that either of the following two inequalities be fulfilled: where U λ f := U λ + f is said to be the f -weighted potential of λ.If (5.3) or (5.4) holds true, then actually c A,f being referred to as the inner f -weighted equilibrium constant. 18emark 5.5.If f is l.s.c. on A (which occurs e.g. in case (P 3 )), then, by (5.4), which combined with (5.3) gives

On the existence of λ A,f and its properties
In all that follows, except for Corollary 6.8, we assume A and f to satisfy the permanent requirements, reminded at the beginning of Section 5.
Proof.This is obvious since for quasicompact A, both (P 1 ) and (6.1) hold true, by virtue of [33, Theorem 3.9] and [11, Definition 2.1], respectively.Theorem 6.3.For λ A,f to exist, it is sufficient that ωA being the inner pseudo-balayage of ω onto A. Furthermore, then c A,f being the inner f -weighted equilibrium constant.
• On account of Theorem 6.1, in the rest of this section we assume that • Unless (P 2 ) holds, assume additionally that Theorem 6.4.Problem 5.1 is unsolvable whenever Corollary 6.5.If ω + = 0, then Problem 5.1 is always unsolvable.
Corollary 6.6.If U ω is u.s.c. on A, then Problem 5.1 is unsolvable whenever Proof.This follows from Theorem 6.4 by use of (3.32).
Theorem 6.7.Unless (P 2 ) holds, assume U ω is continuous 20 on A and Corollary 6.8.Dropping assumption (6.3) as well as all those imposed on ω, consider signed ω ∈ M, compactly supported in A c .Then λ A,f exists if and only Proof.This follows by combining Theorems 6.1, 6.7 and Corollary 6.6.Remark 6.9.Thus, if ω ∈ M is compactly supported in A c while c * (A) = ∞, then, according to Corollary 6.8, Problem 5.1 is unsolvable whenever ω + (R n ) is small enough, whereas ω − (R n ), the total amount of the negative charge, has no influence on this phenomenon.(Note that, when appealing to the electrostatic interpretation of the problem, the fact just observed agrees with our physical intuition.) 20When speaking of a continuous function, we understand that the values are finite numbers. 21Compare with Theorem 6.3 as well as with Remark 8.3.

6.2.
On the description of the support S(λ A,f ).The following Theorem 6.10 establishes sufficient conditions for the minimizer λ A,f to be of compact support, whereas Examples 6.11 and 6.12 analyze their sharpness.
Theorem 6.10.Under the hypotheses of Theorem 6.7, assume moreover that A is not inner α-thin at infinity.Then S(λ A,f ) is compact whenever (6.7) is fulfilled.Examples 6.11 and 6.12 provide explicit formulae for S(λ A,f ) for some specific A and f .The latter equality in (6.9) as well as both equalities in (6.13) show that Theorem 6.10 would fail in general if ωA (R n ) > 1 were replaced by ωA (R n ) = 1.
In a manner similar to that in Example 4.1, we see that γ * Bz,r being the Kelvin transform of γ Bz,r , the capacitary measure on B z,r , with respect to S 1 .Noting that γ * Bz,r (R n ) = U γ B z,r (0) (cf.(4.9)), whereas U γ B z,r (0) = 1, we obtain (compare with (4.5)).Applying Theorem 6.3 we therefore infer that the solution λ A 0 ,f to Problem 5.1 with A := A 0 and f := −U ε 0 does exist, and moreover λ A 0 ,f = εA 0 0 .(6.12) By virtue of the description of S(γ B z,r ) [17, Section II.3.13],(6.10) and (6.12) yield S(λ A 0 ,f ) = ∂A 0 if α 2, A 0 otherwise.(6.13) 6.3.Remark.The results thereby obtained improve substantially many recent ones from [8,33], by strengthening their formulations and/or by extending the areas of their applications.For instance, [8, Corollary 2.6] only deals with closed sets A that are not thin at infinity, and with external fields f of the form −U ω , where ω := cε x 0 , c ∈ (0, ∞), ε x 0 being the unit Dirac measure at x 0 ∈ A. However, even for these very particular A and ω, all the assertions in [8, Corollary 2.6] are in general weaker than the relevant ones, established above.This is caused, in particular, by the fact that those assertions from [8] are given in terms of ω(R n ), whereas ours -in terms of ωA (R n ) (see Section 4 for the relations between these two values).
Regarding the advantages of our current approach in comparison with that suggested in [33], see Remark 5.2 above.

Proof of Theorem 6.3
Due to condition (6.2), the inner pseudo-balayage ωA , minimizing I f (µ) over the class E + f (A), actually belongs to its proper subclass Ȇ+ (A), see (5.1).Therefore, which combined with (5.2) gives Thus ωA serves as the (unique) solution to Problem 5.1, i.e. ωA = λ A,f .Substituting this equality into (3.8),we get This ξ := ξ A,f will be referred to as the extremal measure (in Problem 5.1).Due to (8.2), we have E + (A) being strongly closed by (P 1 ), and moreover and in the affirmative case λ A,f = ξ A,f .( Proof.The "if" part is evident by (8.3), whereas the opposite is implied by the fact that the trivial net (λ A,f ) is obviously minimizing, and hence converges strongly to both λ A,f and ξ A,f .Since the strong topology on E is Hausdorff, (8.6)By [17,Theorem 2.6] applied to A \ K, K ∈ C being arbitrarily chosen, there exists the (unique) inner capacitary measure γ A\K , minimizing the energy µ 2 over the (convex) set Γ A\K consisting of all µ ∈ E + with U µ 1 n.e. on A \ K.
For any K ′ ∈ C such that K ⊂ K ′ , we have Γ A\K ⊂ Γ A\K ′ , and [17, Lemma 2.2] therefore gives 3), there are mutually nonintersecting, compact sets K j ⊂ A, j ∈ N, such that |x| j for all x ∈ K j and c(K j ) j.If λ j := γ K j /c(K j ) ∈ Ȇ+ (K j ) denotes the normalized capacitary measure on K j , then ) the latter being implied by the fact that for any compact subset K of R n , we have K ∩ S(λ j ) = ∅ for all j large enough.Define Noting from (6.5) that 0 < c j 1 for all j, we get µ j ∈ Ȇ+ (A) for all j, whence where Let first (P 2 ) take place.Applying the Cauchy-Schwarz inequality to the measures ω − , λ j ∈ E + , and then letting j → ∞, we infer from (8.16) that L ∞ = 0. (8.22) Otherwise, (P 3 ) and hence (6.4) must be fulfilled, which again results in (8.22), λ j being the unit measure supported by A ∩ {|x| j}.Substituting (8.22) into (8.21), and then combining the inequality thus obtained with (8.20) and (5.2), we get which shows that the sequence (µ j ) is, in fact, minimizing in Problem 5.1, and hence converges both strongly and vaguely to the extremal measure ξ A,f : µ j → ξ A,f strongly and vaguely in E + as j → ∞.
On account of (8.17)- (8.19), this gives ωA = ξ A,f , the vague topology on M being Hausdorff.Therefore, by (6.5), and an application of Lemma 8.1 shows that Problem 5.1 is indeed unsolvable.
• Unless case (P 2 ) takes place, assume in what follows that the external field f = −U ω is continuous on A, and that (6.6) is fulfilled.Lemma 9.2.For the extremal measure ξ A,f , we have Proof.By virtue of (8.2) and (9.1), λ K,f → ξ A,f strongly and vaguely in E + as K ↑ A, If case (P 2 ) takes place, then the strong convergence of (λ K,f ) K K 0 to ξ A,f yields, by applying (3.12) to each of λ K,f and ξ A,f , whence, by Lemma 9.1, In the remaining case (P 3 ), for any t > 0 choose r so that |f | < t/2 on A ∩ B c r , which is possible in view of (6.6).On account of (8.4), The above r can certainly be chosen so that Furthermore, by combining (9.3) with (9.6), C ξ being the (finite) constant appearing in (9.9).Fix K * ∈ C A .The strong topology on E + being first-countable, one can choose a subsequence (λ K j ,f ) j∈N of the net (λ K,f ) K∈C A such that λ K j ,f → ξ strongly (hence vaguely) in E + as j → ∞. (

9.13)
There is certainly no loss of generality in assuming that K * ⊂ K j for all j, for if not, we replace K j by K ′ j := K j ∪ K * ; then, by the monotonicity of w f (K) , the sequence (λ K ′ j ,f ) j∈N remains minimizing, and hence also converges strongly to ξ. Due to the arbitrary choice of K * ∈ C A , (9.7) will follow once we show that U ξ f C ξ n.e. on K * .(9.14) Passing if necessary to a subsequence and changing the notations, we conclude from (9.13), by virtue of [10,p. 166,Remark], that U ξ = lim j→∞ U λ K j ,f n.e. on R n .(9.15) Applying now (9.10) to each K j , and then letting j → ∞, on account of (9.12) and (9.15) we arrive at (9.14).(Here the countable subadditivity of inner capacity on Borel sets has been utilized.)Since (λ K j ,f ) converges to ξ vaguely, see (9.13), for every x ∈ S(ξ) there exist a subsequence (K j k ) of the sequence (K j ) and points x j k ∈ S(λ K j k ,f ), k ∈ N, such that x j k approach x as k → ∞.Thus, according to (9.11), Letting here k → ∞, and applying (9.12), the continuity of f on A, and the lower semicontinuity of the mapping (x, µ) → U µ (x) on R n ×M + , M + being equipped with the vague topology [10, Lemma 2.2.1(b)], we get the remaining inequality (9.8).9.2.Proof of Theorem 6.7.We first remark from Theorems 6.3 and 6.4 that it is actually enough to consider the case when (6.7) is fulfilled.
Applying Corollary 9.3 we see that under assumption (6.7), λ A,f does indeed exist, and moreover λ A,f = ξ.The equality λ A,f = ξ implies, by use of (5.5), (9.9), and (9.17), that which proves the third relation in (6.8).The first is obvious since, by (6.7), Finally, the first relation implies the second, for if not, then w f (A) = ŵf (A), whence λ A,f = ωA , by the uniqueness of ωA and the inclusion Ȇ+ (A) ⊂ E + f (A), cf.(5.1).9.3.Proof of Theorem 6.10.According to Theorem 6.7, under the stated assumptions there exists the solution λ A,f to Problem 5.1, and moreover, by (6.8), c A,f = 0. (9.18) Assume to the contrary that S(λ A,f ) is noncompact.As seen from Remark 5.5, then there exists a sequence (x j ) ⊂ A such that |x j | → ∞ as j → ∞, and U λ A,f f (x j ) = c A,f for all j ∈ N.
On account of (6.6), this yields lim inf j→∞ U λ A,f (x j ) = c A,f , whence c A,f 0, which in view of (9.18) shows that, actually, However, this is impossible in consequence of [16, Remark 4.12(i)], the set A not being inner α-thin at infinity.

(3. 1 )
Then (and only then) the class E + (A) is not reduced to {0}, see [10, Lemma 2.3.1], and the problems in question become nontrivial.Fix a (signed) measure ω ∈ M (not necessarily of finite energy), and define the external field f : R n → [−∞, ∞] by means of the formula

Example 4 . 2 .
A slight modification of arguments in Example 4.1 shows that if G ⊂ R n is an open, relatively compact set, then for any α ∈ (2, n) and any x 0 ∈ G, 1 < εG c x 0 (R n ) 2 n−α .Example 4.3.Let α ∈ (2, n), A := B c R := {|x| R}, R ∈ (0, ∞), and let ω := ε 0 , where ε 0 denotes the unit Dirac measure at x = 0.As follows from Example 4.1, and let ω := ε 0 .As shown in Example 4.3, by virtue of Theorem 6.3, the solution λ B c R ,f to Problem 5.1 with f := −U ω does exist, and moreover λ B c R ,f = ωB c R = εB c R 0 /q.In view of what was pointed out in Examples 4.1 and 4.3, this gives

8 . 4 8. 1 .
0, which according to Theorem 5.4 establishes the remaining relation c A,f = 0. Proofs of Theorems 6.1 and 6.Extremal measures.Let M f (A) stand for the (nonempty) set of all nets (µ s ) s∈S ⊂ Ȇ+ (A) having the propertylim s∈S I f (µ s ) = w f (A); (8.1)those nets (µ s ) s∈S are said to be minimizing (in Problem 5.1).Using the finiteness of w f (A) (Lemma 5.3), the convexity of Ȇ+ (A), and the perfectness of the α-Riesz kernel, one can see with the aid of arguments similar to those in[33, Lemma 4.2]   that there exists the unique ξ A,f ∈ E + such that, for every (µ s ) s∈S ∈ M f (A), µ s → ξ A,f strongly and vaguely in E + (as s ranges through S).(8.2)

Lemma 8 . 1 .
) the mapping µ → µ(R n ) being vaguely l.s.c. on M + [3, Section IV.1, Proposition 4].The following simple observation is crucial to the proofs given below.Problem 5.1 is solvable if and only if

(
j,K)∈N×C ˆ1A\K dµ j , N × C being the directed product of the directed sets N and C := C R n [15, p. 68].The former relation in (8.5) will therefore follow once we establish the equality lim inf (j,K)∈N×C ˆ1A\K dµ j = 0. (8.10)

9
[26, occurs e.g. if ω itself is of finite energy, or if ω is bounded and meets the separation condition of minimizing the Gauss functional µ 2 − 2 ´Uω dµ, µ ranging over E + (A).Alternatively, ω A is uniquely characterized within E + (A) by U ω A = U ω n.e. on A.Proof.Since ω A = (ω A ) A (see[26, Corollary 4.2]), Corollary 2.4 applied to ω A ∈ E + shows that, indeed, ω A ∈ E + (A), and hence ω A is the orthogonal projection of itself onto E + (A); or equivalently, it is the (unique) solution to the problem of minimizing the functional µ 2 −2 ´Uω A dµ, µ ranging over E + (A).This implies the former part of the claim by noting that ´Uω A dµ = ´Uω dµ for all µ ∈ E + (A), which, in turn, is derived from (2.2) by use of the fact, to be often used in what follows, that any µ-measurable subset of A with c * (•) = 0 is µ-negligible for any µ ∈ E + (A).For the latter part, assume (2.2) holds for some µ 0 ∈ E + (A) in place of ω A .By the strengthened version of countable subadditivity for inner capacity (Lemma 2.7), problem where M + is equipped with the vague topology [10, Lemma 2.2.1(b)], we obtain (3.10) for ζ in place of ωA .