Minimum energy problems with external fields
on locally compact spaces

Natalia Zorii

_In memory of Makoto Ohtsuka (1922–2007)_

Abstract. The paper deals with minimum energy problems in the presence of
external fields on a locally compact space $X$ with respect to a function
kernel $\kappa$ satisfying the energy and consistency principles. For quite a
general (not necessarily lower semicontinuous) external field $f$, we
establish sufficient and/or necessary conditions for the existence of
$\lambda_{A,f}$ minimizing the Gauss functional

$\int\kappa(x,y)\,d(\mu\otimes\mu)(x,y)+2\int f\,d\mu$

over all positive Radon measures $\mu$ with $\mu(X)=1$, concentrated on quite
a general (not necessarily closed or bounded) $A\subset X$, thereby giving an
answer to a question raised by M. Ohtsuka (J. Sci. Hiroshima Univ., 1961).
Such results are specified for the Riesz kernels $|x-y|^{\alpha-n}$,
$0<\alpha<n$, on $\mathbb{R}^{n}$, $n\geqslant 2$, and are illustrated by some
examples. Furthermore, we provide various alternative characterizations of the
minimizer $\lambda_{A,f}$, and as a by-product we analyze the strong and vague
continuity of $\lambda_{A,f}$ under the exhaustion of $A$ by compact $K\subset
A$. The results obtained hold true and are new for many interesting kernels in
classical and modern potential theory.  000 2010 Mathematics Subject
Classification: Primary 31C15. 000 Key words: Radon measures on locally
compact spaces, minimum energy problems, external fields, perfect kernels,
maximum principles, inner capacitary measures, inner balayage.

##  1\. Statement of the problem. Main results

C.F. Gauss investigated the variational problem of minimizing the Newtonian
energy evaluated in the presence of an external field, nowadays called the
Gauss functional (or, in constructive function theory, the weighted energy),
over positive charges $\varphi\,ds$ on the boundary surface of a bounded
domain in $\mathbb{R}^{3}$ (see [17]).

A far-reaching generalization of the original Gauss variational problem,
employing vector-valued Radon measures $\boldsymbol{\mu}=(\mu^{i})_{i\in I}$
on a locally compact space $X$ as charges and replacing the Newtonian kernel
by a function kernel $\kappa$ on $X$, has grown into an eminent branch of
modern potential theory, initiated in the fundamental work by M. Ohtsuka [26].
See e.g. the author’s papers [27]–[30], where vector-valued Radon measures
$\boldsymbol{\mu}$ were even allowed to be infinite dimensional. Regarding the
analytic, constructive, and numerical analysis of the Gauss variational
problem for scalar Borel measures on $\mathbb{R}^{n}$, $n\geqslant 2$, with
respect to the logarithmic or Riesz kernels, see the monographs [2, 25] and
numerous references therein, as well as [10], [18]–[20].

Throughout the present paper, $X$ denotes a locally compact (Hausdorff) space,
$\mathfrak{M}$ the linear space of all (real-valued scalar Radon) measures
$\mu$ on $X$ equipped with the vague (= weak∗) topology of pointwise
convergence on the class $C_{0}(X)$ of all continuous functions
$\varphi:X\to\mathbb{R}$ of compact support, and $\mathfrak{M}^{+}$ the cone
of all positive $\mu\in\mathfrak{M}$, where $\mu\in\mathfrak{M}$ is positive
if and only if $\mu(\varphi)\geqslant 0$ for all positive $\varphi\in
C_{0}(X)$.

A kernel $\kappa$ on $X$ is meant to be a symmetric function from
$\Phi(X\times X)$, where $\Phi(Y)$ consists of all lower semicontinuous
(l.s.c.) functions $g:Y\to(-\infty,\infty]$ such that $g\geqslant 0$ unless
the topological space $Y$ is compact. Then the energy

$I(\mu):=\int\kappa(x,y)\,d(\mu\otimes\mu)(x,y),\quad\mu\in\mathfrak{M},$

with respect to the kernel $\kappa$ is well defined (as a finite number or
$+\infty$) on all of $\mathfrak{M}^{+}$, and represents there a vaguely l.s.c.
function (see Section 2 below for more details). We denote by
$\mathcal{E}^{+}$ the set of all $\mu\in\mathfrak{M}^{+}$ with
$I(\mu)<\infty$.

For any $A\subset X$, denote by $\mathcal{E}^{+}(A)$ the class of all
$\mu\in\mathcal{E}^{+}$ concentrated on $A$ [4, Section V.5.7], and by
$\breve{\mathcal{E}}^{+}(A)$ its subclass consisting of all $\mu$ with
$\mu(X)=1$.

###  1.1. Statement of the problem

Fix a universally measurable function $f:X\to[-\infty,\infty]$, to be treated
as an external field acting on charges (measures) on $X$. Given $A\subset X$,
let $\breve{\mathcal{E}}^{+}_{f}(A)$ stand for the class of all
$\mu\in\breve{\mathcal{E}}^{+}(A)$ such that $f$ is $\mu$-integrable [4]
(Chapter IV, Sections 3, 4). Then the $f$-weighted energy (= the Gauss
functional)

$I_{f}(\mu):=I(\mu)+2\int f\,d\mu$ (1.1)

is well defined and finite for all $\mu\in\breve{\mathcal{E}}^{+}_{f}(A)$, and
one can introduce the extremal value111As usual, the infimum over the empty
set is interpreted as $+\infty$. We also agree that $1/(+\infty)=0$ and
$1/0=+\infty$.

$w_{f}(A):=\inf_{\mu\in\breve{\mathcal{E}}^{+}_{f}(A)}\,I_{f}(\mu)\in[-\infty,\infty].$
(1.2)

In the remainder of the present section, we shall always assume that

$-\infty<w_{f}(A)<\infty.$ (1.3)

(See Lemmas 3.2, 3.3 and Corollary 3.4 for necessary and/or sufficient
conditions for this to hold.) Then
$\breve{\mathcal{E}}^{+}_{f}(A)\neq\varnothing$, and hence the following
problem makes sense.

######  Problem 1.1.

Does there exist $\lambda=\lambda_{A,f}\in\breve{\mathcal{E}}^{+}_{f}(A)$ with

$I_{f}(\lambda_{A,f})=w_{f}(A)?$

Problem 1.1 is often referred to as the inner Gauss variational problem [26].
We call its solutions $\lambda_{A,f}$ (if they exist) the inner $f$-weighted
equilibrium measures of $A$.

Assume for a moment that $A=K\subset X$ is compact, and that $f\in\Phi(X)$.
Then the solutions $\lambda_{K,f}$ do exist,222In general, such
$\lambda_{K,f}$ are not unique (unless of course the kernel $\kappa$ satisfies
the energy principle). which follows easily from the vague compactness of the
class of admissible measures, and the vague lower semicontinuity of the
$f$-weighted energy $I_{f}(\mu)$ on $\mathfrak{M}^{+}$. However, this fails to
hold if either $A$ is noncompact or $f\not\in\Phi(X)$, and then Problem 1.1
becomes "rather difficult" (Ohtsuka [26, Section 2.2]).

In the remainder of this subsection as well as throughout Section 1.2, we
impose on $\kappa$ the following permanent requirement:

  * $(H_{1})$

The kernel $\kappa$ is perfect, or equivalently it satisfies the energy and
consistency principles (see [13], cf. Section 2.2 below).

Then all (signed) measures $\mu\in\mathfrak{M}$ with $I(\mu)<\infty$ form a
pre-Hilbert space $\mathcal{E}$ with the inner product
$\langle\mu,\nu\rangle:=\int\kappa(x,y)\,d(\mu\otimes\nu)(x,y)$ and the energy
norm $\|\mu\|:=\sqrt{I(\mu)}$. Moreover, the cone $\mathcal{E}^{+}$ then
becomes complete in the strong topology, determined by this norm, and the
strong topology on $\mathcal{E}^{+}$ is finer than the (induced) vague
topology.

These facts made it possible to develop the theory of inner capacitary
measures as well as that of inner balayage, the latter however additionally
requiring the (perfect) kernel $\kappa$ to satisfy the domination principle
(B. Fuglede [13] and N. Zorii [33]–[36], cf. Sections 2.4, 2.5 below). Along
with the advantages of using perfect kernels, these two theories are crucial
to the analysis of Problem 1.1, performed in the present work.

###  1.2. Main results

Maintaining the interaction between the strong and the vague topologies on the
(strongly complete) cone $\mathcal{E}^{+}$ as the main tool in the present
study, we obtain sufficient and/or necessary conditions for the solvability of
Problem 1.1 for noncompact (and even nonclosed) sets $A\subset X$ and for
quite general (not necessarily lower semicontinuous) external fields $f$,
thereby giving an answer to the above-quoted question by Ohtsuka [26, Section
2.2] (see Theorems 1.2, 1.5, 1.10 and Corollaries 1.3, 1.8 below).
Furthermore, we establish various alternative characterizations of the
solution $\lambda_{A,f}$ to Problem 1.1 (see Theorems 1.2, 1.5), and as a by-
product we prove its strong and vague continuity under the exhaustion of $A$
by compact $K\subset A$ (Theorem 1.4), thereby justifying the term "inner
$f$-weighted equilibrium measure".

For this purpose, we impose on $f$ the following permanent requirement:

  * $(H_{2})$

The external field $f$ is representable in the form333In the recent works on
Problem 1.1, mainly dealing with the logarithmic or Riesz kernels on
$\mathbb{R}^{n}$ (see [2, 10, 25] and references therein), the external field
$f$ is always required to be of the class $\Phi(\mathbb{R}^{n})$, whereas the
presence of an alternative/additional source of energy, generated by a signed
charge $\vartheta\in\mathcal{E}$, cf. (1.4) or (1.5), agrees well with the
original electrostatic nature of the problem.

$f=\psi+U^{\vartheta},\text{ \ where $\psi\in\Phi(X)$ and
$\vartheta\in\mathcal{E}$,}$ (1.4)

$U^{\vartheta}(\cdot):=\int\kappa(\cdot,y)\,d\vartheta(y)$ being the potential
with respect to the kernel $\kappa$.

(As shown in Lemma 3.3 below, assumption (1.3) is then equivalent to
$w_{f}(A)<\infty$, which in turn holds if and only if ${\rm
cap}_{*}\bigl{(}\\{x\in A:\ \psi(x)<\infty\\}\bigr{)}>0$. Here and in the
sequel, ${\rm cap}_{*}(\cdot)$ denotes the inner capacity of a set with
respect to the kernel $\kappa$.)

Along with $(H_{1})$ and $(H_{2})$, some/all of the following three hypotheses
on $\kappa$, $A$, and $f$ will often be required:

  * $(H_{3})$

The class $\mathcal{E}^{+}(A)$ is closed in the strong topology on
$\mathcal{E}^{+}$. (This in particular occurs if $A\subset X$ is quasiclosed
(quasicompact), that is, if $A$ can be approximated in outer capacity by
closed (compact) sets [14]; see Section 2.3 below.)

  * $(H_{4})$

The (perfect) kernel $\kappa$ satisfies the first and the second maximum
principles (for definitions see Section 2.2).

  * $(H_{2}^{\prime})$

The following particular case of (1.4) takes place:

$f=-U^{\zeta},\text{ \ where $\zeta\in\mathcal{E}^{+}$ and $\zeta(X)\leqslant
1$}.$ (1.5)

######  Theorem 1.2.

Assume that $(H_{1})$–$(H_{3})$ are fulfilled, and that

${\rm cap}_{*}(A)<\infty.$ (1.6)

Then the solution $\lambda=\lambda_{A,f}$ to Problem 1.1 does exist.(444The
solution to Problem 1.1 is unique (if it exists), which follows easily from
the convexity of the class of admissible measures and the energy principle
(see [27, Lemma 6], cf. Section 3 below).)(555$\lambda_{A,f}$ would not
necessarily exist if $(H_{3})$ were omitted from the hypotheses (cf. footnote
19, pertaining to the unweighted case $f=0$ and the Newtonian kernel
$|x-y|^{2-n}$ on $\mathbb{R}^{n}$, $n\geqslant 3$).) Furthermore,
$\lambda_{A,f}$ is uniquely determined within $\breve{\mathcal{E}}^{+}_{f}(A)$
by either of the two characteristic properties666For more details about such
characteristic inequalities for $\lambda_{A,f}$, see the author’s earlier
paper [27] (cf. Theorems 3.5 and 3.6 below). As seen from there, the latter
part of Theorem 1.2 actually holds true under much more general assumptions
than stated.

$\displaystyle U_{f}^{\lambda}$ $\displaystyle\geqslant c_{A,f}\text{ \ n.e.\
on\/ $A$},$ $\displaystyle U_{f}^{\lambda}$ $\displaystyle\leqslant
c_{A,f}\text{ \ $\lambda$-a.e.\ on\/ $X$},$

where $U_{f}^{\lambda}:=U^{\lambda}+f$ denotes the $f$-weighted potential of
$\lambda$, and

$c_{A,f}:=\int U_{f}^{\lambda}\,d\lambda=w_{f}(A)-\int
f\,d\lambda\in(-\infty,\infty)$ (1.7)

is said to be the inner $f$-weighted equilibrium constant for the set $A$.

Here and in the sequel, the abbreviation n.e. (nearly everywhere) means, as
usual, that the set of all $x\in A$ where the inequality fails is of inner
capacity zero.

######  Corollary 1.3.

Under (permanent) requirements $(H_{1})$ and $(H_{2})$, Problem 1.1 is
(uniquely) solvable for any quasicompact $A\subset X$.

###### Proof.

For quasicompact $A$, (1.6) necessarily holds, the capacity of a compact set
with respect to a strictly positive definite kernel being finite. Since
$(H_{3})$ is fulfilled as well (see Theorem 2.13 below), the corollary follows
directly from Theorem 1.2. ∎

Given $A\subset X$, denote by $\mathfrak{C}_{A}$ the upward directed set of
all compact subsets $K$ of $A$, where $K_{1}\leqslant K_{2}$ if and only if
$K_{1}\subset K_{2}$. If a net $(x_{K})_{K\in\mathfrak{C}_{A}}\subset Y$
converges to $x_{0}\in Y$, $Y$ being a topological space, then we shall
indicate this fact by writing

$x_{K}\to x_{0}\text{ \ in $Y$ as $K\uparrow A$}.$

######  Theorem 1.4.

Under (permanent) requirements $(H_{1})$ and $(H_{2})$, assume that the
solution $\lambda_{A,f}$ to Problem 1.1 exists.777See Theorems 1.2, 1.5, 1.10
and Corollaries 1.3, 1.8 for sufficient conditions for this to occur. Then

$\lambda_{K,f}\to\lambda_{A,f}\text{ \ strongly and vaguely in\/
$\mathcal{E}^{+}$ as\/ $K\uparrow A$},$ (1.8)

$\lambda_{K,f}$ being the solution to Problem 1.1 with $A:=K$ (such solutions
$\lambda_{K,f}$ do exist for all $K\in\mathfrak{C}_{A}$ large enough). If
moreover $f=U^{\vartheta}$ with $\vartheta\in\mathcal{E}$, then also

$\lim_{K\uparrow A}\,c_{K,f}=c_{A,f},$ (1.9)

$c_{A,f}$ and $c_{K,f}$ being introduced by formula (1.7) applied to $A$ and
$K$, respectively.

######  Theorem 1.5.

Let $(H_{1})$, $(H_{3})$, $(H_{4})$, and $(H_{2}^{\prime})$ be fulfilled, and
let $\zeta^{A}$ denote the inner balayage of $\zeta$ onto $A$, the measure
$\zeta$ appearing in $(H_{2}^{\prime})$. If moreover888Under hypotheses
$(H_{1})$, $(H_{3})$, and $(H_{4})$, assumption $\zeta^{A}(X)=1$ is fulfilled,
for instance, if $\zeta\in\mathcal{E}^{+}$ is a measure of unit total mass
concentrated on $A$, i.e. $\zeta\in\breve{\mathcal{E}}^{+}(A)$. Yet another
possibility, pertaining to the $\alpha$-Riesz kernels $|x-y|^{\alpha-n}$ on
$\mathbb{R}^{n}$, $n\geqslant 2$, where $0<\alpha<n$ and $\alpha\leqslant 2$,
requires $\zeta\in\mathcal{E}^{+}$ to be an arbitrary measure on
$\mathbb{R}^{n}$ with $\zeta(\mathbb{R}^{n})=1$, and $A$ to be not inner
$\alpha$-thin at infinity, which according to [22, 32] means that
$\sum_{k\in\mathbb{N}}\,\frac{{\rm cap}_{*}(A_{k})}{q^{k(n-\alpha)}}=\infty,$
where $q\in(1,\infty)$ and $A_{k}:=A\cap\\{x\in\mathbb{R}^{n}:\
q^{k}\leqslant|x|<q^{k+1}\\}$. For the latter, see [32, Corollary 5.3].

either ${\rm cap}_{*}(A)<\infty$, or $\zeta^{A}(X)=1$, (1.10)

then, and only then, the solution $\lambda_{A,f}$ to Problem 1.1 does exist,
and it has the form999Representation (1.11) is particularly useful in
applications. For instance, for the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$
on $\mathbb{R}^{n}$, $n\geqslant 2$, of order $0<\alpha\leqslant 2$,
$\alpha<n$, it enables us to give an answer to Open question 2.1 raised by
Ohtsuka in [26, Section 2.12]. In view of [31, Theorem 8.5] and [32, Theorem
4.2, Corollary 5.4], such answer is in the positive if $\alpha<2$, and it is
in the negative otherwise.

$\lambda_{A,f}=\left\\{\begin{array}[]{cl}\zeta^{A}+\eta_{A,f}\gamma_{A}&\text{if
\ ${\rm cap}_{*}(A)<\infty$},\\\ \zeta^{A}&\text{otherwise},\\\
\end{array}\right.$ (1.11)

where $\gamma_{A}$ denotes the inner capacitary measure on $A$, while

$\eta_{A,f}:=\frac{1-\zeta^{A}(X)}{{\rm cap}_{*}(A)}\in[0,\infty).$ (1.12)

Furthermore, the above $\lambda_{A,f}$ can alternatively be characterized by
means of any one of the following three assertions:

  * (i)

$\lambda_{A,f}$ is the unique measure of minimum energy norm in the class

$\Lambda_{A,f}:=\bigl{\\{}\mu\in\mathcal{E}^{+}:\
U^{\mu}_{f}\geqslant\eta_{A,f}\text{ \ n.e.\ on\ $A$}\bigr{\\}},$ (1.13)

$\eta_{A,f}$ being introduced by formula (1.12). That is,
$\lambda_{A,f}\in\Lambda_{A,f}$ and

$\|\lambda_{A,f}\|=\min_{\mu\in\Lambda_{A,f}}\,\|\mu\|.$

  * (ii)

$\lambda_{A,f}$ is the unique measure of minimum potential in the class
$\Lambda_{A,f}$, introduced by means of (1.13). That is,
$\lambda_{A,f}\in\Lambda_{A,f}$ and101010This implies immediately that
$\lambda_{A,f}$ can also be characterized as the unique measure of minimum
$f$-weighted potential in the class $\Lambda_{A,f}$ — now, however, nearly
everywhere on $X$:
$U^{\lambda_{A,f}}_{f}=\min_{\mu\in\Lambda_{A,f}}\,U^{\mu}_{f}\text{ \ n.e.\
on\ $X$}\vspace{-5mm}.$

$U^{\lambda_{A,f}}=\min_{\mu\in\Lambda_{A,f}}\,U^{\mu}\text{ \ on\ $X$}.$

  * (iii)

$\lambda_{A,f}$ is the only measure in $\mathcal{E}^{+}(A)$ having the
property

$U^{\lambda_{A,f}}_{f}=\eta_{A,f}\text{ \ n.e.\ on\ $A$}.$

In addition, the inner $f$-weighted equilibrium constant $c_{A,f}$, introduced
by (1.7), admits an alternative representation

$c_{A,f}=\eta_{A,f},$ (1.14)

and hence (1.9) can be specified as follows:

$c_{K,f}\downarrow c_{A,f}\text{ \ in $\mathbb{R}$ as $K\uparrow A$}.$ (1.15)

######  Corollary 1.6.

Under the assumptions of Theorem 1.5, if moreover $X$ is
$\sigma$-compact,111111A locally compact space is said to be $\sigma$-compact
if it is representable as a countable union of compact sets [3, Section I.9,
Definition 5]. then $\lambda_{A,f}$ is a measure of minimum total mass in the
class $\Lambda_{A,f}$, i.e.

$\lambda_{A,f}(X)=\min_{\mu\in\Lambda_{A,f}}\,\mu(X)\quad\bigl{(}{}=1\bigr{)}.$
(1.16)

######  Remark 1.7.

However, extremal property (1.16) cannot serve as an alternative
characterization of the inner $f$-weighted equilibrium measure
$\lambda_{A,f}$, for it does not determine $\lambda_{A,f}$ uniquely within
$\Lambda_{A,f}$. Indeed, consider the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$
of order $\alpha\leqslant 2$, $\alpha<n$, on $\mathbb{R}^{n}$, $n\geqslant 2$,
a proper, closed subset $A$ of $\mathbb{R}^{n}$ that is not $\alpha$-thin at
infinity (take, for instance, $A:=\\{|x|\geqslant 1\\}$), and let $f$ be given
by (1.5) with $\zeta\in\breve{\mathcal{E}}^{+}(\mathbb{R}^{n}\setminus A)$.
Applying [32, Corollary 5.3] we get

$\zeta^{A}(\mathbb{R}^{n})=\zeta(\mathbb{R}^{n})=1.$ (1.17)

Noting that then $\eta_{A,f}=0$, and hence $\zeta,\zeta^{A}\in\Lambda_{A,f}$
(cf. (2.11)), we conclude from (1.17) that there are actually infinitely many
measures of minimum total mass in $\Lambda_{A,f}$, for so is every measure of
the form $a\zeta+b\zeta^{A}$, where $a,b\in[0,1]$ and $a+b=1$.

Let hypotheses $(H_{1})$, $(H_{3})$, $(H_{4})$, and $(H_{2}^{\prime})$ be
fulfilled; and let ${\rm cap}_{*}(A)=\infty$, for if not, the solution
$\lambda_{A,f}$ to Problem 1.1 does exist by Theorem 1.2 or 1.5. Then the
following corollary to Theorem 1.5 holds, where $\zeta$ is the measure
appearing in (1.5).

######  Corollary 1.8.

The minimizer $\lambda_{A,f}$ does not exist if $\zeta(X)<1$, and it exists if
$\zeta$ is a measure of unit total mass concentrated on $A$.121212The latter
assertion would fail in general if $\zeta$ were not required to be
concentrated on $A$. See Theorem 1.10(e); for illustration, see the last claim
in Example 1.11 (pertaining to the set $F_{2}$). In the latter case,
$\lambda_{A,f}=\zeta$.

######  Remark 1.9.

All the above-quoted results hold true and are new for the Green kernels
associated with the Laplacian on Greenian sets in $\mathbb{R}^{n}$,
$n\geqslant 2$ (thus in particular for the Newtonian kernel $|x-y|^{2-n}$ on
$\mathbb{R}^{n}$, $n\geqslant 3$), as well as for the $\alpha$-Riesz kernels
$|x-y|^{\alpha-n}$ and the associated $\alpha$-Green kernels on
$\mathbb{R}^{n}$, where $0<\alpha<2\leqslant n$. Furthermore, Theorems 1.2,
1.4 and Corollary 1.3 also hold true and are new for the $\alpha$-Riesz
kernels of order $2<\alpha<n$ as well as for the logarithmic kernel
$-\log\,|x-y|$ on a closed disc in $\mathbb{R}^{2}$ of radius ${}<1$, and for
the Deny kernels on $\mathbb{R}^{n}$, defined with the aid of Fourier
transformation (see condition $(A)$ in [7, Section 1], cf. [23, Section
VI.1.2]). For more details, see Example 2.11 below.

###  1.3. Applications to the Riesz kernels

Let $X:=\mathbb{R}^{n}$, $n\geqslant 2$, and let $\kappa(x,y)$ be the
$\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order $0<\alpha<n$. Then the
above-quoted results on the (un)solvability of Problem 1.1 can be specified as
follows.

######  Theorem 1.10.

Assume that an external field $f$ is of form (1.4), that
$A\subset\mathbb{R}^{n}$ is quasiclosed (or, more generally, that
$\mathcal{E}^{+}(A)$ is strongly closed), and that $w_{f}(A)<\infty$.

  * (a)

If moreover ${\rm cap}_{*}(A)<\infty$, then the solution $\lambda_{A,f}$ to
Problem 1.1 does exist.

In the remaining case ${\rm cap}_{*}(A)=\infty$, assume $\alpha\leqslant 2$,
and consider $f$ of form (1.5). Then the following (b)–(e) hold true, where
$\zeta$ is the measure appearing in (1.5).

  * (b)

$\lambda_{A,f}$ exists if and only if $\zeta^{A}(\mathbb{R}^{n})=1$, and in
the affirmative case $\lambda_{A,f}=\zeta^{A}$.

  * (c)

$\lambda_{A,f}$ does not exist if $\zeta(\mathbb{R}^{n})<1$, and it exists if
$\zeta$ is a measure of unit total mass concentrated on $A$. In the latter
case, $\lambda_{A,f}=\zeta$.

  * (d)

If $A$ is not inner $\alpha$-thin at infinity, then $\lambda_{A,f}$ exists if
and only if $\zeta(\mathbb{R}^{n})=1$, and in the affirmative case
$\lambda_{A,f}=\zeta^{A}$.131313If $A$ is not inner $\alpha$-thin at infinity,
then the requirement ${\rm cap}_{*}(A)=\infty$ does hold automatically, cf.
[32, Section 2], and hence can be dropped.

  * (e)

Assume that $\overline{A}:={\rm Cl}_{\mathbb{R}^{n}}A$ is $\alpha$-thin at
infinity, and that $D:=\mathbb{R}^{n}\setminus\overline{A}$ is connected
unless $\alpha<2$. Then $\lambda_{A,f}$ does not exist whenever
$\zeta(D)>0$.141414Actually, we require $D$ to be connected (when $\alpha=2$)
only in order to simplify the formulation. Compare with [31] (Lemma 7.1 and
Theorems 7.2, 8.7).

The above results on the solvability of Problem 1.1 for the $\alpha$-Riesz
kernels, included in the present work rather for illustration purposes, can
actually be much strengthened, which we plan to pursue in a subsequent paper.

######  Example 1.11.

On $\mathbb{R}^{3}$, consider the kernel $1/|x-y|$ and the rotation bodies

$F_{i}:=\bigl{\\{}x\in\mathbb{R}^{3}:\ 0\leqslant x_{1}<\infty,\
x_{2}^{2}+x_{3}^{2}\leqslant\varrho_{i}^{2}(x_{1})\bigr{\\}},\ i=1,2,3,$

where

$\displaystyle\varrho_{1}(x_{1})$ $\displaystyle:=x_{1}^{-s}\text{ \ with\
}s\in[0,\infty),$ $\displaystyle\varrho_{2}(x_{1})$
$\displaystyle:=\exp(-x_{1}^{s})\text{ \ with\ }s\in(0,1],$
$\displaystyle\varrho_{3}(x_{1})$ $\displaystyle:=\exp(-x_{1}^{s})\text{ \
with\ }s\in(1,\infty).$

As can be derived from estimates in [23, Section V.1, Example], $F_{1}$ is not
$2$-thin at infinity, $F_{2}$ is $2$-thin at infinity, though has infinite
Newtonian capacity, whereas $F_{3}$ is of finite Newtonian capacity.
Therefore, by Theorem 1.10, $\lambda_{F_{3},f}$ exists for any $f$ of form
(1.4), see (a). Let now $f$ be of form (1.5) with the measure $\zeta$
involved. Then $\lambda_{F_{1},f}$ exists if and only if
$\zeta(\mathbb{R}^{n})=1$, see (d); whereas $\lambda_{F_{2},f}$ exists if and
only if both $\zeta(\mathbb{R}^{n})=1$ and $S(\zeta)\subset F_{2}$ hold true,
see (c) and (e). (Here $S(\zeta)$ is the support of $\zeta$.)

Figure 1. The set $F_{1}$ in Example 1.11 with $\varrho_{1}(x_{1})=1/x_{1}$.
Figure 2. The set $F_{2}$ in Example 1.11 with
$\varrho_{2}(x_{1})=\exp(-x_{1})$.

The rest of the paper is organized as follows. For the convenience of the
reader, in Section 2 we review some basic facts of the theory of potentials on
locally compact spaces, the main emphasis being placed on the theory of inner
capacitary measures and that of inner balayage. Section 3 provides preliminary
results on Problem 1.1, and Section 4 gives proofs to Theorems 1.2, 1.4, 1.5,
1.10 and Corollaries 1.6, 1.8.

##  2\. On the theory of potentials on locally compact spaces

Throughout the rest of the paper, we shall use the notations and conventions
introduced in Section 1.

For the theory of measures and integration on a locally compact (Hausdorff)
space $X$, we refer to N. Bourbaki [4]. A comprehensive application of this
theory to the theory of potentials on $X$ with respect to function kernels
satisfying the principle of consistency, is presented in the pioneering paper
by Fuglede [13].

######  Lemma 2.1 ([4, Section IV.1, Proposition 4]).

For any l.s.c. function $g:X\to[0,\infty]$, the mapping $\mu\mapsto\int
g\,d\mu$ is vaguely l.s.c. on $\mathfrak{M}^{+}$, the integral here being
understood as an upper integral.

For an arbitrary set $A\subset X$, denote by $\mathfrak{M}^{+}(A)$ the cone of
all $\mu\in\mathfrak{M}^{+}$ concentrated on $A$, which means that
$A^{c}:=X\setminus A$ is locally $\mu$-negligible, or equivalently that $A$ is
$\mu$-measurable and $\mu=\mu|_{A}$, $\mu|_{A}:=1_{A}\cdot\mu$ being the trace
of $\mu$ to $A$ [12, Section IV.14.7]. (Note that for
$\mu\in\mathfrak{M}^{+}(A)$, the indicator function $1_{A}$ of $A$ is locally
$\mu$-integrable.) The total mass of $\mu\in\mathfrak{M}^{+}(A)$ is
$\mu(X)=\mu_{*}(A)$, $\mu_{*}(A)$ and $\mu^{*}(A)$ denoting the inner and the
outer measure of $A$, respectively. If moreover $A$ is closed, or if $A^{c}$
is contained in a countable union of sets $Q_{j}$ with
$\mu^{*}(Q_{j})<\infty$,151515If the latter holds, $A^{c}$ is said to be
$\mu$-$\sigma$-finite [12, Section IV.7.3]. This in particular occurs if the
measure $\mu$ is bounded (that is, with $\mu(X)<\infty$), or if the locally
compact space $X$ is $\sigma$-compact. then for any
$\mu\in\mathfrak{M}^{+}(A)$, $A^{c}$ is $\mu$-negligible, that is,
$\mu^{*}(A^{c})=0$. In particular, if $A$ is closed, $\mathfrak{M}^{+}(A)$
consists of all $\mu\in\mathfrak{M}^{+}$ with support $S(\mu)\subset A$, cf.
[4, Section IV.2.2].

######  Lemma 2.2 (cf. [13, Lemma 1.2.2]).

For any l.s.c. function $g:X\to[0,\infty]$, any measure
$\mu\in\mathfrak{M}^{+}$, and any $\mu$-measurable set $A\subset X$,

$\int g\,d\mu|_{A}=\lim_{K\uparrow A}\,\int g\,d\mu|_{K}.$

A kernel on $X$ is thought of as a symmetric function $\kappa\in\Phi(X\times
X)$ (Section 1); thus, either $\kappa(x,y)$ is ${}\geqslant 0$ for all
$(x,y)\in X\times X$, or the space $X$ is compact.

######  Remark 2.3.

The case of compact $X$ can mostly be reduced to that of $\kappa\geqslant 0$
by replacing the kernel $\kappa$ by $\kappa^{\prime}:=\kappa+q\geqslant 0$,
where $q\in[0,\infty)$, which is always possible since a lower semicontinuous
function on a compact space is lower bounded.

######  Remark 2.4.

For similar reasons, Lemmas 2.1 and 2.2 actually hold true for any
$g\in\Phi(X)$ — even if $g\ngeqslant 0$. Indeed, then $X$ must be compact, and
hence $g$ can be replaced by $g^{\prime}:=g+q\geqslant 0$, where
$q\in[0,\infty)$. For Lemma 2.1, use the vague continuity of the mapping
$\mu\mapsto\mu(X)$ on $\mathfrak{M}^{+}$, the space $X$ being compact, while
for Lemma 2.2, use the fact that for any $\mu\in\mathfrak{M}^{+}$ and any
$\mu$-measurable set $A\subset X$,

$\lim_{K\uparrow A}\,\mu|_{K}(X)=\mu|_{A}(X)<\infty.$

Given (signed) $\mu,\nu\in\mathfrak{M}$, define the potential and the mutual
energy by

$\displaystyle U^{\mu}(x):=\int\kappa(x,y)\,d\mu(y),\quad x\in X,$
$\displaystyle I(\mu,\nu):=\int\kappa(x,y)\,d(\mu\otimes\nu)(x,y),$

respectively, provided the right-hand side is well defined as a finite number
or $\pm\infty$ (for more details see [13, Section 2.1]). For $\mu=\nu$,
$I(\mu,\nu)$ defines the energy $I(\mu):=I(\mu,\mu)$ of $\mu\in\mathfrak{M}$.
In particular, if $\mu,\nu\geqslant 0$, then $U^{\mu}(x)$, resp. $I(\mu,\nu)$,
is always well defined and represents a l.s.c. function of $(x,\mu)\in
X\times\mathfrak{M}^{+}$, resp. of
$(\mu,\nu)\in\mathfrak{M}^{+}\times\mathfrak{M}^{+}$ (the principle of descent
[13, Lemma 2.2.1], cf. Lemma 2.1 and Remark 2.4). Thus

$U^{\mu}\in\Phi(X)\text{ \ for all $\mu\in\mathfrak{M}^{+}$}.$ (2.1)

######  Lemma 2.5.

For any $A\subset X$ and any $\mu\in\mathfrak{M}^{+}(A)$,

$I(\mu)=\lim_{K\uparrow A}\,I(\mu|_{K}).$

###### Proof.

This is seen from Lemma 2.2 (cf. Remark 2.4) in view of the fact that for any
$\mu$-measurable set $Q\subset X$, $(\mu\otimes\mu)|_{Q\times
Q}=\mu|_{Q}\otimes\mu|_{Q}$, cf. [13, Lemma 1.2.5]. ∎

In what follows, we always assume a kernel $\kappa$ to satisfy the energy
principle (or equivalently, to be strictly positive definite), which means
that $I(\mu)\geqslant 0$ for all (signed) $\mu\in\mathfrak{M}$ whenever
$I(\mu)$ is well defined, and moreover $I(\mu)=0$ only for zero measure. Then
all $\mu\in\mathfrak{M}$ with $I(\mu)<\infty$ form a pre-Hilbert space
$\mathcal{E}$ with the inner product $\langle\mu,\nu\rangle:=I(\mu,\nu)$ and
the energy norm $\|\mu\|:=\sqrt{I(\mu)}$ [13, Section 3.1]. The topology on
$\mathcal{E}$ determined by this norm is said to be strong.

###  2.1. Capacities of a set

For any $A\subset X$, denote

$\displaystyle\breve{\mathfrak{M}}^{+}(A):=\bigl{\\{}\mu\in\mathfrak{M}^{+}(A):\
\mu(X)=1\bigr{\\}},$
$\displaystyle\mathcal{E}^{+}(A):=\mathcal{E}\cap\mathfrak{M}^{+}(A),\quad\breve{\mathcal{E}}^{+}(A):=\mathcal{E}\cap\breve{\mathfrak{M}}^{+}(A),$
$\displaystyle
w(A):=\inf_{\mu\in\breve{\mathcal{E}}^{+}(A)}\,I(\mu)\in[0,\infty].$ (2.2)

The value

${\rm cap}_{*}(A):=1/w(A)\in[0,\infty]$

is said to be the (Wiener) inner capacity of the set $A$ (with respect to the
kernel $\kappa$).

It is seen from Lemma 2.5 that ${\rm cap}_{*}(A)$ would be the same if the
admissible measures in (2.2) were required to be of compact support. This in
turn yields161616We write ${\rm cap}(A)$ in place of ${\rm cap}_{*}(A)$ if $A$
is capacitable, that is, if ${\rm cap}_{*}(A)={\rm cap}^{*}(A)$, where ${\rm
cap}^{*}(A)$ is the outer capacity of $A$, defined as $\inf\,{\rm
cap}_{*}(D)$, $D$ ranging over all open sets containing $A$. This occurs, for
instance, if $A$ is compact [13, Lemma 2.3.4] or open.

${\rm cap}(K)\uparrow{\rm cap}_{*}(A)\text{ \ as $K\uparrow A$}.$ (2.3)

######  Lemma 2.6 (cf. [13, Lemma 2.3.1]).

For any $A\subset X$,

${\rm
cap}_{*}(A)=0\iff\mathcal{E}^{+}(A)=\\{0\\}\iff\mathcal{E}^{+}(K)=\\{0\\}\text{
\ for all\/ $K\in\mathfrak{C}_{A}$}.$

######  Lemma 2.7.

Given $\mu\in\mathcal{E}^{+}$, let $A\subset X$ be a $\mu$-measurable and
$\mu$-$\sigma$-finite set with ${\rm cap}_{*}(A)=0$. Then $A$ is
$\mu$-negligible.

###### Proof.

As $A$ is $\mu$-measurable and $\mu$-$\sigma$-finite, $\mu^{*}(A)=0$ will
follow once we show that $\mu(K)=0$ for every compact $K\subset A$, which
however is obvious by Lemma 2.6. ∎

A proposition $\mathcal{P}$ involving a variable point $x\in X$ is said to
hold nearly everywhere (n.e.) on $A\subset X$ if ${\rm cap}_{*}(E)=0$, $E$
being the set of all $x\in A$ where $\mathcal{P}(x)$ fails. Replacing here
${\rm cap}_{*}(E)$ by ${\rm cap}^{*}(E)$ leads to the concept of quasi-
everywhere (q.e.).

The potential $U^{\mu}$ of any $\mu\in\mathcal{E}$ is (well defined and)
finite quasi-everywhere on $X$ [13, Corollary to Lemma 3.2.3]. Furthermore,
for any two given $\mu,\nu\in\mathcal{E}$,

$\mu=\nu\iff U^{\mu}=U^{\nu}\text{ \ n.e.\ on $X$}\iff U^{\mu}=U^{\nu}\text{ \
q.e.\ on $X$},$ (2.4)

which follows from [13, Lemma 3.2.1] by making use of the energy principle.

In the study of inner potential theoretical concepts, the following
strengthened version of countable subadditivity for inner capacity is
particularly useful.

######  Lemma 2.8.

For arbitrary $A\subset X$ and universally measurable $U_{j}\subset X$,
$j\in\mathbb{N}$,

${\rm cap}_{*}\Bigl{(}\bigcup_{j\in\mathbb{N}}\,A\cap
U_{j}\Bigr{)}\leqslant\sum_{j\in\mathbb{N}}\,{\rm cap}_{*}(A\cap U_{j}).$

###### Proof.

Noting that a strictly positive definite kernel is pseudo-positive, cf. [13,
p. 150], we derive the lemma from [13] (see Lemma 2.3.5 and the remark
following it). For the Newtonian kernel $|x-y|^{2-n}$ on $\mathbb{R}^{n}$,
$n\geqslant 3$, this goes back to H. Cartan [6, p. 253].∎

###  2.2. Potential-theoretic principles

The following two maximum principles are often used throughout the paper. A
kernel $\kappa$ is said to satisfy Frostman’s maximum principle (= the first
maximum principle) if for any $\mu\in\mathcal{E}^{+}$ with $U^{\mu}\leqslant
1$ on $S(\mu)$, the same inequality holds everywhere on $X$; and it is said to
satisfy the domination principle (= the second maximum principle) if for any
$\mu,\nu\in\mathcal{E}^{+}$ with $U^{\mu}\leqslant U^{\nu}$ $\mu$-a.e., the
same inequality is fulfilled on all of $X$.

For classical kernels, the principle of positivity of mass goes back to J.
Deny [8]. In the generality stated below, it was established in [35] (Theorem
2.1 and Remark 2.1).

######  Theorem 2.9.

Assume that a locally compact space $X$ is $\sigma$-compact, and that a
(strictly positive definite) kernel $\kappa$ satisfies the first and second
maximum principles. For any $\mu,\nu\in\mathcal{E}^{+}$ with $U^{\mu}\leqslant
U^{\nu}$ n.e. on $X$, we have $\mu(X)\leqslant\nu(X)$. In the case where $X$
is compact, this remains valid with the second maximum principle dropped.

Unless explicitly stated otherwise, from now on a (strictly positive definite)
kernel $\kappa$ will always be assumed to satisfy the consistency principle,
or equivalently to be perfect, which means that the cone $\mathcal{E}^{+}$ is
complete in the induced strong topology, and moreover that the strong topology
on $\mathcal{E}^{+}$ is finer than the (induced) vague topology on
$\mathcal{E}^{+}$ [13, Section 3.3]. Thus, for a perfect kernel $\kappa$, any
strong Cauchy net $(\mu_{s})\subset\mathcal{E}^{+}$ converges both strongly
and vaguely to the same unique limit $\mu_{0}\in\mathcal{E}^{+}$, the strong
and the vague topologies on $\mathcal{E}^{+}$ being Hausdorff.171717Since the
space $\mathfrak{M}$ equipped with the vague topology does not necessarily
satisfy the first axiom of countability, the vague convergence cannot in
general be described in terms of sequences. We follow Moore and Smith’s theory
of convergence [24], based on the concept of nets. However, if a locally
compact space $X$ is second-countable, then the space $\mathfrak{M}$ is first-
countable [36, Lemma 4.4], and the use of nets in $\mathfrak{M}$ may often be
avoided.

######  Remark 2.10.

As seen from a well-known counterexample by Cartan [5], the whole pre-Hilbert
space $\mathcal{E}$ is in general strongly incomplete, and this is the case
even for the Newtonian kernel $|x-y|^{2-n}$ on $\mathbb{R}^{n}$, $n\geqslant
3$, despite the fact that the Newtonian kernel is perfect (cf. also [23,
Theorems 1.18, 1.19]).

######  Example 2.11.

The $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order $\alpha\in(0,2]$,
$\alpha<n$, on $\mathbb{R}^{n}$, $n\geqslant 2$ (thus in particular the
Newtonian kernel $|x-y|^{2-n}$ on $\mathbb{R}^{n}$, $n\geqslant 3$), is
perfect, and it satisfies the first and second maximum principles [23,
Theorems 1.10, 1.15, 1.18, 1.27, 1.29]. The same holds true for the associated
$\alpha$-Green kernel on an arbitrary open subset of $\mathbb{R}^{n}$,
$n\geqslant 2$ [16, Theorems 4.6, 4.9, 4.11]. The ($2$-)Green kernel on a
planar Greenian set is likewise strictly positive definite [9, Section
I.XIII.7] and perfect [11], and it fulfills the first and the second maximum
principles (see [1, Theorem 5.1.11] or [9, Section I.V.10]). The restriction
of the logarithmic kernel $-\log\,|x-y|$ to a closed disc in $\mathbb{R}^{2}$
of radius ${}<1$ satisfies the energy principle as well as Frostman’s maximum
principle [23, Theorems 1.6, 1.16], and hence it is perfect [13, Theorem
3.4.2]. (However, the domination principle then fails in general; it does hold
only in a weaker sense where $\mu,\nu$ involved in the above-quoted definition
meet the additional requirement that
$\nu(\mathbb{R}^{2})\leqslant\mu(\mathbb{R}^{2})$, cf. [25, Theorem 3.2].)
Thus all the above-mentioned kernels also satisfy the principle of positivity
of mass (Theorem 2.9). Also note that the $\alpha$-Riesz kernels of order
$2<\alpha<n$ on $\mathbb{R}^{n}$, $n\geqslant 2$, are likewise perfect [23,
Theorems 1.15, 1.18],181818However, these kernels do not satisfy any of the
two maximum principles, since the $\alpha$-Riesz potentials $U^{\mu}$, where
$2<\alpha<n$ and $\mu\geqslant 0$, are superharmonic on $\mathbb{R}^{n}$ (see
[23, Theorems 1.4, 1.5]). and so are the Deny kernels, defined with the aid of
Fourier transformation (see condition $(A)$ in [7, Section 1], cf. [23,
Section VI.1.2]).

###  2.3. Quasiclosed sets

For arbitrary $A\subset X$, denote by $\mathcal{E}^{\prime}(A)$ the closure of
the cone $\mathcal{E}^{+}(A)$ in the strong topology on $\mathcal{E}^{+}$.
Being a strongly closed subset of the strongly complete cone
$\mathcal{E}^{+}$, $\mathcal{E}^{\prime}(A)$ is likewise strongly complete.
Furthermore,

$\mathcal{E}^{\prime}(A)\subset\mathcal{E}^{+}(\overline{A}),$ (2.5)

for $\mathcal{E}^{+}(\overline{A})$ is strongly closed (see Theorem 2.13
below). Here $\overline{A}:={\rm Cl}_{X}A$.

######  Definition 2.12 (Fuglede [14]).

A set $A\subset X$ is said to be quasiclosed if

$\inf\,\bigl{\\{}{\rm cap}^{*}(A\bigtriangleup F):\ F\text{ closed, }F\subset
X\bigr{\\}}=0,$

$\bigtriangleup$ being the symmetric difference. Replacing here "closed" by
"compact", we arrive at the concept of a quasicompact set.

######  Theorem 2.13.

If $A\subset X$ is quasiclosed (or in particular quasicompact), then the cone
$\mathcal{E}^{+}(A)$ is strongly closed, and hence

$\mathcal{E}^{\prime}(A)=\mathcal{E}^{+}(A).$

###### Proof.

Given a net $(\mu_{s})\subset\mathcal{E}^{+}(A)$ converging strongly (hence
vaguely) to $\mu_{0}\in\mathcal{E}^{+}$, we only need to show that $\mu_{0}$
is concentrated on $A$. For closed $A$, the cone $\mathfrak{M}^{+}(A)$ is
vaguely closed according to [4, Section III.2, Proposition 6], whence
$\mu_{0}\in\mathfrak{M}^{+}(A)$.

For quasiclosed $A$, note that for every $q\in(0,\infty)$,
$\mathcal{E}^{+}_{q}:=\\{\mu\in\mathcal{E}^{+}:\ \|\mu\|\leqslant q\\}$ is
hereditary [15, Definition 5.2] and vaguely compact [13, Lemma 2.5.1]. Since
$(\mu_{s})$ can certainly be chosen to be strongly bounded,
$(\mu_{s})\subset\mathcal{E}^{+}_{q^{\prime}}$ for some
$q^{\prime}\in(0,\infty)$. Applying now [15, Corollary 6.2] with
$\mathcal{J}:=\mathcal{E}^{+}_{q^{\prime}}$ and $H:=A$, we infer that
$\widetilde{\mathcal{J}}:=\mathcal{E}^{+}_{q^{\prime}}\cap\mathfrak{M}^{+}(A)$
is vaguely compact, and consequently, $(\mu_{s})$
$({}\subset\widetilde{\mathcal{J}})$ has a vague limit point
$\nu_{0}\in\widetilde{\mathcal{J}}$. The vague topology being Hausdorff,
$\nu_{0}=\mu_{0}$, whence
$\mu_{0}\in\widetilde{\mathcal{J}}\subset\mathfrak{M}^{+}(A)$ as desired.∎

###  2.4. Inner capacitary measures

Assume for a moment that a set $A=K\subset X$ is compact, and $w(K)<\infty$;
then the infimum in (2.2) is an actual minimum, the energy being vaguely
l.s.c. on $\mathfrak{M}^{+}$ (the principle of descent), whereas the class
$\breve{\mathfrak{M}}^{+}(K)$ being vaguely compact (cf. [4, Section III.1,
Corollary 3]). But if $A\subset X$ is noncompact, the class
$\breve{\mathfrak{M}}^{+}(A)$ is no longer vaguely compact, and the infimum in
(2.2) is in general not attained among the admissible measures — not even for
perfect kernels.

To overcome this inconvenience, Fuglede [13, Theorem 4.1] has formulated a
dual minimum energy problem, which leads to the same concept of inner
capacity, but it is already solvable. Under the assumptions of the present
study, where the kernel $\kappa$ is assumed throughout to be perfect,
Fuglede’s result can be specified as follows.

######  Theorem 2.14 ([35, Theorem 6.1]).

For any $A\subset X$,

$\inf_{\nu\in\Gamma_{A}}\,\|\nu\|^{2}={\rm cap}_{*}(A),$ (2.6)

where

$\Gamma_{A}:=\bigl{\\{}\nu\in\mathcal{E}^{+}:\ U^{\nu}\geqslant 1\text{ \
n.e.\ on $A$}\bigr{\\}}.$

If ${\rm cap}_{*}(A)<\infty$, the infimum in (2.6) is an actual minimum with
the unique minimizer $\gamma_{A}\in\Gamma_{A}$, called the inner capacitary
measure of $A$ and having the properties

$\displaystyle\gamma_{A}(X)=\|\gamma_{A}\|^{2}={\rm cap}_{*}(A),$ (2.7)
$\displaystyle U^{\gamma_{A}}\geqslant 1\text{ \ n.e.\ on\/ $A$},$ (2.8)
$\displaystyle U^{\gamma_{A}}\leqslant 1\text{ \ on\/ $S(\gamma_{A})$},$
$\displaystyle U^{\gamma_{A}}=1\text{ \ $\gamma_{A}$-a.e.\ on\/ $X$.}$

If moreover Frostman’s maximum principle is fulfilled, then also

$U^{\gamma_{A}}=1\text{ \ n.e.\ on $A$.}$ (2.9)

The following convergence theorem is often useful in applications.

######  Theorem 2.15 ([35, Theorem 8.1]).

For any $A\subset X$ with ${\rm cap}_{*}(A)<\infty$,

$\gamma_{K}\to\gamma_{A}\text{ \ strongly and vaguely in\/ $\mathcal{E}^{+}$
as\/ $K\uparrow A$.}$

If moreover the first and the second maximum principles both hold, then also

$U^{\gamma_{K}}\uparrow U^{\gamma_{A}}\text{ \ pointwise on\/ $X$ as\/
$K\uparrow A$.}$

In general, $\gamma_{A}$ is not concentrated on the set $A$ itself.191919For
instance, if $A:=B_{r}$ is an open ball $\\{|x|<r\\}$, $r>0$, in
$\mathbb{R}^{n}$, $n\geqslant 3$, and if $\kappa(x,y)$ is the Newtonian kernel
$|x-y|^{2-n}$, then $\gamma_{B_{r}}$ is the positive measure of total mass
$r^{n-2}$, uniformly distributed over the sphere $\\{|x|=r\\}$. Thus
$S(\gamma_{B_{r}})\cap B_{r}=\varnothing$. (Compare with Theorem 2.16.)
However, $\gamma_{A}\in\mathcal{E}^{+}(A)$ would necessarily hold if
$\mathcal{E}^{+}(A)$ were assumed to be strongly closed. (Indeed, then
$\gamma_{A}\in\mathcal{E}^{\prime}(A)=\mathcal{E}^{+}(A)$, the former relation
being implied by Theorem 2.15.)

######  Theorem 2.16 (cf. [35, Theorem 7.2]).

Given $A\subset X$ with ${\rm cap}_{*}(A)<\infty$, assume that the class
$\mathcal{E}^{+}(A)$ is strongly closed (or in particular that the set $A$ is
quasiclosed). Then necessarily $\gamma_{A}\in\mathcal{E}^{+}(A)$. If moreover
Frostman’s maximum principle is fulfilled, then $\gamma_{A}$ is uniquely
characterized within $\mathcal{E}^{+}(A)$ by property (2.9).202020We emphasize
that, if the requirement of the strong closedness of the class
$\mathcal{E}^{+}(A)$ is dropped, then there is in general no
$\nu\in\mathcal{E}^{+}(A)$ with $U^{\nu}=1$ n.e. on $A$ (see [35, footnote
3]).

For other alternative characterizations of the inner capacitary measure
$\gamma_{A}$ which hold true even for arbitrary $A$, we refer to [35, Theorems
9.1–9.3].

###  2.5. Inner balayage

Throughout this subsection, a (perfect) kernel $\kappa$ is additionally
required to satisfy the domination principle.

Assume for a moment that $A=K\subset X$ is compact. Then one can easily prove
by generalizing the classical Gauss variational method (see [6], cf. also [23,
Section IV.5.23]) that for any given $\mu\in\mathcal{E}^{+}$, there exists
$\mu^{K}\in\mathcal{E}^{+}(K)$ uniquely determined within $\mathcal{E}^{+}(K)$
by the equality $U^{\mu^{K}}=U^{\mu}$ n.e. on $K$, the uniqueness being
obvious from (2.4) applied to the space $X:=K$ and the kernel
$\kappa^{\prime}:=\kappa|_{K\times K}$. This $\mu^{K}$ is said to be the
balayage of $\mu\in\mathcal{E}^{+}$ onto (compact) $K$.

If now $A\subset X$ is arbitrary, there is in general no
$\nu\in\mathcal{E}^{+}(A)$ having the property $U^{\nu}=U^{\mu}$ n.e. on $A$.
Nevertheless, a substantial theory of inner balayage (sweeping), generalizing
Cartan’s theory [6] of Newtonian balayage to a suitable kernel $\kappa$ on a
locally compact space $X$, was developed (see [33]–[36]), and this was
performed by means of several alternative approaches described in Theorem 2.17
below.

For any $\mu\in\mathcal{E}^{+}$ and any $A\subset X$, denote

$\Gamma_{A,\mu}:=\bigl{\\{}\nu\in\mathcal{E}^{+}:\ U^{\nu}\geqslant
U^{\mu}\text{ \ n.e.\ on $A$}\bigr{\\}}.$

######  Theorem 2.17 ([35, Theorem 3.1]).

There exists precisely the same unique measure
$\mu^{A}\in\Gamma_{A,\mu}\cap\mathcal{E}^{\prime}(A)$, called the inner
balayage of $\mu\in\mathcal{E}^{+}$ to $A\subset X$, and solving each of the
following three extremal problems:

$\displaystyle\|\mu^{A}\|=\min_{\nu\in\Gamma_{A,\mu}}\,\|\nu\|,$
$\displaystyle U^{\mu^{A}}=\min_{\nu\in\Gamma_{A,\mu}}\,U^{\nu}\text{ \ on\
$X$},$
$\displaystyle\|\mu-\mu^{A}\|=\min_{\nu\in\mathcal{E}^{\prime}(A)}\,\|\mu-\nu\|.$
(2.10)

This $\mu^{A}$ is said to be the inner balayage of $\mu$ to $A$, and it has
the properties

$\displaystyle U^{\mu^{A}}$ $\displaystyle=U^{\mu}\text{ \ n.e.\ on\ }A,$
(2.11) $\displaystyle U^{\mu^{A}}$ $\displaystyle=U^{\mu}\text{ \
$\mu^{A}$-a.e.,}$ (2.12) $\displaystyle U^{\mu^{A}}$ $\displaystyle\leqslant
U^{\mu}\text{ \ on $X$.}$

Moreover, (2.11) characterizes the inner balayage $\mu^{A}$ uniquely within
$\mathcal{E}^{\prime}(A)$.

Thus, according to (2.10), the inner balayage $\mu^{A}$ can in particular be
characterized as the (unique) orthogonal projection of $\mu\in\mathcal{E}^{+}$
in the pre-Hilbert space $\mathcal{E}$ onto the (strongly complete, convex)
cone $\mathcal{E}^{\prime}(A)$, cf. [12, Theorem 1.12.3].

######  Corollary 2.18.

For arbitrary $A$ with ${\rm cap}_{*}(A)<\infty$,

$\gamma_{A}=(\gamma_{A})^{A}.$ (2.13)

###### Proof.

Since $\gamma_{A}\in\mathcal{E}^{\prime}(A)$ (cf. Section 2.4), whereas the
orthogonal projection of any $\nu\in\mathcal{E}^{\prime}(A)$ onto
$\mathcal{E}^{\prime}(A)$ is certainly the same $\nu$, (2.10) applied to
$\gamma_{A}$ gives (2.13). ∎

In general, $\mu^{A}\not\in\mathcal{E}^{+}(A)$. (Indeed, for
$A:=B_{r}\subset\mathbb{R}^{n}$, $\kappa(x,y):=|x-y|^{2-n}$, and
$\mu:=\gamma_{B_{r}}$, where $n\geqslant 3$ and $r\in(0,\infty)$, this is
obvious from Corollary 2.18 and footnote 19.) Nonetheless, the following
assertion holds true.

######  Corollary 2.19.

If $\mathcal{E}^{+}(A)$ is strongly closed (or in particular if $A$ is
quasiclosed), then the inner balayage $\mu^{A}$ can alternatively be
characterized as the orthogonal projection of $\mu$ in the pre-Hilbert space
$\mathcal{E}$ onto the cone $\mathcal{E}^{+}(A)$; i.e.
$\mu^{A}\in\mathcal{E}^{+}(A)$ and

$\|\mu-\mu^{A}\|=\min_{\nu\in\mathcal{E}^{+}(A)}\,\|\mu-\nu\|.$

The same $\mu^{A}$ is uniquely determined within $\mathcal{E}^{+}(A)$ by
property (2.11).

###### Proof.

Since then $\mathcal{E}^{\prime}(A)=\mathcal{E}^{+}(A)$, this is obvious from
Theorem 2.17. ∎

The following Theorem 2.20 justifies the term "inner balayage". (It is worth
emphasizing here that, although a kernel $\kappa$ was assumed in [33] to be
positive, this restriction on $\kappa$ is unnecessary for the validity of the
results from [33] quoted below.)

######  Theorem 2.20 ([33, Theorem 4.8]).

For any $\mu\in\mathcal{E}^{+}$ and any $A\subset X$,

$\displaystyle\mu^{K}\to\mu^{A}\text{ \ strongly and vaguely in\/
$\mathcal{E}^{+}$ as\/ $K\uparrow A$},$ $\displaystyle U^{\mu^{K}}\uparrow
U^{\mu^{A}}\text{ \ pointwise on\/ $X$ as\/ $K\uparrow A$}.$

We complete this brief review of the theory of inner balayage with the
following useful properties of inner swept measures (see [33, Propositions
7.2, 7.4], cf. [34]).

######  Proposition 2.21 (Balayage "with a rest"212121Regarding the
terminology used here, we follow N.S. Landkof [23, p. 264]. ).

If $A\subset Q$, then for any $\mu\in\mathcal{E}^{+}$,

$\mu^{A}=(\mu^{Q})^{A}.$

######  Proposition 2.22.

If Frostman’s maximum principle is fulfilled, then222222It is worth noting
that, if moreover $\kappa\geqslant 0$ and ${\rm cap}_{*}(A)<\infty$, or if $A$
is compact, then actually $\mu^{A}(X)=\int U^{\mu}\,d\gamma_{A}$. See [33]
(Eq. (7.3) and Proposition 7.6), cf. [34].

$\mu^{A}(X)\leqslant\mu(X)\text{ \ for any\/ $\mu\in\mathcal{E}^{+}$ and\/
$A\subset X$}.$ (2.14)

##  3\. Preliminary results on Problem 1.1

Keeping the (permanent) assumption of the energy principle, in the present
section we do not require a kernel $\kappa$ to be perfect.

Fix an external field $f$ admitting the representation232323In particular,
(3.1) necessarily holds for $f$ of form (1.4). In fact, in view of (2.1) one
can take $f_{1}:=\psi+U^{\vartheta^{+}}$ and $f_{2}:=U^{\vartheta^{-}}$, where
$\vartheta^{+}$ and $\vartheta^{-}$ denote the positive and negative parts of
$\vartheta\in\mathcal{E}$ in the Hahn–Jordan decomposition. The required
inequality $\int U^{\vartheta^{-}}\,d\mu<\infty$ is indeed fulfilled for all
$\mu\in\mathcal{E}^{+}$ since, by virtue of the positive definiteness of the
kernel, $\mathcal{E}=\mathcal{E}^{+}-\mathcal{E}^{+}$ (see [13, Section 3.1]).

$f=f_{1}-f_{2},\text{ \ where $f_{1},f_{2}\in\Phi(X)$ and $\int
f_{2}\,d\mu<\infty$ for all $\mu\in\mathcal{E}^{+}$.}$ (3.1)

Observing with the aid of Lemma 2.6 that then necessarily $f_{2}<\infty$ n.e.
on $X$, we infer that such $f$ is well defined and takes values in
$(-\infty,\infty]$ n.e. on $X$. Also note that $f$ is universally measurable,
i.e. $\nu$-measurable for all $\nu\in\mathfrak{M}^{+}$.

For any $\mu\in\mathcal{E}^{+}$, define the $f$-weighted potential
$U_{f}^{\mu}$ by means of the formula

$U_{f}^{\mu}:=U^{\mu}+f;$

then $U_{f}^{\mu}$ is well defined and takes values in $(-\infty,\infty]$ n.e.
on $X$. This follows from the fact that the same holds true for both $U^{\mu}$
and $f$, by making use of the countable subadditivity of inner capacity on
universally measurable sets (Lemma 2.8).

Let $\mathcal{E}^{+}_{f}$ stand for the (convex) class of all
$\mu\in\mathcal{E}^{+}$ such that the function $f$ (or, equivalently, $f_{1}$)
is $\mu$-integrable. Then for every $\mu\in\mathcal{E}^{+}_{f}$, the
$f$-weighted energy $I_{f}(\mu)$, introduced by (1.1), is finite and, by
Lebesgue–Fubini’s theorem [4, Section V.8, Theorem 1], is representable in the
form

$I_{f}(\mu)=\int\bigl{(}U_{f}^{\mu}+f\bigr{)}\,d\mu\in(-\infty,\infty).$

For arbitrary $A\subset X$, denote

$\mathcal{E}^{+}_{f}(A):=\mathcal{E}^{+}_{f}\cap\mathfrak{M}^{+}(A),\quad\breve{\mathcal{E}}^{+}_{f}(A):=\mathcal{E}^{+}_{f}\cap\breve{\mathfrak{M}}^{+}(A),$

and let $w_{f}(A)$ stand for the infimum of $I_{f}(\mu)$, where $\mu$ ranges
over $\breve{\mathcal{E}}^{+}_{f}(A)$, see (1.2). If this infimum is finite,
see (1.3),242424It follows immediately from (1.3) that ${\rm cap}_{*}(A)>0$
(for if not, $\breve{\mathcal{E}}^{+}(A)=\varnothing$ by Lemma 2.6). Actually,
even a stronger assertion then necessarily holds — see Lemma 3.2. one can
consider the inner Gauss variational problem on the existence of
$\lambda=\lambda_{A,f}\in\breve{\mathcal{E}}^{+}_{f}(A)$ with
$I_{f}(\lambda_{A,f})=w_{f}(A)$, see Problem 1.1.

The present section contains preliminary results on Problem 1.1, some of them
being established in a much more general form in the author’s earlier paper
[27].

To begin with, we note that a solution $\lambda_{A,f}$ to Problem 1.1 is
unique (if it exists), which follows easily from the convexity of the class
$\breve{\mathcal{E}}^{+}_{f}(A)$ and the energy principle, by use of the
parallelogram identity in the pre-Hilbert space $\mathcal{E}$ (cf. [27, Lemma
6]).

######  Lemma 3.1 (cf. [27, Lemma 4]).

Given $\kappa$, $A$, and $f$,

$w_{f}(K)\downarrow w_{f}(A)\text{ \ as\/ $K\uparrow A$}.$ (3.2)

In fact, (3.2) can be derived from the relation

$I_{f}(\mu)=\lim_{K\uparrow A}\,I_{f}(\mu|_{K})\text{ \ for all
$\mu\in\mathcal{E}^{+}_{f}(A)$},$ (3.3)

which in turn follows by applying Lemma 2.2 to each of the $\mu$-integrable
functions $\kappa$, $f_{1}$, and $f_{2}$ (cf. Remark 2.4), $f_{1}$ and $f_{2}$
appearing in (3.1). Formula (3.3) also implies that the extremal value
$w_{f}(A)$ would be the same if the admissible measures $\mu$ in (1.2) were
required to be of compact support $S(\mu)\subset A$ (cf. [27, Remark 2]).

######  Lemma 3.2 (cf. [27, Lemma 5]).

$w_{f}(A)<\infty$ holds if and only if

${\rm cap}_{*}\bigl{(}\\{x\in A:\ f_{1}(x)<\infty\\}\bigr{)}>0.$ (3.4)

######  Lemma 3.3.

Assume an external field $f$ is of form (1.4), that is,
$f=\psi+U^{\vartheta}$, where $\psi\in\Phi(X)$ and $\vartheta\in\mathcal{E}$.
Then (1.3) is equivalent to $w_{f}(A)<\infty$, and hence to

${\rm cap}_{*}\bigl{(}\\{x\in A:\ \psi(x)<\infty\\}\bigr{)}>0.$ (3.5)

###### Proof.

The first part of the claim will follow once we show that

$w_{f}(A)>-\infty.$

For any given $\vartheta\in\mathcal{E}$,

$I_{U^{\vartheta}}(\mu)=\|\mu\|^{2}+2\int
U^{\vartheta}\,d\mu=\|\mu+\vartheta\|^{2}-\|\vartheta\|^{2}\text{ \ for all
$\mu\in\mathcal{E}^{+}$},$ (3.6)

and hence, by the energy principle,

$w_{U^{\vartheta}}(A)\geqslant-\|\vartheta\|^{2}>-\infty.$

It thus remains to prove that for any given $\psi\in\Phi(X)$,

$\inf_{\mu\in\breve{\mathcal{E}}^{+}(A)}\,\int\psi\,d\mu>-\infty.$

This however is obvious if $\psi\geqslant 0$, while the remaining case of
compact $X$ is treated as described in Remark 2.4, by use of the equality
$\mu(X)=1$ for all $\mu\in\breve{\mathcal{E}}^{+}(A)$.252525 The above proof
also implies the following observation, to be useful in the sequel: if an
external field $f$ is of form (1.4), then $I_{f}(\mu)>-\infty$ for all
$\mu\in\mathcal{E}^{+}$.

In view of Lemma 3.2, the latter part of the claim is reduced to the
equivalence of (3.4) and (3.5), which however is obvious from Lemma 2.8 and
the fact that the potential of a measure of finite energy is finite q.e. on
$X$ (cf. footnote 23).∎

######  Corollary 3.4.

Assume that $f=U^{\vartheta}$, where $\vartheta\in\mathcal{E}$. Then

$\mathcal{E}^{+}_{f}(A)=\mathcal{E}^{+}(A).$ (3.7)

Furthermore, assumption (1.3) is fulfilled if and only if ${\rm
cap}_{*}(A)>0$.

###### Proof.

Noting from (3.6) that $I_{f}(\mu)$ is finite for all $\mu\in\mathcal{E}^{+}$,
we obtain (3.7), while the latter part of the claim is an immediate
consequence of Lemma 3.3 with $\psi=0$. ∎

###  3.1. Characteristic properties of $\lambda_{A,f}$

In Theorems 3.5 and 3.6, assume again an external field $f$ to be of form
(3.1). Also assume (1.3) to be fulfilled (see Lemmas 3.2, 3.3 and Corollary
3.4 for necessary and/or sufficient conditions for this to hold).

######  Theorem 3.5 (cf. [27, Theorems 1, 2]).

If the solution $\lambda=\lambda_{A,f}$ to Problem 1.1 exists, then its
$f$-weighted potential $U_{f}^{\lambda}$ has the properties

$\displaystyle U_{f}^{\lambda}$ $\displaystyle\geqslant c_{A,f}\text{ \ n.e.\
on\/ $A$},$ (3.8) $\displaystyle U_{f}^{\lambda}$ $\displaystyle=c_{A,f}\text{
\ $\lambda$-a.e.\ on\/ $X$},$ (3.9)

where

$c_{A,f}:=\int U_{f}^{\lambda}\,d\lambda=w_{f}(A)-\int
f\,d\lambda\in(-\infty,\infty)$ (3.10)

is said to be the inner $f$-weighted equilibrium constant for $A$.

If moreover $f\in\Phi(X)$, then also

$U_{f}^{\lambda}\leqslant c_{A,f}\text{ \ on\/ $S(\lambda)$},$

and hence

$U_{f}^{\lambda}=c_{A,f}\text{ \ n.e.\ on\/ $S(\lambda)\cap A$}.$

Relations (3.8) and (3.10), resp. (3.9) and (3.10), characterize the minimizer
$\lambda_{A,f}$ uniquely within $\breve{\mathcal{E}}^{+}_{f}(A)$. In more
detail, the following theorem holds true.

######  Theorem 3.6 (cf. [27, Proposition 1]).

For $\mu\in\breve{\mathcal{E}}^{+}_{f}(A)$ to be the (unique) solution
$\lambda_{A,f}$ to Problem 1.1, it is necessary and sufficient that either of
the following two characteristic inequalities be fulfilled:

$\displaystyle U_{f}^{\mu}$ $\displaystyle\geqslant\int
U_{f}^{\mu}\,d\mu=:c_{1}\text{ n.e.\ on\/ $A$},$ (3.11) $\displaystyle
U_{f}^{\mu}$ $\displaystyle\leqslant w_{f}(A)-\int f\,d\mu=:c_{2}\text{ \
$\mu$-a.e.\ on\/ $X$.}$ (3.12)

If (3.11) or (3.12) holds true, then equality actually prevails in (3.12), and
moreover

$c_{1}=c_{2}=c_{A,f},$

the inner $f$-weighted equilibrium constant $c_{A,f}$ being introduced by
formula (3.10).

##  4\. Proofs of the main results

Throughout this section,

$w_{f}(A)<\infty,$ (4.1)

and so the class $\breve{\mathcal{E}}^{+}_{f}(A)$ of admissible measures in
Problem 1.1 is nonempty.

Assume for a moment that $A=K\subset X$ is compact, and that $f\in\Phi(X)$.
Then Problem 1.1 is solvable for an arbitrary (not necessarily perfect) kernel
$\kappa$ on $X$, which follows from the vague compactness of the class
$\breve{\mathfrak{M}}^{+}(K)$ [4, Section III.1, Corollary 3] and the vague
lower semicontinuity of the $f$-weighted energy $I_{f}(\mu)$ on
$\mathfrak{M}^{+}$, the latter being obvious from Lemma 2.1 applied to each of
$\kappa\in\Phi(X\times X)$ and $f\in\Phi(X)$ (cf. also Remark 2.4). However,
this fails to hold if either of the above two requirements is omitted, and
moreover Problem 1.1 then becomes in general unsolvable.

To analyze Problem 1.1 for quite a general (not necessarily of the class
$\Phi(X)$) external field $f$ and for quite a general (not necessarily closed)
set $A\subset X$, from now on we assume that the kernel $\kappa$ is perfect,
and that $f$ is of form (1.4), i.e.

$f=\psi+U^{\vartheta},\text{ \ where $\psi\in\Phi(X)$ and
$\vartheta\in\mathcal{E}$}$

(see assumptions $(H_{1})$ and $(H_{2})$ in Section 1). This enables us to
develop an approach based on the close interaction between the strong and the
vague topologies on the (strongly complete) cone $\mathcal{E}^{+}$, which
turned out to be efficient for proving Theorems 1.2, 1.4, 1.5, 1.8, 1.10 and
Corollary 1.6 (see Sections 4.2–4.7).

Our choice of $f$ is basically caused by the fact that, if a net
$(\mu_{s})_{s\in S}\subset\mathcal{E}^{+}$ converges to
$\mu_{0}\in\mathcal{E}^{+}$ both strongly and vaguely, then (see the proof of
Lemma 4.4)

$I_{f}(\mu_{0})\leqslant\liminf_{s\in S}\,I_{f}(\mu_{s}),$

which is one of the crucial points in the analysis performed below.

###  4.1. Preparatory lemmas

A net $(\mu_{s})_{s\in S}\subset\breve{\mathcal{E}}^{+}_{f}(A)$ is said to be
minimizing (in Problem 1.1) if

$\lim_{s\in S}\,I_{f}(\mu_{s})=w_{f}(A).$ (4.2)

The (nonempty) set of all those $(\mu_{s})_{s\in S}$ is denoted by
$\mathbb{M}_{f}(A)$.

######  Lemma 4.1.

There is the unique $\xi_{A,f}\in\mathcal{E}^{+}$ such that, for all
$(\mu_{s})_{s\in S}\in\mathbb{M}_{f}(A)$,

$\mu_{s}\to\xi_{A,f}\text{ \ strongly and vaguely}.$ (4.3)

This $\xi_{A,f}$ is said to be the extremal measure (in Problem 1.1).

###### Proof.

We shall first show that for any $(\mu_{s})_{s\in S}$ and $(\nu_{t})_{t\in T}$
from $\mathbb{M}_{f}(A)$,

$\lim_{(s,t)\in S\times T}\,\|\mu_{s}-\nu_{t}\|=0,$ (4.4)

$S\times T$ being the directed product of the directed sets $S$ and $T$ (see
e.g. [21, p. 68]). In fact, due to the convexity of the class
$\breve{\mathcal{E}}^{+}_{f}(A)$, for any $(s,t)\in S\times T$ we have

$4w_{f}(A)\leqslant
4I_{f}\Bigl{(}\frac{\mu_{s}+\nu_{t}}{2}\Bigr{)}=\|\mu_{s}+\nu_{t}\|^{2}+4\int
f\,d(\mu_{s}+\nu_{t}).$

Applying the parallelogram identity in the pre-Hilbert space $\mathcal{E}$
therefore gives

$0\leqslant\|\mu_{s}-\nu_{t}\|^{2}\leqslant-4w_{f}(A)+2I_{f}(\mu_{s})+2I_{f}(\nu_{t}),$

which yields (4.4) when combined with (4.2).

Taking the two nets in (4.4) to be equal, we see that every $(\nu_{t})_{t\in
T}\in\mathbb{M}_{f}(A)$ is strong Cauchy. The cone $\mathcal{E}^{+}$ being
strongly complete, $(\nu_{t})_{t\in T}$ must converge strongly to some
$\xi_{A,f}\in\mathcal{E}^{+}$. The same unique $\xi_{A,f}$ also serves as the
strong limit of any other $(\mu_{s})_{s\in S}\in\mathbb{M}_{f}(A)$, which is
obvious from (4.4). The strong topology on $\mathcal{E}^{+}$ being finer than
the vague topology on $\mathcal{E}^{+}$ by virtue of the perfectness of the
kernel, $(\mu_{s})_{s\in S}$ must converge to $\xi_{A,f}$ also vaguely. ∎

######  Remark 4.2.

In general, the extremal measure $\xi_{A,f}$ is not concentrated on $A$. What
is clear so far is that

$\xi_{A,f}\in\mathcal{E}^{\prime}(A)\subset\mathcal{E}^{+}(\overline{A}),$
(4.5)

the former relation being clear from (4.3), and the latter from (2.5).

######  Remark 4.3.

Another consequence of (4.3) is that

$\xi_{A,f}(X)\leqslant 1,$ (4.6)

the map $\mu\mapsto\mu(X)$ being vaguely l.s.c. on $\mathfrak{M}^{+}$ (Lemma
2.1 with $g=1$). Equality necessarily prevails in (4.6) if $A=K$ is compact,
cf. [4, Section III.1, Corollary 3].

If the unweighted case $f=0$ takes place, we shall drop the index $f$, and
shall simply write $\lambda_{A}$, $\mathbb{M}(A)$, and $\xi_{A}$ in place of
$\lambda_{A,f}$, $\mathbb{M}_{f}(A)$, and $\xi_{A,f}$, respectively.

######  Lemma 4.4.

For the extremal measure $\xi:=\xi_{A,f}$, we have

$-\infty<I_{f}(\xi_{A,f})\leqslant w_{f}(A)<\infty.$ (4.7)

###### Proof.

The first inequality in (4.7) is fulfilled by footnote 25, and the last by
(permanent) assumption (4.1). In view of (4.2), it is therefore enough to show
that

$I_{f}(\xi)\leqslant\lim_{s\in S}\,I_{f}(\mu_{s}),$ (4.8)

$(\mu_{s})_{s\in S}\in\mathbb{M}_{f}(A)$ being fixed.

Assume first that $f\in\Phi(X)$. Since $\mu_{s}\to\xi$ both strongly and
vaguely, Lemma 2.1 applied to $f$ gives (4.8), for (cf. Remark 2.4)

$I_{f}(\xi)=\|\xi\|^{2}+2\int f\,d\xi\leqslant\liminf_{s\in
S}\,\bigl{(}\|\mu_{s}\|^{2}+2\int f\,d\mu_{s}\bigr{)}=\lim_{s\in
S}\,I_{f}(\mu_{s}).$

Otherwise, $f=\psi+U^{\vartheta}$, where $\psi\in\Phi(X)$ and
$\vartheta\in\mathcal{E}$. Similarly as in the preceding paragraph, (4.8) will
follow once we verify the inequality

$\int\bigl{(}\psi+U^{\vartheta}\bigr{)}\,d\xi\leqslant\liminf_{s\in
S}\,\int\bigl{(}\psi+U^{\vartheta}\bigr{)}\,d\mu_{s},$

which, again by Lemma 2.1 applied to $\psi$, is reduced to the equality

$\langle\vartheta,\xi\rangle=\lim_{s\in S}\,\langle\vartheta,\mu_{s}\rangle.$
(4.9)

Applying the Cauchy–Schwarz (Bunyakovski) inequality to the (signed) measures
$\vartheta$ and $\xi-\mu_{s}$, $s\in S$, elements of the pre-Hilbert space
$\mathcal{E}$, we get

$\left|\langle\vartheta,\xi-\mu_{s}\rangle\right|\leqslant\|\vartheta\|\cdot\|\xi-\mu_{s}\|,$

which by the strong convergence of $(\mu_{s})_{s\in S}$ to $\xi$ proves (4.9),
whence the lemma.∎

######  Lemma 4.5.

The following two assertions are equivalent:

  * (i)

There exists the solution $\lambda=\lambda_{A,f}$ to Problem 1.1.

  * (ii)

The extremal measure $\xi=\xi_{A,f}$ belongs to the class
$\breve{\mathcal{E}}^{+}(A)$.262626Compare with (4.5) and (4.6).

If either of (i) or (ii) is fulfilled, then actually

$\xi_{A,f}=\lambda_{A,f}.$ (4.10)

###### Proof.

Assume (i) holds true. The trivial sequence $(\lambda)$ being obviously
minimizing:

$(\lambda)\in\mathbb{M}_{f}(A),$

it must converge strongly (and vaguely) to the extremal measure $\xi$, see
Lemma 4.1. The strong topology on $\mathcal{E}$ being Hausdorff, this yields
(4.10), whence (ii).

Assuming now that (ii) holds, we note from (4.7) that
$\xi\in\breve{\mathcal{E}}^{+}_{f}(A)$, whence

$I_{f}(\xi)\geqslant w_{f}(A).$

The opposite being valid again by (4.7), this establishes (i) with
$\lambda:=\xi$. ∎

######  Lemma 4.6.

For each $K\in\mathfrak{C}_{A}$ large enough, there exists the (unique)
solution $\lambda_{K,f}$ to Problem 1.1 with $A:=K$, and moreover

$\lambda_{K,f}\to\xi_{A,f}\text{ \ strongly and vaguely as\/ $K\uparrow A$}.$
(4.11)

###### Proof.

For each $K\in\mathfrak{C}_{A}$ large enough ($K\geqslant K_{0}$), we have
$w_{f}(K)<\infty$, cf. (4.1) and (3.2). Therefore, by Lemma 4.1 with $A:=K$,
there is the unique $\xi_{K,f}\in\mathcal{E}^{+}$ such that every
$(\mu_{s})_{s\in S}\in\mathbb{M}_{f}(K)$ converges to $\xi_{K,f}$ strongly and
vaguely. Noting that $\xi_{K,f}\in\breve{\mathcal{E}}^{+}(K)$ (see Remarks 4.2
and 4.3), we now derive from Lemma 4.5 that $\xi_{K,f}$ serves as the (unique)
solution $\lambda_{K,f}$ to Problem 1.1 with $A:=K$, which establishes the
first part of the lemma. But $(\lambda_{K,f})_{K\geqslant
K_{0}}\subset\breve{\mathcal{E}}^{+}_{f}(A)$, and moreover, by Lemma 3.1,

$\lim_{K\uparrow A}\,I_{f}(\lambda_{K,f})=\lim_{K\uparrow
A}\,w_{f}(K)=w_{f}(A),$

which shows that, actually,

$(\lambda_{K,f})_{K\geqslant K_{0}}\in\mathbb{M}_{f}(A).$

Applying Lemma 4.1 once again, we obtain (4.11), whence the whole lemma. ∎

###  4.2. Proof of Theorem 1.2

Let the assumptions of the theorem be fulfilled. Due to $(H_{3})$,
$\mathcal{E}^{\prime}(A)=\mathcal{E}^{+}(A)$, which substituted into (4.5)
gives

$\xi\in\mathcal{E}^{+}(A),$ (4.12)

$\xi=\xi_{A,f}$ being the extremal measure in Problem 1.1 (cf. Lemma 4.1).

We aim to show that in the case ${\rm cap}_{*}(A)<\infty$, $\xi$ serves as the
solution $\lambda_{A,f}$ to Problem 1.1. By Lemma 4.5, this will follow once
we verify that $\xi\in\breve{\mathcal{E}}^{+}(A)$, which in view of (4.12) is
equivalent to the assertion

$\xi(X)=1.$ (4.13)

Fix a minimizing net $(\mu_{s})_{s\in S}\in\mathbb{M}_{f}(A)$. Taking a subnet
(if necessary) and changing notation, we can certainly assume $(\mu_{s})_{s\in
S}$ to be strongly bounded:

$\sup_{s\in S}\,\|\mu_{s}\|<\infty.$ (4.14)

Since $\mu_{s}\to\xi$ vaguely, cf. (4.3), applying [4, Section IV.4, Corollary
3] gives

$\int 1_{K}\,d\xi\geqslant\limsup_{s\in S}\,\int 1_{K}\,d\mu_{s}\text{ \ for
every compact $K\subset X$},$ (4.15)

the indicator function $1_{K}$ being bounded, of compact support, and upper
semicontinuous on $X$. On the other hand,

$\xi(X)=\lim_{K\uparrow X}\,\xi(K)=\lim_{K\uparrow X}\,\int 1_{K}\,d\xi,$

which together with (4.6) and (4.15) results in

$1\geqslant\xi(X)\geqslant\limsup_{(s,K)\in S\times\mathfrak{C}_{X}}\,\int
1_{K}\,d\mu_{s}=1-\liminf_{(s,K)\in S\times\mathfrak{C}_{X}}\,\int
1_{A\setminus K}\,d\mu_{s},$

the equality being implied by the fact that every $\mu_{s}$ is a positive
measure of unit total mass concentrated on $A$. (Here $\mathfrak{C}_{X}$ is
the upward directed set of all compact subsets $K$ of $X$.) The proof of
(4.13) is thus reduced to that of

$\liminf_{(s,K)\in S\times\mathfrak{C}_{X}}\,\int 1_{A\setminus
K}\,d\mu_{s}=0.$ (4.16)

By Theorem 2.14 applied to $A\setminus K$, $K\in\mathfrak{C}_{X}$ being
arbitrarily chosen, there exists the (unique) inner capacitary measure
$\gamma_{A\setminus K}$, minimizing the energy $\|\mu\|^{2}$ over the (convex)
set $\Gamma_{A\setminus K}\subset\mathcal{E}^{+}$. For any
$K^{\prime}\in\mathfrak{C}_{X}$ such that $K\subset K^{\prime}$, we have
$\Gamma_{A\setminus K}\subset\Gamma_{A\setminus K^{\prime}}$, and applying
[13, Lemma 4.1.1] with $\mathcal{H}:=\mathcal{E}$ and
$\Gamma:=\Gamma_{A\setminus K^{\prime}}$ gives

$\|\gamma_{A\setminus K}-\gamma_{A\setminus
K^{\prime}}\|^{2}\leqslant\|\gamma_{A\setminus K}\|^{2}-\|\gamma_{A\setminus
K^{\prime}}\|^{2}.$ (4.17)

Since $\|\gamma_{A\setminus K}\|^{2}={\rm cap}_{*}(A\setminus K)$ by (2.7),
$\|\gamma_{A\setminus K}\|^{2}$ decreases as $K$ ranges through
$\mathfrak{C}_{X}$, which together with (4.17) implies that the net
$(\gamma_{A\setminus K})_{K\in\mathfrak{C}_{X}}\subset\mathcal{E}^{+}$ is
Cauchy in the strong topology on $\mathcal{E}^{+}$. Noting that
$(\gamma_{A\setminus K})_{K\in\mathfrak{C}_{X}}$ converges vaguely to
zero,272727Indeed, for any given $\varphi\in C_{0}(X)$, there exists a
relatively compact open set $G\subset X$ such that $\varphi(x)=0$ for all
$x\not\in\overline{G}$. Hence, $\gamma_{A\setminus K}(\varphi)=0$ for all
$K\in\mathfrak{C}_{X}$ with $K\supset\overline{G}$, and the claim follows. we
get

$\gamma_{A\setminus K}\to 0\text{ \ strongly in $\mathcal{E}^{+}$ as
$K\uparrow X$,}$ (4.18)

the kernel $\kappa$ being perfect.

But, by (2.8) applied to $A\setminus K$,

$U^{\gamma_{A\setminus K}}\geqslant 1_{A\setminus K}\text{ \ n.e.\ on
$A\setminus K$},$ (4.19)

hence $\mu_{s}$-a.e. for all $s\in S$, the latter being derived from Lemma 2.7
due to the fact that $A\setminus K$ along with $A$ is $\mu_{s}$-measurable,
whereas $\mu_{s}\in\mathcal{E}^{+}$ is bounded. Integrating (4.19) with
respect to $\mu_{s}$ we therefore obtain, by the Cauchy–Schwarz inequality,

$\int 1_{A\setminus K}\,d\mu_{s}\leqslant\int U^{\gamma_{A\setminus
K}}\,d\mu_{s}\leqslant\|\gamma_{A\setminus K}\|\cdot\|\mu_{s}\|\text{ \ for
all $K\in\mathfrak{C}_{X}$ and $s\in S$},$

which combined with (4.14) and (4.18) results in (4.16).

Thus, under the assumptions made, the solution $\lambda_{A,f}$ to Problem 1.1
does indeed exist. The remaining part of the theorem, uniquely characterizing
$\lambda_{A,f}$ within $\breve{\mathcal{E}}^{+}_{f}(A)$, follows immediately
by applying Theorem 3.6.

###  4.3. Proof of Theorem 1.4

Under (permanent) hypotheses $(H_{1})$ and $(H_{2})$, assume that the solution
$\lambda_{A,f}$ to Problem 1.1 exists; then by Lemma 4.5, it must coincide
with the extremal measure $\xi_{A,f}$, determined by Lemma 4.1. On the other
hand, according to Lemma 4.6, for every $K\in\mathfrak{C}_{A}$ large enough
($K\geqslant K_{0}$), there is the solution $\lambda_{K,f}$ to Problem 1.1
with $A:=K$; and moreover the net $(\lambda_{K,f})_{K\geqslant K_{0}}$
converges strongly and vaguely to $\xi_{A,f}$, see (4.11). Substituting
$\lambda_{A,f}=\xi_{A,f}$ into (4.11) we get (1.8).

Assume now that $f=U^{\vartheta}$, where $\vartheta\in\mathcal{E}$. By Theorem
3.5, the inner $f$-weighted equilibrium constant $c_{A,f}$ can be written in
the form

$c_{A,f}=\int U_{f}^{\lambda_{A,f}}\,d\lambda_{A,f}=\|\lambda_{A,f}\|^{2}+\int
U^{\vartheta}\,d\lambda_{A,f}=\|\lambda_{A,f}\|^{2}+\langle\vartheta,\lambda_{A,f}\rangle,$

and likewise

$c_{K,f}=\|\lambda_{K,f}\|^{2}+\langle\vartheta,\lambda_{K,f}\rangle\quad(K\geqslant
K_{0}).$

Since $\lambda_{K,f}\to\lambda_{A,f}$ strongly in $\mathcal{E}^{+}$ as
$K\uparrow A$, see (1.8), applying the Cauchy–Schwarz inequality to
$\vartheta$ and $\lambda_{K,f}-\lambda_{A,f}$ gives
$\langle\vartheta,\lambda_{K,f}\rangle\to\langle\vartheta,\lambda_{A,f}\rangle$
(as $K\uparrow A$), whence (1.9).

###  4.4. Proof of Theorem 1.5

Let the assumptions of the theorem be fulfilled, and let $\zeta$ be the
measure appearing in $(H_{2}^{\prime})$. The proof is divided into four steps.

Step 1. Assume first that ${\rm cap_{*}}(A)<\infty$. Since
$\mathcal{E}^{+}(A)$ is strongly closed according to $(H_{3})$, $\zeta^{A}$
and $\gamma_{A}$, the inner balayage of $\zeta$ to $A$ and the inner
capacitary measure of $A$, respectively, are both concentrated on $A$, i.e.

$\zeta^{A},\,\gamma_{A}\in\mathcal{E}^{+}(A)$ (4.20)

(see Corollary 2.19 and Theorem 2.16). We aim to show that

$\omega:=\zeta^{A}+\eta_{A,f}\gamma_{A},$ (4.21)

the constant $\eta_{A,f}$ being introduced by the equality in (1.12), serves
as the (unique) solution $\lambda_{A,f}$ to Problem 1.1. This will provide an
alternative proof of the solvability of Problem 1.1 (compare with Theorem 1.2
and its proof, given in Section 4.2).

Indeed, applying (2.14) gives

$0\leqslant\zeta^{A}(X)\leqslant\zeta(X)\leqslant 1,$ (4.22)

the last inequality being valid by virtue of (1.5). Thus
$\eta_{A,f}\in[0,\infty)$, and combining (1.12), (2.7), (4.20), and (4.21)
shows that, actually, $\omega\in\breve{\mathcal{E}}^{+}(A)$. Hence, by (3.7),

$\omega\in\breve{\mathcal{E}}^{+}_{f}(A).$

According to Theorem 3.6, $\omega=\lambda_{A,f}$ will therefore follow once we
verify the inequality

$U_{f}^{\omega}\geqslant\int U_{f}^{\omega}\,d\omega\text{ \ n.e.\ on $A$}.$
(4.23)

By use of Lemma 2.8, we derive from (2.9), (2.11), and (4.21) that the
relation

$U_{f}^{\omega}=U^{\omega}-U^{\zeta}=U^{\zeta^{A}-\zeta}+\eta_{A,f}U^{\gamma_{A}}=\eta_{A,f}$
(4.24)

holds true n.e. on $A$, hence $\omega$-a.e. (see Lemma 2.7), and consequently

$\int U_{f}^{\omega}\,d\omega=\eta_{A,f}\omega(X)=\eta_{A,f}.$

Substituted into (4.24), this gives (4.23). Thus the solution $\lambda_{A,f}$
to Problem 1.1 does exist, and moreover $\lambda_{A,f}=\omega$ and
$\eta_{A,f}=c_{A,f}$, $c_{A,f}$ being the inner $f$-weighted equilibrium
constant. This establishes (1.14) as well as the representation

$\lambda_{A,f}=\zeta^{A}+\eta_{A,f}\gamma_{A}.$ (4.25)

As $(\gamma_{A})^{A}=\gamma_{A}$ (Corollary 2.18), identity (4.25) can be
rewritten in the form

$\lambda_{A,f}=(\zeta+\eta_{A,f}\gamma_{A})^{A},$ (4.26)

the inner balayage being additive on positive measures of finite energy. By
virtue of Theorem 2.17 applied to
$\mu:=\zeta+\eta_{A,f}\gamma_{A}\in\mathcal{E}^{+}$, $\lambda_{A,f}$ can
therefore be characterized as the unique measure of minimum energy norm, resp.
of minimum potential, within the class of all $\nu\in\mathcal{E}^{+}$ having
the property

$U^{\nu}\geqslant U^{\zeta}+\eta_{A,f}U^{\gamma_{A}}\text{ \ n.e.\ on $A$}.$

Noting from (2.9) with the aid of Lemma 2.8 that this inequality is equivalent
to

$U^{\nu}_{f}\geqslant\eta_{A,f}\text{ \ n.e.\ on $A$},$

we arrive at assertion (i), resp. (ii), of the theorem.

Finally, by making use of Corollary 2.19 we derive from (4.26) that
$\lambda_{A,f}$ is the only measure in $\mathcal{E}^{+}(A)$ having the
property

$U^{\lambda_{A,f}}=U^{\zeta}+\eta_{A,f}U^{\gamma_{A}}\text{ \ n.e.\ on $A$},$

or equivalently (cf. Lemma 2.8)

$U^{\lambda_{A,f}}_{f}=\eta_{A,f}\text{ \ n.e.\ on $A$}.$

This establishes assertion (iii).

Step 2. Let now ${\rm cap}_{*}(A)=\infty$. By virtue of assumption (1.10),
then necessarily $\zeta^{A}(X)=1$. The class $\mathcal{E}^{+}(A)$ being
strongly closed according to $(H_{3})$, we actually have
$\zeta^{A}\in\breve{\mathcal{E}}^{+}(A)$ (Corollary 2.19), whence (Corollary
3.4)

$\zeta^{A}\in\breve{\mathcal{E}}^{+}_{f}(A).$ (4.27)

Our aim is to show that $\zeta^{A}$ serves as the (unique) solution to Problem
1.1, i.e.

$\zeta^{A}=\lambda_{A,f}.$ (4.28)

By (2.12) with $\mu:=\zeta$, $U_{f}^{\zeta^{A}}=U^{\zeta^{A}}-U^{\zeta}=0$
$\zeta^{A}$-a.e., whence

$\int U_{f}^{\zeta^{A}}\,d\zeta^{A}=0,$

which combined with (2.11) applied to $\mu:=\zeta$ gives

$U_{f}^{\zeta^{A}}=\int U_{f}^{\zeta^{A}}\,d\zeta^{A}\text{ \ n.e.\ on $A$}.$

By Theorem 3.6, this together with (4.27) implies (4.28) as well as
$c_{A,f}=0$. Furthermore, noting from (1.12) that $\eta_{A,f}$ also equals
$0$, we obtain (1.14).

We finally observe that, due to the equalities $\zeta^{A}=\lambda_{A,f}$ and
$\eta_{A,f}=0$ thereby established, assertions (i)–(iii), providing
alternative characterizations of $\lambda_{A,f}$, can be derived directly from
Theorem 2.17 with $\mu:=\zeta$.

Step 3. The aim of this step is to verify that, under hypotheses $(H_{1})$,
$(H_{3})$, $(H_{4})$, and $(H_{2}^{\prime})$, assumption (1.10) is not only
sufficient, but also necessary for the existence of the solution
$\lambda_{A,f}$. Assume to the contrary that $\lambda_{A,f}$ exists, but
(1.10) does not hold; in view of (1.5) and (2.14), then necessarily

${\rm cap}_{*}(A)=\infty\text{ \ and \ }\zeta^{A}(X)<1.$ (4.29)

According to Lemma 4.6, for each $K\in\mathfrak{C}_{A}$ large enough
($K\geqslant K_{0}$), there is the solution $\lambda_{K,f}$ to Problem 1.1
with $A:=K$; and moreover the net $(\lambda_{K,f})_{K\geqslant K_{0}}$
converges strongly and vaguely to the extremal measure $\xi_{A,f}$, determined
by Lemma 4.1. We assert that, due to the former relation in (4.29),

$\xi_{A,f}=\zeta^{A}.$ (4.30)

Note that the capacity of any compact set is finite, the kernel $\kappa$ being
strongly positive definite by assumption. Therefore, by (1.12) and (4.25)
applied to $K$,

$\lambda_{K,f}=\zeta^{K}+\widetilde{\eta}_{K,f}\lambda_{K}\text{ \ for all
$K\geqslant K_{0}$},$ (4.31)

where $\lambda_{K}:=\gamma_{K}/{\rm cap}(K)$ is the (unique) solution to
problem (2.2) with $A:=K$, and

$\widetilde{\eta}_{K,f}:=1-\zeta^{K}(X).$

But the net $(\widetilde{\eta}_{K,f})_{K\geqslant K_{0}}\subset\mathbb{R}$ is
bounded since, by (4.22) with $A:=K$,

$0\leqslant\zeta^{K}(X)\leqslant\zeta(X)\leqslant 1\text{ \ for all
$K\geqslant K_{0}$}.$

Furthermore, by Theorem 2.20,

$\zeta^{K}\to\zeta^{A}\text{ \ strongly (and vaguely) in $\mathcal{E}^{+}$ as
$K\uparrow A$}.$

Thus, if we show that

$\lambda_{K}\to 0\text{ \ strongly in $\mathcal{E}^{+}$ as $K\uparrow A$},$
(4.32)

identity (4.30) will follow from (4.31) by passing to the limit as $K\uparrow
A$, and making use of the triangle inequality in the pre-Hilbert space
$\mathcal{E}$.

It is seen from (2.3) that the net $(\lambda_{K})_{K\geqslant K_{0}}$ is
minimizing in Problem 1.1 with $f=0$, i.e. $(\lambda_{K})_{K\geqslant
K_{0}}\in\mathbb{M}(A)$. Applying Lemmas 4.1 and 4.6, we therefore conclude
that there exists the unique extremal measure $\xi_{A}$ in Problem 1.1 with
$f=0$, and moreover $\lambda_{K}\to\xi_{A}$ strongly in $\mathcal{E}^{+}$ as
$K\uparrow A$. This yields

$\|\xi_{A}\|^{2}=\lim_{K\uparrow A}\,\|\lambda_{K}\|^{2}=\lim_{K\uparrow
A}\,w(K)=0,$

the last equality being valid due to the assumption ${\rm cap}_{*}(A)=\infty$.
By virtue of the energy principle, $\xi_{A}=0$, which proves (4.32), whence
(4.30).

Since $\lambda_{A,f}$ exists by assumption, applying Lemma 4.5 therefore gives

$\lambda_{A,f}=\xi_{A,f}=\zeta^{A},$

which however is impossible, for $\zeta^{A}(X)<1$ by (4.29). Contradiction.

Step 4. To complete the proof of the theorem, it remains to establish (1.15).
Applying (1.12) and (1.14) to each $K\in\mathfrak{C}_{A}$ large enough
($K\geqslant K_{0}$), we get

$c_{K,f}=\frac{1-\zeta^{K}(X)}{{\rm cap}(K)}.$

In view of (1.9), (1.15) will therefore follow once we show that the net
$(c_{K,f})_{K\geqslant K_{0}}$ decreases, or equivalently, that the net
$\bigl{(}\zeta^{K}(X)\bigr{)}_{K\in\mathfrak{C}_{A}}$ increases, cf. (2.3).
But for any $K,K^{\prime}\in\mathfrak{C}_{A}$ such that $K^{\prime}\geqslant
K$, we have $\zeta^{K}=(\zeta^{K^{\prime}})^{K}$ by Proposition 2.21, hence

$\zeta^{K}(X)\leqslant\zeta^{K^{\prime}}(X)$

by Proposition 2.22, whence the claim.

###  4.5. Proof of Corollary 1.6

As pointed out in Theorem 1.5, $\lambda_{A,f}\in\Lambda_{A,f}$, the class
$\Lambda_{A,f}$ being introduced by (1.13). We thus only need to show that in
the case of $\sigma$-compact $X$,

$\lambda_{A,f}(X)\leqslant\mu(X),$ (4.33)

$\mu\in\Lambda_{A,f}$ being arbitrarily given. But according to Theorem
1.5(ii), then necessarily

$U^{\lambda_{A,f}}\leqslant U^{\mu}\text{ \ everywhere on $X$},$

and (4.33) follows by making use of the principle of positivity of mass
(Theorem 2.9).

###  4.6. Proof of Corollary 1.8

Let hypotheses $(H_{1})$, $(H_{3})$, $(H_{4})$, and $(H_{2}^{\prime})$ be
fulfilled, and let ${\rm cap}_{*}(A)=\infty$. To verify the first part of the
corollary, assume moreover that $\zeta(X)<1$. In view of (2.14), then

$\zeta^{A}(X)\leqslant\zeta(X)<1,$

which implies, by use of Theorem 1.5, that Problem 1.1 indeed has no solution.

It remains to consider the case of $\zeta$ concentrated on $A$. Then the
orthogonal projection of $\zeta$ onto the (strongly closed by $(H_{3})$, hence
strongly complete) cone $\mathcal{E}^{+}(A)$ is certainly the same $\zeta$,
which implies, by virtue of Corollary 2.19, that

$\zeta^{A}=\zeta.$

Applying Theorem 1.5 once again, we therefore conclude that the solution
$\lambda_{A,f}$ to Problem 1.1 exists if and only if
$\zeta(X)=\zeta^{A}(X)=1$, and in the affirmative case

$\lambda_{A,f}=\zeta^{A}=\zeta,$

cf. the latter formula in representation (1.11). This completes the whole
proof.

###  4.7. Proof of Theorem 1.10

Consider the $\alpha$-Riesz kernel of order $\alpha\in(0,n)$ on
$\mathbb{R}^{n}$, $n\geqslant 2$, a set $A\subset\mathbb{R}^{n}$ with strongly
closed $\mathcal{E}^{+}(A)$, and an external field $f$ of form (1.4) such that
$w_{f}(A)<\infty$, or equivalently with ${\rm cap}_{*}\bigl{(}\\{x\in A:\
\psi(x)<\infty\\}\bigr{)}>0$. If moreover ${\rm cap}_{*}(A)<\infty$, then
Problem 1.1 is solvable according to Theorem 1.2, the $\alpha$-Riesz kernel of
arbitrary order $\alpha$ being perfect. This establishes assertion (a).

If now ${\rm cap}_{*}(A)=\infty$, assume that $\alpha\leqslant 2$; and let $f$
be of form (1.5) — then $w_{f}(A)<\infty$ is equivalent to ${\rm
cap}_{*}(A)>0$. Furthermore, then the first and second maximum principles hold
true, hence Theorem 1.5 and Corollary 1.8 can be utilized. By a direct
application of Theorem 1.5, we thus see that $\lambda_{A,f}$ exists if and
only if $\zeta^{A}(\mathbb{R}^{n})=1$, and in the affirmative case
$\lambda_{A,f}=\zeta^{A}$, cf. (1.11). This validates (b).

Assertion (c) is obtained as a direct application of Corollary 1.8, or
alternatively, it can be derived from (b) with the aid of arguments similar to
those in Section 4.6.

The "only if" part of (d) follows directly from (c). For the "if" part, let
$A$ be not inner $\alpha$-thin at infinity, and let $\zeta(\mathbb{R}^{n})=1$.
Applying [32, Corollary 5.3] gives

$\zeta^{A}(\mathbb{R}^{n})=\zeta(\mathbb{R}^{n})=1,$

which implies, by making use of (b), that $\lambda_{A,f}$ does exist, and
moreover $\lambda_{A,f}=\zeta^{A}$.

To verify (e), assume that $\overline{A}$ is $\alpha$-thin at infinity. In the
Newtonian case $\alpha=2$, also assume for the sake of simplicity that
$D:=(\overline{A})^{c}$ is connected. We aim to show that in the case
$\zeta(D)>0$, $\lambda_{A,f}$ does not exist. As seen from (b), this will
follow once we prove that $\zeta^{A}(\mathbb{R}^{n})<1$, which in turn is
reduced to proving

$(\zeta|_{D})^{A}(\mathbb{R}^{n})<\zeta|_{D}(\mathbb{R}^{n}).$ (4.34)

Due to the $\alpha$-thinness of $\overline{A}$ at infinity, there exists the
$\alpha$-Riesz equilibrium measure $\gamma$ of $\overline{A}$, treated in an
extended sense where $I(\gamma)$ as well as $\gamma(\mathbb{R}^{n})$ may be
infinite (for details, see [23, Section V.1.1], cf. [31, Section 5] and [32,
Sections 1.3, 2.1]). Applying [31, Theorem 8.7], we therefore get

$(\zeta|_{D})^{\overline{A}}(\mathbb{R}^{n})<\zeta|_{D}(\mathbb{R}^{n}),$

whence (4.34), for, in consequence of Propositions 2.21 and 2.22,

$(\zeta|_{D})^{A}(\mathbb{R}^{n})=\bigl{(}(\zeta|_{D})^{\overline{A}}\bigr{)}^{A}(\mathbb{R}^{n})\leqslant(\zeta|_{D})^{\overline{A}}(\mathbb{R}^{n}).$

##  5\. Acknowledgements

The author is deeply indebted to Bent Fuglede and Douglas P. Hardin for
reading and commenting on the manuscript. The present research was partially
supported by the NAS of Ukraine under grant 0122U000670.

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