# Zorii N. V.

### Extremal problems dual to the Gauss variational problem

Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 747–764

We formulate and solve extremal problems of potential theory that are dual to the Gauss variational problem but, unlike the latter, are always solvable. Statements on the compactness of classes of solutions and the continuity of extremals are also established.

### Necessary and Sufficient Conditions for the Solvability of the Gauss Variational Problem

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 60–83

We investigate the well-known Gauss variational problem considered over classes of Radon measures associated with a system of sets in a locally compact space. Under fairly general assumptions, we obtain necessary and sufficient conditions for its solvability. As an auxiliary result, we describe potentials of vague and (or) strong limit points of minimizing sequences of measures. The results obtained are also specified for the Newton kernel in $\mathbb{R}^n$.

### Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1513-1526

The concept of consistent kernels introduced by Fuglede in 1960 is widely used in extremal problems of the theory of potential on classes of positive measures. In the present paper, we show that this concept is also efficient for the investigation of extremal problems on fairly broad classes of signed measures. In particular, for an arbitrary consistent kernel in a locally compact space, we prove a theorem on the strong completeness of fairly general subspaces *E* of all measures with finite energy. (Note that, according to the well-known Cartan counterexample, the entire space *E* is strongly incomplete even in the classical case of the Newton kernel in ℝ^{n} Using this theorem, we obtain new results for the Gauss variational problem, namely, in the non-compact case, we give a description of vague and (or) strong limiting measures of minimizing sequences and obtain sufficient solvability conditions.

### Equilibrium Problems for Potentials with External Fields

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1315-1339

We investigate the problem on the minimum of energy over fairly general (generally speaking, noncompact) classes of real-valued Radon measures associated with a system of sets in a locally compact space in the presence of external fields. The classes of admissible measures are determined by a certain normalization or by a normalization and a certain majorant measure σ. In both cases, we establish sufficient conditions for the existence of minimizing measures and prove that, under fairly general assumptions, these conditions are also necessary. We show that, for sufficiently large σ, there is a close correlation between the facts of unsolvability (or solvability) of both variational problems considered.

### Equilibrium Potentials with External Fields

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1178-1195

We investigate the Gauss variational problem over fairly general classes of Radon measures in a locally compact space **X**. We describe potentials of minimizing measures, establish their characteristic properties, and prove the continuity of extremals. Extremal problems dual to the original one are formulated and solved. The results obtained are new even in the case of classical kernels and the Euclidean space \(\mathbb{R}^n \) .

### Extremal Problems in Logarithmic Potential Theory

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1220-1236

We pose and solve an extremal problem of logarithmic potential theory that is dual to the main minimum problem in the theory of interior capacities of condensers but, in contrast to the latter, it is solvable even in the case of a nonclosed condenser. Its solution is a natural generalization of the classical notion of interior equilibrium measure of a set. A condenser is treated as a finite collection of signed sets such that the closures of sets with opposite signs are pairwise disjoint. We also prove several assertions on the continuity of extremals.

### Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. III

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 758-782

We complete the construction of the theory of interior capacities of condensers in locally compact spaces begun in the previous two parts of the work. A condenser is understood as an ordered finite collection of sets each of which is marked with the sign + or − so that the closures of sets with opposite signs are mutually disjoint. The theory developed here is rich in content for arbitrary (not necessarily compact or closed) condensers. We obtain sufficient and (or) necessary conditions for the solvability of the main minimum problem of the theory of capacities of condensers and show that, under fairly general assumptions, these conditions form a criterion. For the main minimum problem (generally speaking, unsolvable even for a closed condenser), we pose and solve dual problems that are always solvable (even in the case of a nonclosed condenser). For all extremal problems indicated, we describe the potentials of minimal measures and investigate properties of extremals. As an auxiliary result, we solve the well-known problem of the existence of a condenser measure. The theory developed here includes (as special cases) the main results of the theory of capacities of condensers in \(\mathbb{R}^n\) , *n* ≥ 2, with respect to the classical kernels.

### Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. II

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 466-488

We continue the investigation of the problem of energy minimum for condensers began in the first part of the present work. Condensers are treated in a certain generalized sense. The main attention is given to the case of classes of measures noncompact in the vague topology. In the case of a positive-definite kernel, we develop an approach to this minimum problem based on the use of both strong and vague topologies in the corresponding semimetric spaces of signed Radon measures. We obtain necessary and (or) sufficient conditions for the existence of minimal measures. We describe potentials for properly determined extremal measures.

### Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. I

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 168-189

The present paper is the first part of a work devoted to the development of the theory of κ-capacities of condensers in a locally compact space ** X**; here, κ:

**×**

*X***→ (−∞, +∞] is a lower-semicontinuous function. Condensers are understood in a generalized sense. We investigate the corresponding problem on the minimum of energy on fairly general classes of normalized signed Radon measures. We describe potentials of minimal measures, establish their characteristic properties, and study the uniqueness problem. (The subsequent two parts of this work are devoted to the problem of existence of minimal measures in the noncompact case and to the development of the corresponding approaches and methods.) As an auxiliary result, we investigate the continuity of the mapping $$\left( {x,{\mu }} \right) \mapsto \int {\kappa \left( {x,y} \right)} d{\mu }\left( y \right),\quad \left( {x,{\mu }} \right) \in X \times \mathfrak{M}^ + \left( X \right),$$ where \(\mathfrak{M}^ +\) is the cone of positive measures in**

*X***equipped with the topology of vague convergence.**

*X*### A noncompact variational problem in the theory of riesz potentials. II

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 603-613

We study some generalizations of the well-known problem of minimization of the Riesz energy on condensers. Under fairly general assumptions, we establish necessary and sufficient conditions for the existence of minimal measures.

### Estimates of capacities of plane condensers

Ukr. Mat. Zh. - 1991. - 43, № 2. - pp. 193–199

### A variational problem in the theory of green potential. I

Ukr. Mat. Zh. - 1990. - 42, № 4. - pp. 494–500

### Extremal lengths and green capacities of condensers

Ukr. Mat. Zh. - 1990. - 42, № 3. - pp. 317–323

### Precise estimate of the 2-capacity of a condenser

Ukr. Mat. Zh. - 1990. - 42, № 2. - pp. 253–257

### Moduli of families of surfaces and the Green capacity of condensers

Ukr. Mat. Zh. - 1990. - 42, № 1. - pp. 64–69

### Functional characteristics of space condensers: Their properties and relations among them

Ukr. Mat. Zh. - 1987. - 39, № 5. - pp. 565–573

### An extremal problem on the minimum of energy for space condensers

Ukr. Mat. Zh. - 1986. - 38, № 4. - pp. 431–437

### Estimates of capacities and energies under reconstruction of condensers

Ukr. Mat. Zh. - 1980. - 32, № 6. - pp. 811–813