arXiv subjects

Natalia Zorii

Publications and source records attributed to Natalia Zorii.

At least 19 recordsLinked to original sources

arXiv2025

Weighted minimum $α$-Green energy problems

For the $α$-Green kernel $g^α_D$ on a domain $D\subset\mathbb R^n$, $n\geqslant2$, associated with the $α$-Riesz kernel $|x-y|^{α-n}$, where $α\in(0,n)$ and $α\leqslant2$, and a relatively closed set $F\subset D$, we investigate the problem on minimizing the Gauss functional \[\int g^α_D(x,y)\,d(μ\otimesμ)(x,y)-2\int g^α_D(x,y)\,d(\vartheta\otimesμ)(x,y),\] $\vartheta$ being a given positive (Radon) measure concentrated on $D\setminus F$, and $μ$ ranging over all probability measures of finite energy, supported in $D$ by $F$. For suitable $\vartheta$, we find necessary and/or sufficient conditions for the existence of the solution to the problem, give a description of its support, provide various alternative characterizations, and prove convergence theorems when $F$ is approximated by partially ordered families of sets. The analysis performed is substantially based on the perfectness of the $α$-Green kernel, discovered by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018).

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arXiv2025

On Fuglede's problem on pseudo-balayage for signed Radon measures of infinite energy

For suitable kernels on a locally compact space, we develop a theory of inner (outer) pseudo-balayage of quite general signed Radon measures (not necessarily of finite energy) onto quite general sets (not necessarily closed). Such investigations were initiated in Fuglede's study (Anal. Math., 2016), which was, however, mainly concerned with the outer pseudo-balayage of positive measures of finite energy. The results thereby obtained solve Fuglede's problem, posed to the author in a private correspondence (2016), whether his theory could be extended to measures of infinite energy. An application of this theory to weighted minimum energy problems is also given.

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arXiv2025

General theory of balayage on locally compact spaces. Applications to weighted minimum energy problems

Under suitable requirements on a kernel on a locally compact space, we develop a theory of inner (outer) balayage of quite general Radon measures $ω$ (not necessarily of finite energy) onto quite general sets (not necessarily closed). We prove the existence and the uniqueness of inner (outer) swept measures, analyze their properties, and provide a number of alternative characterizations. In spite of being in agreement with Cartan's theory of Newtonian balayage, the results obtained require essentially new methods and approaches, since in the case in question, useful specific features of Newtonian potentials may fail to hold. The theory thereby established extends considerably that by Fuglede (Anal. Math., 2016) and that by the author (Anal. Math., 2022), these two dealing with $ω$ of finite energy. Such a generalization enables us to improve substantially our recent results on the Gauss variational problem (Constr. Approx., 2024), by strengthening their formulations and/or by extending the area of their validity. This study covers many interesting kernels in classical and modern potential theory, which also looks promising for other applications.

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arXiv2024

Balayage, equilibrium measure, and Deny's principle of positivity of mass for $α$-Green potentials

In the theory of $g_α$-potentials on a domain $D\subset\mathbb R^n$, $n\geqslant2$, $g_α$ being the $α$-Green kernel associated with the $α$-Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,n)$, $α\leqslant2$, we establish the existence and uniqueness of the $g_α$-balayage $μ^F$ of a positive Radon measure $μ$ onto a relatively closed set $F\subset D$, we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for $μ^F(D)=μ(D)$ to hold, given in terms of the $α$-harmonic measure of suitable Borel subsets of $\overline{\mathbb R^n}$, the one-point compactification of $\mathbb R^n$. As a by-product, we find necessary and/or sufficient conditions for the existence of the $g_α$-equilibrium measure $γ_F$, $γ_F$ being understood in an extended sense where $γ_F(D)$ might be infinite. We also discover quite a surprising version of Deny's principle of positivity of mass for $g_α$-potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed.

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arXiv2024

Integral representation of balayage on locally compact spaces and its application

In the theory of inner and outer balayage of positive Radon measures on a locally compact space $X$ to arbitrary $A\subset X$ with respect to suitable, quite general function kernels, developed in a series of the author's recent papers, we find conditions ensuring the validity of the integral representations. The results thereby obtained do hold and seem to be largely new even for several interesting kernels in classical and modern potential theory, which looks promising for possible applications. As an example of such applications, we analyze how the total mass of a measure varies under its balayage with respect to fractional Green kernels.

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arXiv2024

Balayage of Radon measures of infinite energy on locally compact spaces

For suitable kernels on a locally compact space $X$, we develop a theory of inner balayage of quite general Radon measures $ω$ (not necessarily of finite energy) to arbitrary $A\subset X$. In the case where $A$ is Borel, this theory provides, as a by-product, a theory of outer balayage. We prove the existence and the uniqueness of inner (outer) swept measures, analyze their properties, and provide a number of alternative characterizations. In spite of being in agreement with Cartan's theory of Newtonian balayage, the results obtained require essentially new methods and approaches, since in the case in question, useful specific features of Newtonian potentials may fail to hold. The theory thereby established generalizes substantially the existing ones, pertaining either to $ω$ of finite energy, or to some particular $A$ (e.g. quasiclosed). This work covers many interesting kernels in classical and modern potential theory, which looks promising for possible applications.

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arXiv2023

Inner Riesz balayage in minimum energy problems with external fields

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arXiv2023

Fractional harmonic measure in minimum Riesz energy problems with external fields

For the Riesz kernel $κ_α(x,y):=|x-y|^{α-n}$ on $\mathbb R^n$, where $n\geqslant2$, $α\in(0,2]$, and $α<n$, we consider the problem of minimizing the Gauss functional \[\intκ_α(x,y)\,d(μ\otimesμ)(x,y)+2\int f_{q,z}\,dμ,\quad\text{where $f_{q,z}:=-q\intκ_α(\cdot,y)\,d\varepsilon_z(y)$},\] $q$ being a positive number, $\varepsilon_z$ the unit Dirac measure at $z\in\mathbb R^n$, and $μ$ ranging all probability measures of finite energy, concentrated on quasiclosed $A\subset\mathbb R^n$. For any $z\in A^u\cup(\mathbb R^n\setminus{\rm Cl}_{\mathbb R^n}A)$, where $A^u$ is the set of all inner $α$-ultrairregular points for $A$, we provide necessary and sufficient conditions for the existence of the minimizer $λ_{A,f_{q,z}}$, establish its alternative characterizations, and describe its support, thereby discovering new interesting phenomena. In detail, $z\in\partial_{\mathbb R^n}A$ is said to be inner $α$-ultrairregular if the inner $α$-harmonic measure $\varepsilon_z^A$ of $A$ is of finite energy. We show that for any $z\in A^u\cup(\mathbb R^n\setminus{\rm Cl}_{\mathbb R^n}A)$, $λ_{A,f_{q,z}}$ exists if and only if either $A$ is of finite inner capacity, or $q\geqslant H_z$, where $H_z:=1/\varepsilon_z^A(\mathbb R^n)\in[1,\infty)$. Thus, for any closed $A$, any $z\in A^u$, and any $q\geqslant H_z$ -- even arbitrarily large, no compensation effect occurs between the two oppositely signed charges, $-q\varepsilon_z$ and $λ_{A,f_{q,z}}$, carried by the same conductor $A$, which seems to contradict our physical intuition.

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arXiv2023

On the role of the point at infinity in Deny's principle of positivity of mass for Riesz potentials

First introduced by J. Deny, the classical principle of positivity of mass states that if $κ_αμ\leqslantκ_αν$ everywhere on $\mathbb{R}^n$, then $μ(\mathbb{R}^n)\leqslantν(\mathbb{R}^n)$. Here $μ,ν$ are positive Radon measures on $\mathbb{R}^n$, $n\geqslant2$, and $κ_αμ$ is the potential of $μ$ with respect to the Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,2]$, $α 0$, then $κ_αξ>0$ everywhere on $\mathbb{R}^n$, except for a subset which is inner $α$-thin at infinity. The analysis performed is based on the author's recent theories of inner Riesz balayage and inner Riesz equilibrium measures (Potential Anal., 2022), the inner equilibrium measure being understood in an extended sense where both the energy and the total mass may be infinite.

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arXiv2023

Minimum Riesz energy problems with external fields

The paper deals with minimum energy problems in the presence of external fields with respect to the Riesz kernels $|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$. For quite a general (not necessarily lower semicontinuous) external field $f$, we obtain necessary and/or sufficient conditions for the existence of $λ_{A,f}$ minimizing the Gauss functional \[\int|x-y|^{α-n}\,d(μ\otimesμ)(x,y)+2\int f\,dμ\] over all positive Radon measures $μ$ with $μ(\mathbb R^n)=1$, concentrated on quite a general (not necessarily closed) $A\subset\mathbb R^n$. We also provide various alternative characterizations of the minimizer $λ_{A,f}$, analyze the continuity of both $λ_{A,f}$ and the modified Robin constant for monotone families of sets, and give a description of the support of $λ_{A,f}$. The significant improvement of the theory in question thereby achieved is due to a new approach based on the close interaction between the strong and the vague topologies, as well as on the theory of inner balayage, developed recently by the author.

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arXiv2023

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

For the Riesz kernel $κ_α(x,y):=|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$, we introduce the inner pseudo-balayage $\hatω^A$ of a (Radon) measure $ω$ on $\mathbb R^n$ to a set $A\subset\mathbb R^n$ as the (unique) measure minimizing the Gauss functional \[\intκ_α(x,y)\,d(μ\otimesμ)(x,y)-2\intκ_α(x,y)\,d(ω\otimesμ)(x,y)\] over the class $\mathcal E^+(A)$ of all positive measures $μ$ of finite energy, concentrated on $A$. For quite general signed $ω$ (not necessarily of finite energy) and $A$ (not necessarily closed), such $\hatω^A$ does exist, and it maintains the basic features of inner balayage for positive measures (defined when $α\leqslant2$), 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 $\hatω^A$ is further shown to be a powerful tool in the problem of minimizing the Gauss functional over all $μ\in\mathcal E^+(A)$ with $μ(\mathbb R^n)=1$, 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 $ω$ 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 $\hatω^A(\mathbb R^n)\geqslant1$.

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arXiv2022

Minimum energy problems with external fields on locally compact spaces

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 $κ$ 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 $λ_{A,f}$ minimizing the Gauss functional \[\intκ(x,y)\,d(μ\otimesμ)(x,y)+2\int f\,dμ\] over all positive Radon measures $μ$ with $μ(X)=1$, 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 $|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$, and are illustrated by some examples. Furthermore, we provide various alternative characterizations of the minimizer $λ_{A,f}$, and as a by-product we analyze the strong and vague continuity of $λ_{A,f}$ 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.

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arXiv2022

On the theory of capacities on locally compact spaces and its interaction with the theory of balayage

The paper deals with the theory of inner (outer) capacities on locally compact spaces with respect to general function kernels, the main emphasis being placed on the establishment of alternative characterizations of inner (outer) capacities and inner (outer) capacitary measures for arbitrary sets. The analysis is substantially based on the close interaction between the theory of capacities and that of balayage. As a by-product, we provide a rigorous justification of Fuglede's theories of inner and outer capacitary measures and capacitability (Acta Math., 1960). The results obtained are largely new even for the logarithmic, Newtonian, Green, $α$-Riesz, and $α$-Green kernels.

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arXiv2021

On the theory of balayage on locally compact spaces

The paper deals with the theory of balayage of Radon measures $μ$ of finite energy on a locally compact space $X$ with respect to a consistent kernel $κ$ satisfying the domination principle. Such theory is now specified for the case where the topology on $X$ has a countable base, while any $f\in C_0(X)$, a continuous function on $X$ of compact support, can be approximated in the inductive limit topology on the space $C_0(X)$ by potentials $κλ:=\intκ(\cdot,y)\,dλ(y)$ of measures $λ$ of finite energy. In particular, we show that then the inner balayage can always be reduced to balayage to Borel sets. In more details, for arbitrary $A\subset X$, there exists a $K_σ$-set $A_0\subset A$ such that $μ^A=μ^{A_0}=μ^{*A_0}$ for all $μ$, $μ^A$ and $μ^{*A}$ denoting the inner and the outer balayage of $μ$ to $A$, respectively. Furthermore, $μ^A$ is now uniquely determined by the symmetry relation $\intκμ^A\,dλ=\intκλ^A\,dμ$, $λ$ ranging over a certain countable family of measures depending on $X$ and $κ$ only. As an application of these theorems, we analyze the convergence of inner and outer swept measures and their potentials. The results obtained do hold for many interesting kernels in classical and modern potential theory on $\mathbb R^n$, $n\geqslant2$.

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arXiv2020

Balayage of measures on a locally compact space

We develop a theory of inner balayage of a positive Radon measure $μ$ of finite energy on a locally compact space $X$ to arbitrary $A\subset X$, generalizing Cartan's theory of Newtonian inner balayage on $\mathbb R^n$, $n\geqslant3$, to a suitable function kernel on $X$. As an application of the theory thereby established, we show that if the space $X$ is perfectly normal and of class $K_σ$, then a recent result by Bent Fuglede (Anal. Math., 2016) on outer balayage of $μ$ to quasiclosed $A$ remains valid for arbitrary Borel $A$. We give in particular various alternative definitions of inner (outer) balayage, provide a formula for evaluation of its total mass, and prove convergence theorems for inner (outer) swept measures and their potentials. The results obtained do hold (and are new in part) for most classical kernels on $\mathbb R^n$, $n\geqslant2$, which is important in applications.

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arXiv2020

Harmonic measure, equilibrium measure, and thinness at infinity in the theory of Riesz potentials

Focusing first on the inner $α$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $μ^A$ the inner $α$-Riesz balayage of a Radon measure $μ$ to $A\subset\mathbb R^n$ arbitrary), we describe its Euclidean support, provide a formula for evaluation of its total mass, establish the vague continuity of the map $y\mapsto\varepsilon_y^A$ outside the inner $α$-irregular points for $A$, and obtain necessary and sufficient conditions for $\varepsilon_y^A$ to be of finite energy (more generally, for $\varepsilon_y^A$ to be absolutely continuous with respect to inner capacity) as well as for $\varepsilon_y^A(\mathbb R^n)\equiv1$ to hold. Those criteria are given in terms of the newly defined concepts of $α$-thinness and $α$-ultrathinness at infinity that generalize the concepts of thinness at infinity by Doob and Brelot, respectively. Further, we extend some of these results to $μ^A$ general by verifying the formula $μ^A=\int\varepsilon_y^A\,dμ(y)$. We also show that there is a $K_σ$-set $A_0\subset A$ such that $μ^A=μ^{A_0}$ for all $μ$, and give various applications of this theorem. In particular, we prove the vague and strong continuity of the inner swept, resp. equilibrium, measure under the approximation of $A$ arbitrary, thereby strengthening Fuglede's result established for $A$ Borel (Acta Math., 1960). Being new even for $α=2$, the results obtained also present a further development of the theory of inner Newtonian capacities and of inner Newtonian balayage, originated by Cartan.

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arXiv2019

A concept of weak Riesz energy with application to condensers with touching plates

We proceed further with the study of minimum weak Riesz energy problems for condensers with touching plates, initiated jointly with Bent Fuglede (Potential Anal. 51 (2019), 197--217). Having now added to the analysis constraint and external source of energy, we obtain a Gauss type problem, but with weak energy involved. We establish sufficient and/or necessary conditions for the existence of solutions to the problem and describe their potentials. Treating the solution as a function of the condenser and the constraint, we prove its continuity relative to the vague topology and the topologies determined by the weak and standard energy norms. We show that the criteria for the solvability thus obtained fail in general once the problem is reformulated in the setting of standard energy, thereby justifying an advantage of weak energy when dealing with condensers with touching plates.

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arXiv2019

A theory of inner Riesz balayage and its applications

We establish the theory of balayage for the Riesz kernel $|x-y|^{α-n}$, $α\in(0,2]$, on $\mathbb R^n$, $n\geqslant3$, alternative to that suggested in the book by Landkof. A need for that is caused by the fact that the balayage in that book is defined by means of the integral representation, which, however, so far is not completely justified. Our alternative approach is mainly based on Cartan's ideas concerning inner balayage, formulated by him for the Newtonian kernel. Applying the theory of inner Riesz balayage thereby developed, we obtain a number of criteria for the existence of an inner equilibrium measure $γ_A$ for $A\subset\mathbb R^n$ arbitrary, in particular given in terms of the total mass of the inner swept measure $μ^A$ with $μ$ suitably chosen. For example, $γ_A$ exists if and only if $\varepsilon^{A^*}\ne\varepsilon$, where $\varepsilon$ is a Dirac measure at $x=0$ and $A^*$ the inverse of $A$ relative to the sphere $|x|=1$, which leads to a Wiener type criterion of inner $α$-irregularity. The results obtained are illustrated by examples.

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