arXiv subjects

Natalia Zorii

Publications and source records attributed to Natalia Zorii.

32 records · Page 2Linked to original sources

arXiv2019

Condensers with infinitely many touching Borel plates and minimum energy problems

Defining a condenser in a locally compact space as a locally finite, countable collection of Borel sets $A_i$, $i\in I$, with the sign $s_i=\pm1$ prescribed such that $A_i\cap A_j=\varnothing$ whenever $s_is_j=-1$, we consider a minimum energy problem with an external field over infinite dimensional vector measures $(μ^i)_{i\in I}$, where $μ^i$ is a suitably normalized positive Radon measure carried by $A_i$ and such that $μ^i\leqslantξ^i$ for all $i\in I_0$, $I_0\subset I$ and constraints $ξ^i$, $i\in I_0$, being given. If $I_0=\varnothing$, the problem reduces to the (unconstrained) Gauss variational problem, which is in general unsolvable even for a condenser of two closed, oppositely signed plates in $\mathbb R^3$ and the Coulomb kernel. Nevertheless, we provide sufficient conditions for the existence of solutions to the stated problem in its full generality, establish the vague compactness of the solutions, analyze their uniqueness, describe their weighted potentials, and single out their characteristic properties. The strong and the vague convergence of minimizing nets to the minimizers is also studied. The phenomena of non-existence and non-uniqueness of solutions to the problem are illustrated by examples. The results obtained are new even for the classical kernels on $\mathbb R^n$, $n\geqslant2$, and closed $A_i$, $i\in I$, which is important for applications.

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arXiv2018

Various concepts of Riesz energy of measures and application to condensers with touching plates

We investigate minimum weak $α$-Riesz energy problems with external fields in both the unconstrained and constrained settings for generalized condensers $(A_1,A_2)$ such that the closures of $A_1$ and $A_2$ in $\mathbb R^n$ are allowed to intersect one another. (Such problems with the standard $α$-Riesz energy in place of the weak one would be unsolvable, which justifies the need for the concept of weak energy when dealing with condenser problems.) We obtain sufficient and/or necessary conditions for the existence of minimizers, provide descriptions of their supports and potentials, and single out their characteristic properties. To this end we have discovered an intimate relation between minimum weak $α$-Riesz energy problems over signed measures associated with $(A_1,A_2)$ and minimum $α$-Green energy problems over positive measures carried by $A_1$. Crucial for our analysis of the latter problems is the perfectness of the $α$-Green kernel, established in our recent paper. As an application of the results obtained, we describe the support of the $α$-Green equilibrium measure.

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arXiv2018

An alternative concept of Riesz energy of measures with application to generalized condensers

In view of a recent example of a positive Radon measure $μ$ on a domain $D\subset\mathbb R^n$, $n\geqslant3$, such that $μ$ is of finite energy $E_g(μ)$ relative to the $α$-Green kernel $g$ on $D$, though the energy of $μ-μ^{D^c}$ relative to the $α$-Riesz kernel $|x-y|^{α-n}$, $0<α\leqslant2$, is not well defined (here $μ^{D^c}$ is the $α$-Riesz swept measure of $μ$ onto $D^c=\mathbb R^n\setminus D$), we propose a weaker concept of $α$-Riesz energy for which this defect has been removed. This concept is applied to the study of a minimum weak $α$-Riesz energy problem over (signed) Radon measures on $\mathbb R^n$ associated with a (generalized) condenser ${\mathbf A}=(A_1,D^c)$, where $A_1$ is a relatively closed subset of $D$. A solution to this problem exists if and only if the $g$-capacity of $A_1$ is finite, which in turn holds if and only if there exists a so-called measure of the condenser $\mathbf A$, whose existence was analyzed earlier in different settings by Beurling, Deny, Kishi, Bliedtner, and Berg. Our analysis is based particularly on our recent result on the completeness of the cone of all positive Radon measures $μ$ on $D$ with finite $E_g(μ)$ in the metric determined by the norm $\|μ\|_g:=\sqrt{E_g(μ)}$. We also show that the pre-Hilbert space of Radon measures on $\mathbb R^n$ with finite weak $α$-Riesz energy is isometrically imbedded into its completion, the Hilbert space of real-valued tempered distributions with finite energy, defined with the aid of Fourier transformation. This gives an answer in the negative to a question raised by Deny in 1950.

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arXiv2018

Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser

For a finite collection $\mathbf A=(A_i)_{i\in I}$ of locally closed sets in $\mathbb R^n$, $n\geqslant3$, with the sign $\pm1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $α$-Riesz kernel $|x-y|^{α-n}$, $α\in(0,2]$, over positive vector Radon measures $\boldsymbolμ=(μ^i)_{i\in I}$ such that each $μ^i$, $i\in I$, is carried by $A_i$ and normalized by $μ^i(A_i)=a_i\in(0,\infty)$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution $\boldsymbolλ^{\boldsymbolξ}_{\mathbf A}=(λ^i_{\mathbf A})_{i\in I}$ (also in the presence of an external field) if we restrict ourselves to $\boldsymbolμ$ with $μ^i\leqslantξ^i$, $i\in I$, where the constraint $\boldsymbolξ=(ξ^i)_{i\in I}$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted vector $α$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $λ^i_{\mathbf A}$, $i\in I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $α$-Riesz energy on a set of vector measures associated with $\mathbf A$, as well as on the establishment of an intimate relationship between the constrained minimum $α$-Riesz energy problem and a constrained minimum $α$-Green energy problem, suitably formulated. The results are illustrated by examples.

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arXiv2017

Balayage for Riesz kernels with application to potential theory for the associated Green kernels

We study properties of the $α$-Green kernel $g_D^α$ of order $0<α\leqslant2$ for a domain $D\subset\mathbb R^n$, $n\geqslant3$. This kernel is associated with the $α$-Riesz kernel $|x-y|^{α-n}$, $x,y\in\mathbb R^n$, in a manner particularly well known in the case $α=2$. Besides the usual principles of potential theory, we establish for the $α$-Green kernel the property of consistency. This allows us to prove the completeness of the cone of positive measures $μ$ on $D$ with finite energy $g_D^α(μ,μ):=\iint g_D^α(x,y)\,dμ(x)\,dμ(y)$ in the topology defined by the energy norm $\|μ\|_{g_D^α}=\sqrt{g_D^α(μ,μ)}$, as well as the existence of the $α$-Green equilibrium measure for a relatively closed set in $D$ of finite $α$-Green capacity. The main tool is a generalization of Cartan's theory of balayage (sweeping) for the Newtonian kernel to the $α$-Riesz kernels with $0<α<2$.

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arXiv2016

Minimal energy problems for strongly singular Riesz kernels

We study minimal energy problems for strongly singular Riesz kernels on a manifold. Based on the spatial energy of harmonic double layer potentials, we are motivated to formulate the natural regularization of such problems by switching to Hadamard's partie finie integral operator which defines a strongly elliptic pseudodifferential operator on the manifold. The measures with finite energy are shown to be elements from the corresponding Sobolev space, and the associated minimal energy problem admits a unique solution. We relate our continuous approach also to the discrete one, which has been worked out earlier by D.P. Hardin and E.B. Saff.

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arXiv2015

Constrained Gauss variational problem for condensers with touching plates

We study a constrained minimum energy problem with an external field relative to the Riesz kernel of an arbitrary order for a generalized condenser with touching oppositely-charged plates. Conditions sufficient for the solvability of the problem are obtained. Our arguments are mainly based on the definition of an appropriate metric structure on a set of vector measures associated with a generalized condenser and the establishment of a completeness theorem for the corresponding metric space.

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arXiv2012

Necessary and sufficient conditions for the solvability of the Gauss variational problem for infinite dimensional vector measures

We continue our investigation of the Gauss variational problem for infinite dimensional vector measures associated with a condenser $(A_i)_{i\in I}$. It has been shown in Potential Anal., DOI:10.1007/s11118-012-9279-8 that, if some of the plates (say $A_\ell$ for $\ell\in L$) are noncompact then, in general, there exists a vector $\mathbf a=(a_i)_{i\in I}$, prescribing the total charges on $A_i$, $i\in I$, such that the problem admits no solution. Then, what is a description of all the vectors $\mathbf a$ for which the Gauss variational problem is nevertheless solvable? Such a characterization is obtained for a positive definite kernel satisfying Fuglede's condition of perfectness; it is given in terms of a solution to an auxiliary extremal problem intimately related to the operator of orthogonal projection onto the cone of all positive scalar measures supported by $\bigcup_{\ell\in L}A_\ell$. The results are illustrated by examples pertaining to the Riesz kernels.

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arXiv2010

Constrained energy problems with external fields for infinite dimensional vector measures

We consider a constrained minimal energy problem with an external field over noncompact classes of infinite dimensional vector measures on a locally compact space. The components are positive measures (charges) that are constrained from above, satisfy some normalizing assumptions, and are supported by given closed sets (plates) with the sign +1 or -1 prescribed such that the oppositely signed plates are mutually disjoint and the interaction matrix for the charges corresponds to an electrostatic interpretation of a condenser. For all positive definite kernels satisfying Fuglede's condition of consistency between the weak* and strong topologies, sufficient conditions for the existence of minimizers are established and their uniqueness and weak* compactness are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also analyze continuity properties of minimizers in both the weak* and strong topologies when the constraints and the plates are varied simultaneously. The results are new even for classical kernels in an Euclidean space, which is important in applications.

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arXiv2010

Constrained energy problems with external fields

Given a positive definite kernel in a locally compact space, we study a minimal energy problem in the presence of an external field over the class of all nonnegative Radon measures that are supported by a given closed noncompact set, satisfy certain normalizing assumptions, and do not exceed a fixed measure serving as a constraint. Under general assumptions, we establish the existence of a minimizing measure and analyze its continuity properties in the weak* and strong topologies when the set and the constraint are both varied. We also give a description of the weighted potential of a minimizing measure and single out its characteristic properties. Such results are mostly new even for classical kernels, which is important in applications.

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arXiv2010

Minimizing measures on condensers of infinitely many plates

The study deals with a minimal energy problem over noncompact classes of infinite dimensional vector measures in a locally compact space. The components are positive measures (charges) satisfying certain normalizing assumptions and supported by given closed sets (plates) with the sign +1 or -1 prescribed such that oppositely signed sets are mutually disjoint, and the interaction matrix for the charges corresponds to an electrostatic interpretation of a condenser. For all positive definite kernels satisfying Fuglede's condition of consistency between the weak* and strong topologies, sufficient conditions for the existence of equilibrium measures are established and properties of their uniqueness, vague compactness, and continuity are studied.

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arXiv2009

Equilibrium problems for infinite dimensional vector potentials with external fields

The study deals with a minimal energy problem in the presence of an external field over noncompact classes of vector measures of infinite dimension in a locally compact space. The components are positive measures (charges) satisfying certain normalizing assumptions and supported by given closed sets (plates) with the sign +1 or -1 prescribed such that oppositely signed sets are mutually disjoint, and the interaction matrix for the charges corresponds to an electrostatic interpretation of a condenser. For all positive definite kernels satisfying Fuglede's condition of consistency between the weak* and strong topologies, sufficient conditions for the existence of equilibrium measures are established and properties of their uniqueness, vague compactness, and continuity are studied. We also obtain variational inequalities for the weighted equilibrium potentials, single out their characteristic properties, and analyze continuity of the equilibrium constants.

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arXiv2009

Interior capacities of condensers with infinitely many plates in a locally compact space

The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a countable, locally finite collection of arbitrary sets with the sign +1 or -1 prescribed such that the closures of opposite-signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in an Euclidean space and closed condensers of finitely many plates). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing some new equivalent definitions of the capacity), but now always solvable (e.g., even for nonclosed, unbounded condensers of infinitely many plates). For all positive definite kernels satisfying B. Fuglede's condition of consistency between the strong and weak-star topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior capacitary distributions associated with a set. We give a description of those solutions, establish statements on their uniqueness and continuity, and point out their characteristic properties.

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arXiv2009

Interior capacities of condensers in locally compact spaces

The study is motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory of interior capacities of condensers in a locally compact space is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newton, Green, or Riesz kernels in an Euclidean space and closed condensers). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing some new equivalent definitions of the capacity), but always solvable (e.g., even for nonclosed condensers). For all positive definite kernels satisfying B. Fuglede's condition of consistency between the strong and vague topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior capacitary distributions associated with a set. We give a description of those solutions, establish statements on their uniqueness and continuity, and point out their characteristic properties. A condenser is treated as a finite collection of arbitrary sets with sing +1 or -1 prescribed, such that the closures of opposite-signed sets are mutually disjoint.

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