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We investigate minimum weak $\alpha$-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 $\alpha$-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 $\alpha$-Riesz energy problems over signed measures associated with $(A_1,A_2)$ and minimum $\alpha$-Green energy problems over positive measures carried by $A_1$. Crucial for our analysis of the latter problems is the perfectness of the $\alpha$-Green kernel, established in our recent paper. As an application of the results obtained, we describe the support of the $\alpha$-Green equilibrium measure., Comment: 31 pages

In view of a recent example of a positive Radon measure $\mu$ on a domain $D\subset\mathbb R^n$, $n\geqslant3$, such that $\mu$ is of finite energy $E_g(\mu)$ relative to the $\alpha$-Green kernel $g$ on $D$, though the energy of $\mu-\mu^{D^c}$ relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, $0<\alpha\leqslant2$, is not well defined (here $\mu^{D^c}$ is the $\alpha$-Riesz swept measure of $\mu$ onto $D^c=\mathbb R^n\setminus D$), we propose a weaker concept of $\alpha$-Riesz energy for which this defect has been removed. This concept is applied to the study of a minimum weak $\alpha$-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 $\mu$ on $D$ with finite $E_g(\mu)$ in the metric determined by the norm $\|\mu\|_g:=\sqrt{E_g(\mu)}$. We also show that the pre-Hilbert space of Radon measures on $\mathbb R^n$ with finite weak $\alpha$-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., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1802.07171, arXiv:1711.05484

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 $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, $\alpha\in(0,2]$, over positive vector Radon measures $\boldsymbol\mu=(\mu^i)_{i\in I}$ such that each $\mu^i$, $i\in I$, is carried by $A_i$ and normalized by $\mu^i(A_i)=a_i\in(0,\infty)$. 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 $\boldsymbol\lambda^{\boldsymbol\xi}_{\mathbf A}=(\lambda^i_{\mathbf A})_{i\in I}$ (also in the presence of an external field) if we restrict ourselves to $\boldsymbol\mu$ with $\mu^i\leqslant\xi^i$, $i\in I$, where the constraint $\boldsymbol\xi=(\xi^i)_{i\in I}$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted vector $\alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $\lambda^i_{\mathbf A}$, $i\in I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $\alpha$-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 $\alpha$-Riesz energy problem and a constrained minimum $\alpha$-Green energy problem, suitably formulated. The results are illustrated by examples., Comment: 35 pages. arXiv admin note: substantial text overlap with arXiv:1711.05484

We study properties of the $\alpha$-Green kernel $g_D^\alpha$ of order $0<\alpha\leqslant2$ for a domain $D\subset\mathbb R^n$, $n\geqslant3$. This kernel is associated with the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, $x,y\in\mathbb R^n$, in a manner particularly well known in the case $\alpha=2$. Besides the usual principles of potential theory, we establish for the $\alpha$-Green kernel the property of consistency. This allows us to prove the completeness of the cone of positive measures $\mu$ on $D$ with finite energy $g_D^\alpha(\mu,\mu):=\iint g_D^\alpha(x,y)\,d\mu(x)\,d\mu(y)$ in the topology defined by the energy norm $\|\mu\|_{g_D^\alpha}=\sqrt{g_D^\alpha(\mu,\mu)}$, as well as the existence of the $\alpha$-Green equilibrium measure for a relatively closed set in $D$ of finite $\alpha$-Green capacity. The main tool is a generalization of Cartan's theory of balayage (sweeping) for the Newtonian kernel to the $\alpha$-Riesz kernels with $0<\alpha<2$., Comment: 29 pages