We study minimum energy problems relative to the $alpha$-Riesz kernel $|x-y|^{alpha-n}$, $alphain(0,2]$, over signed Radon measures $mu$ on $mathbb R^n$, $ngeqslant3$, associated with a generalized condenser $(A_1,A_2)$, where $A_1$ is a relatively c
losed subset of a domain $D$ and $A_2=mathbb R^nsetminus D$. We show that, though $A_2capmathrm{Cl}_{mathbb R^n}A_1$ may have nonzero capacity, this minimum energy problem is uniquely solvable (even in the presence of an external field) if we restrict ourselves to $mu$ with $mu^+leqslantxi$, where a constraint $xi$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted $alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze their supports. The approach developed is mainly based on the establishment of an intimate relationship between the constrained minimum $alpha$-Riesz energy problem over signed measures associated with $(A_1,A_2)$ and the constrained minimum $alpha$-Green energy problem over positive measures carried by $A_1$. The results are illustrated by examples.
We study the constrained minimum energy problem with an external field relative to the $alpha$-Riesz kernel $|x-y|^{alpha-n}$ of order $alphain(0,n)$ for a generalized condenser $mathbf A=(A_i)_{iin I}$ in $mathbb R^n$, $ngeqslant 3$, whose oppositel
y charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both vary, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our arguments are based particularly on the simultaneous use of the vague topology and a suitable semimetric structure on a set of vector measures associated with $mathbf A$, and the establishment of completeness theorems for proper semimetric spaces. The results remain valid for the logarithmic kernel on $mathbb R^2$ and $mathbf A$ with compact $A_i$, $iin I$. The study is illustrated by several examples.
There are many ways to generate a set of nodes on the sphere for use in a variety of problems in numerical analysis. We present a survey of quickly generated point sets on $mathbb{S}^2$, examine their equidistribution properties, separation, covering
, and mesh ratio constants and present a new point set, equal area icosahedral points, with low mesh ratio. We analyze numerically the leading order asymptotics for the Riesz and logarithmic potential energy for these configurations with total points $N<50,000$ and present some new conjectures.
In terms of the minimal $N$-point diameter $D_d(N)$ for $R^d,$ we determine, for a class of continuous real-valued functions $f$ on $[0,+infty],$ the $N$-point $f$-best-packing constant $min{f(|x-y|), :, x,yin R^d}$, where the minimum is taken over p
oint sets of cardinality $N.$ We also show that $$ N^{1/d}Delta_d^{-1/d}-2le D_d(N)le N^{1/d}Delta_d^{-1/d}, quad Nge 2,$$ where $Delta_d$ is the maximal sphere packing density in $R^d$. Further, we provide asymptotic estimates for the $f$-best-packing constants as $Ntoinfty$.
We investigate the asymptotic behavior, as $N$ grows, of the largest minimal pairwise distance of $N$ points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurat
ions. For this purpose, we compare best-packing configurations with minimal Riesz $s$-energy configurations and determine the $s$-th root asymptotic behavior (as $sto infty)$ of the minimal energy constants. We show that the upper and the lower dimension of a set defined through the Riesz energy or best-packing coincides with the upper and lower Minkowski dimension, respectively. For certain sets in ${rm {bf R}}^d$ of integer Hausdorff dimension, we show that the limiting behavior of the best-packing distance as well as the minimal $s$-energy for large $s$ is different for different subsequences of the cardinalities of the configurations.
For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a la
rge class of manifolds A having finite, positive d-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the d-dimensional sphere.
