With the sphere $mathbb{S}^2 subset mathbb{R}^3$ as a conductor holding a unit charge with logarithmic interactions, we consider the problem of determining the support of the equilibrium measure in the presence of an external field consisting of fini
tely many point charges on the surface of the sphere. We determine that for any such configuration, the complement of the equilibrium support is the stereographic preimage from the plane of a union of classical quadrature domains, whose orders sum to the number of point charges.
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.
Minimum Riesz energy problems in the presence of an external field are analyzed for a condenser with touching plates. We obtain sufficient and/or necessary conditions for the solvability of these problems in both the unconstrained and the constrained
settings, investigate the properties of minimizers, and prove their uniqueness. Furthermore, characterization theorems in terms of variational inequalities for the weighted potentials are established. The results obtained are illustrated by several examples.
