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Constrained minimum Riesz energy problems for a condenser with intersecting plates. (arXiv:1710.01950v1 [math.CA])
来源于:arXiv
We study the constrained minimum energy problem with an external field
relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order
$\alpha\in(0,n)$ for a condenser $\mathbf A=(A_i)_{i\in I}$ in $\mathbb R^n$,
$n\geqslant 3$, whose oppositely 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 are both
varied, describe the weighted equilibrium vector potentials, and single out
their characteristic properties. Our arguments are particularly based 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 obtained remain
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