  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.
