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We study the constrained minimum energy problem with an external fieldrelative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order$\alpha\in(0,n)$ for a generalized condenser $\mathbf A=(A_i)_{i\in I}$ in$\mathbb R^n$, $n\geqslant 3$, whose oppositely charged plates intersect eachother over a set of zero capacity. Conditions sufficient for the existence ofminimizers are found, and their uniqueness and vague compactness are studied.Conditions obtained are shown to be sharp. We also analyze continuity of theminimizers in the vague and strong topologies when the condenser and theconstraint both vary, describe the weighted equilibrium vector potentials, andsingle out their characteristic properties. Our arguments are basedparticularly on the simultaneous use of the vague topology and a suitablesemimetric 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 $A_i$, $i\in I$. The study is illustrated by severalexamples.

Constrained minimum Riesz energy problems for a condenser with intersecting plates