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Defining a condenser in a locally compact space as a locally finite,countable collection of Borel sets $A_i$, $i\in I$, with the sign $s_i=\pm1$prescribed such that $A_i\cap A_j=\varnothing$ whenever $s_is_j=-1$, weconsider a minimum energy problem with an external field over infinitedimensional vector measures $(\mu^i)_{i\in I}$, where $\mu^i$ is a suitablynormalized positive Radon measure carried by $A_i$ and such that$\mu^i\leqslant\xi^i$ for all $i\in I_0$, $I_0\subset I$ and constraints$\xi^i$, $i\in I_0$, being given. If $I_0=\varnothing$, the problem reduces tothe (unconstrained) Gauss variational problem, which is in general unsolvableeven for a condenser of two closed, oppositely signed plates in $\mathbb R^3$and the Coulomb kernel. Nevertheless, we provide sufficient conditions for theexistence of solutions to the stated problem in its full generality, establishthe vague compactness of the solutions, analyze their uniqueness, describetheir weighted potentials, and single out their characteristic properties. Thestrong and the vague convergence of minimizing nets to the minimizers is alsostudied. The phenomena of non-existence and non-uniqueness of solutions to theproblem are illustrated by examples. The results obtained are new even for theclassical kernels on $\mathbb R^n$, $n\geqslant2$, and closed $A_i$, $i\in I$,which is important for applications.

Condensers with infinitely many touching Borel plates and minimum energy problems