  We consider a constrained minimal energy problem with an external field over
noncompact classes of infinite dimensional vector measures on a locally compact
space. The components are positive measures (charges) that are constrained from
above, satisfy some normalizing assumptions, and are supported by given closed
sets (plates) with the sign +1 or -1 prescribed such that the oppositely signed
plates are mutually disjoint and the interaction matrix for the charges
corresponds to an electrostatic interpretation of a condenser. For all positive
definite kernels satisfying Fuglede's condition of consistency between the
weak* and strong topologies, sufficient conditions for the existence of
minimizers are established and their uniqueness and weak* compactness are
studied. Examples illustrating the sharpness of the sufficient conditions are
provided. We also analyze continuity properties of minimizers in both the weak*
and strong topologies when the constraints and the plates are varied
simultaneously. The results are new even for classical kernels in an Euclidean
space, which is important in applications.
