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Condensers with infinitely many touching Borel plates and minimum energy problems

Zorii Natalia

Bulletin of the Polish Academy of Sciences. Mathematics

2019

Vol. 67, no. 2

125--163

Defining a condenser in a locally compact space as a locally finite, countable collection of Borel sets Ai, i ϵ I, with the sign si = ±1 prescribed such that Ai ∩ Aj = Ø; whenever sisj = −1, we consider a minimum energy problem with an external field over infinite-dimensional vector measures (µi) i ϵ I, where µi is a suitably normalized positive Radon measure carried by Ai and such that µi ≤ ξi for all i ϵ I0, I0 ⊂ I and constraints ξi, i ϵ I0, being given. If I0 = Ø, the problem reduces to the (unconstrained) Gauss variational problem, which is in general unsolvable even for a condenser of two closed, oppositely signed plates in R3 and the Coulomb kernel. Nevertheless, we provide sufficient conditions for the existence of solutions to the stated problem in its full generality, establish the vague compactness of the solutions, analyze their uniqueness, describe their weighted potentials, and single out their characteristic properties. The strong and the vague convergence of minimizing nets to the minimizers is studied. The phenomena of non-existence and non-uniqueness of solutions to the problem are illustrated by examples. The results obtained are new even for the classical kernels on Rn, n ≥ 2, and closed Ai, i ϵ I, which is import ant for applications.