Condensers with infinitely many touching Borel plates and minimum energy problems

Natalia Zorii¹

¹ Institute of Mathematics Academy of Sciences of Ukraine Tereshchenkivska 3 01601 Kyiv, Ukraine

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 2, pp. 125-163.

Résumé

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 $s_i=\pm 1$ prescribed such that $A_i\cap A_j=\varnothing $ whenever $s_is_j=-1$, we consider a minimum energy problem with an external field over infinite-dimensional vector measures $(\mu ^i)_{i\in I}$, where $\mu ^i$ is a suitably normalized positive Radon measure carried by $A_i$ and such that $\mu ^i\leq \xi ^i$ 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 to the (unconstrained) Gauss variational problem, which is in general unsolvable even for a condenser of two closed, oppositely signed plates in $\mathbb R^3$ 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 $\mathbb R^n$, $n\geq 2$, and closed $A_i$, $i\in I$, which is important for applications.

Détail

DOI : 10.4064/ba190414-29-11

Keywords: defining condenser locally compact space locally finite countable collection borel sets sign prescribed cap varnothing whenever consider minimum energy problem external field infinite dimensional vector measures where suitably normalized positive radon measure carried leq subset constraints nbsp being given varnothing problem reduces unconstrained gauss variational problem which general unsolvable even condenser closed oppositely signed plates nbsp mathbb coulomb kernel nevertheless provide sufficient conditions existence solutions stated problem its full generality establish vague compactness solutions analyze their uniqueness describe their weighted potentials single out their characteristic properties strong vague convergence minimizing nets minimizers studied phenomena non existence non uniqueness solutions problem illustrated examples results obtained even classical kernels mathbb geq closed which important applications