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Constrained Gauss variational problems for a condenser with intersecting plates

Tom 120 / 2017

Bent Fuglede, Natalia Zorii Annales Polonici Mathematici 120 (2017), 227-260 MSC: Primary 31C15. DOI: 10.4064/ap170619-2-12 Opublikowany online: 14 December 2017


We study a constrained Gauss variational problem relative to a positive definite kernel on a locally compact space for vector measures associated with a condenser $\mathbf A=(A_i)_{i\in I}$ whose oppositely charged plates intersect each other in a set of capacity zero. Sufficient conditions for the existence of minimizers are established, and their uniqueness and vague compactness are studied. Note that the classical (unconstrained) Gauss variational problem would be unsolvable in this formulation. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both vary, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our approach is based on the simultaneous use of the vague topology and a suitable semimetric structure defined in terms of energy on a set of vector measures associated with $\mathbf A$, and on the establishment of completeness results for proper semimetric spaces. The theory developed is valid in particular for the classical kernels, which is important for applications. The study is illustrated by several examples.


  • Bent FugledeDepartment of Mathematical Sciences
    University of Copenhagen
    2100 København, Denmark
  • Natalia ZoriiInstitute of Mathematics
    National Academy of Sciences of Ukraine
    Tereshchenkivska 3
    01601 Kyiv, Ukraine

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