Constrained Gauss variational problems for a condenser with intersecting plates

Bent Fuglede¹ ; Natalia Zorii²

¹ Department of Mathematical Sciences University of Copenhagen 2100 København, Denmark
² Institute of Mathematics National Academy of Sciences of Ukraine Tereshchenkivska 3 01601 Kyiv, Ukraine

Annales Polonici Mathematici, Tome 120 (2017) no. 3, pp. 227-260.

Résumé

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.

Détail

DOI : 10.4064/ap170619-2-12

Keywords: study constrained gauss variational problem relative positive definite kernel locally compact space vector measures associated condenser mathbf whose oppositely charged plates intersect each other set capacity zero sufficient conditions existence minimizers established their uniqueness vague compactness studied note classical unconstrained gauss variational problem would unsolvable formulation analyze continuity minimizers vague strong topologies condenser constraint vary describe weighted equilibrium vector potentials single out their characteristic properties approach based simultaneous vague topology suitable semimetric structure defined terms energy set vector measures associated mathbf establishment completeness results proper semimetric spaces theory developed valid particular classical kernels which important applications study illustrated several examples