Potential Analysis

, Volume 38, Issue 2, pp 397–432 | Cite as

Equilibrium Problems for Infinite Dimensional Vector Potentials with External Fields



We consider a minimal energy problem with an external field over noncompact classes of infinite dimensional vector measures \((\mu^i)_{i\in I}\) on a locally compact space. The components μ i are positive measures normalized by \(\int g_i\,d\mu^i=a_i\) (where a i and g i are given) and supported by closed sets A i with the sign + 1 or − 1 prescribed such that A i  ∩ A j  = ∅ whenever \({\rm sign}\,A_i\ne{\rm sign}\,A_j\), and the law of interaction between μ i , i ∈ I, is determined by the matrix \(\bigl({\rm sign}\,A_i\,{\rm sign}\,A_j\bigr)_{i,j\in I}\). For positive definite kernels satisfying Fuglede’s condition of consistency, sufficient conditions for the existence of equilibrium measures are established and properties of their uniqueness, vague compactness, and strong and vague continuity are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also obtain variational inequalities for the weighted equilibrium potentials, single out their characteristic properties, and analyze continuity of the equilibrium constants. The results hold, e.g., for classical kernels in \(\mathbb R^n\), \(n\geqslant 2\), which is important in applications.


Vector potentials of infinite dimensions Minimal energy problems for vector measures with external fields Condensers Completeness theorem for vector measures 

Mathematics Subject Classification (2010)



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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Institute of MathematicsUkrainian Academy of SciencesKyiv-4Ukraine

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