Equilibrium Problems for Infinite Dimensional Vector Potentials with External Fields

Abstract

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.