Constrained energy problems with external fields for vector measures

✍ Scribed by N. Zorii

• Publisher: John Wiley and Sons
• Year: 2012
• Tongue: English
• Weight: 312 KB
• Volume: 285
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.201000148

✦ Synopsis

Abstract

We consider a constrained minimal energy problem with an external field over noncompact classes of vector measures (μ^i^)~i ∈ I~ of finite or infinite dimensions on a locally compact space. The components μ^i^ are nonnegative Radon measures satisfying normalizing conditions, supported by given A~i~ and such that σ^i^ − μ^i^ ≥ 0, the constraints σ^i^, i ∈ I, being given. For a particular matrix of interaction between μ^i^, i ∈ I, and a rather general class of positive definite kernels, sufficient conditions for the existence of minimizers are established and their uniqueness and vague compactness are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also analyze continuity properties of minimizers in the vague and strong topologies when A~i~ and σ^i^ are varied. Almost all results are valid also for classical kernels in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb R^n$\end{document}, which is important in applications.