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arXiv · 1010.2126

Constrained energy problems with external fields for infinite dimensional vector measures

Natalia Zorii
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Abstract

We consider a constrained minimal energy problem with an external field over noncompact classes of infinite dimensional vector measures on a locally compact space. The components are positive measures (charges) that are constrained from above, satisfy some normalizing assumptions, and are supported by given closed sets (plates) with the sign +1 or -1 prescribed such that the oppositely signed plates are mutually disjoint and the interaction matrix for the charges corresponds to an electrostatic interpretation of a condenser. For all positive definite kernels satisfying Fuglede's condition of consistency between the weak* and strong topologies, sufficient conditions for the existence of minimizers are established and their uniqueness and weak* compactness are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also analyze continuity properties of minimizers in both the weak* and strong topologies when the constraints and the plates are varied simultaneously. The results are new even for classical kernels in an Euclidean space, which is important in applications.

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Natalia Zorii. 2010-10-11. Constrained energy problems with external fields for infinite dimensional vector measures. https://arxiv.org/abs/1010.2126

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