infinite dimensional向量测度的外场约束能量问题
Constrained energy problems with external fields for infinite dimensional vector measures
摘要：
考虑了局部紧空间上无穷维向量测度非紧类上带外场的约束极小能量问题。这些分量是正测度（电荷），它们从上面受到约束，满足一些归一化假设，并由给定的闭集（板）支持，规定符号为+1或-1，使得相反符号的板相互不相交，电荷的相互作用矩阵对应于电容器的静电解释。对于所有满足Fuglede弱拓扑与强拓扑相容条件的正定核，建立了极小解存在的充分条件，并研究了极小解的唯一性和弱紧性.举例说明的清晰度的充分条件提供。我们还分析了在弱 * 和强拓扑中，当约束和板同时变化时极小解的连续性。这些结果对于欧几里得空间中的经典核函数也是新的，具有重要的应用价值。
英文摘要：
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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