广义凝聚向量测度的约束最小Riesz和Green能量问题
Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser
摘要：
对于$\mathbb R^n$，$n\geqslant3$中的局部闭集的有限集合$\mathbf A=(A_i)_{i\in I}$，我们考虑了正向量Radon测度$\boldsymbolμ=(μ^i)_{i\in I}$上相对于$α$ - Riesz核$|x-y|^{α-n}$，$α\in(0,2]$的最小能量问题，使得每个$μ^i$，$i\in I$由$A_i$携带，并由$μ^i(A_i)=a_i\in(0,\infty)$归一化.我们表明，虽然封闭的带相反电荷的板可能会相互交叉，即使在一组非零容量，这个问题有一个解决方案$\boldsymbolλ^{\boldsymbolξ}_{\mathbf A}=(λ^i_{\mathbf A})_{i\in I}$（也存在一个外部字段），如果我们限制自己的$\boldsymbolμ$与$μ^i\leqslantξ^i$，$i\in I$，其中的约束$\boldsymbolξ=(ξ^i)_{i\in I}$是适当的选择。我们建立了清晰的充分条件，从而获得的可解性，提供的加权矢量$α$-Riesz势的解决方案的描述，挑选出他们的特征性质，并分析支持的$λ^i_{\mathbf A}$，$i\in I$。我们的方法是基于同时使用的模糊拓扑结构和适当的半度量结构定义的$α$ -Riesz能量与$\mathbf A$的一组向量的措施，以及建立一个亲密的关系之间的约束最小$α$ -Riesz能量问题和约束最小$α$ -绿色能源问题，适当制定。最后用实例说明了结果。
英文摘要：
For a finite collection $\mathbf A=(A_i)_{i\in I}$ of locally closed sets in $\mathbb R^n$, $n\geqslant3$, with the sign $\pm1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $α$-Riesz kernel $|x-y|^{α-n}$, $α\in(0,2]$, over positive vector Radon measures $\boldsymbolμ=(μ^i)_{i\in I}$ such that each $μ^i$, $i\in I$, is carried by $A_i$ and normalized by $μ^i(A_i)=a_i\in(0,\infty)$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution $\boldsymbolλ^{\boldsymbolξ}_{\mathbf A}=(λ^i_{\mathbf A})_{i\in I}$ (also in the presence of an external field) if we restrict ourselves to $\boldsymbolμ$ with $μ^i\leqslantξ^i$, $i\in I$, where the constraint $\boldsymbolξ=(ξ^i)_{i\in I}$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted vector $α$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $λ^i_{\mathbf A}$, $i\in I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $α$-Riesz energy on a set of vector measures associated with $\mathbf A$, as well as on the establishment of an intimate relationship between the constrained minimum $α$-Riesz energy problem and a constrained minimum $α$-Green energy problem, suitably formulated. The results are illustrated by examples.
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