infinite dimensional向量测度高斯变分问题可解的充要条件
Necessary and sufficient conditions for the solvability of the Gauss variational problem for infinite dimensional vector measures
摘要：
我们继续研究与凝聚体$(A_i)_{i\in I}$相联系的无穷维向量测度的高斯变分问题。它已被显示在潜在的肛门。DOI：10.1007/s11118-012-9279-8指出，如果某些平板（比如$\ell\in L$的$A_\ell$）是非紧的，那么一般来说，存在一个矢量$\mathbf a=(a_i)_{i\in I}$，规定了$A_i$上的总电荷，$i\in I$，这样问题就没有解决方案。那么，对于所有的向量$\mathbf a$，高斯变分问题仍然是可解的描述是什么呢？这样一个表征得到一个正定核满足Fuglede的条件的完美性，它是在一个辅助极值问题的解决方案密切相关的运营商的正交投影到锥上的所有正标量措施支持$\bigcup_{\ell\in L}A_\ell$。结果示出了有关的Riesz内核的例子。
英文摘要：
We continue our investigation of the Gauss variational problem for infinite dimensional vector measures associated with a condenser $(A_i)_{i\in I}$. It has been shown in Potential Anal., DOI:10.1007/s11118-012-9279-8 that, if some of the plates (say $A_\ell$ for $\ell\in L$) are noncompact then, in general, there exists a vector $\mathbf a=(a_i)_{i\in I}$, prescribing the total charges on $A_i$, $i\in I$, such that the problem admits no solution. Then, what is a description of all the vectors $\mathbf a$ for which the Gauss variational problem is nevertheless solvable? Such a characterization is obtained for a positive definite kernel satisfying Fuglede's condition of perfectness; it is given in terms of a solution to an auxiliary extremal problem intimately related to the operator of orthogonal projection onto the cone of all positive scalar measures supported by $\bigcup_{\ell\in L}A_\ell$. The results are illustrated by examples pertaining to the Riesz kernels.
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