论文标题
最低riesz的能源问题与外部田地
Minimum Riesz energy problems with external fields
论文作者
论文摘要
本文在存在外部字段的情况下处理最低能量问题，相对于riesz内核$ | x-y |^{α-n} $，$ 0 <α<n $，on $ \ mathbb r^n $，$ n \ geqslant2 $。对于非常普遍的（不一定要较低的半连续性）外部字段$ f $，我们获得了$λ_{a，f}的存在必要和/或足够的条件$μ（\ Mathbb r^n）= 1 $，集中在一般（不一定关闭）$ a \ subset \ mathbb r^n $上。我们还提供了最小化器$λ_{a，f} $的各种替代性特征，分析$λ_{a，f} $的连续性和集合单调元素的修改后的robin常数，并描述了$λ_{a，f} $的$λ__的支持。因此，该理论的重大改进是由于作者最近开发的，基于强大拓扑与模糊拓扑之间的紧密相互作用以及内在的Balayage理论，这是由于一种新方法。
The paper deals with minimum energy problems in the presence of external fields with respect to the Riesz kernels $|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$. For quite a general (not necessarily lower semicontinuous) external field $f$, we obtain necessary and/or sufficient conditions for the existence of $λ_{A,f}$ minimizing the Gauss functional \[\int|x-y|^{α-n}\,d(μ\otimesμ)(x,y)+2\int f\,dμ\] over all positive Radon measures $μ$ with $μ(\mathbb R^n)=1$, concentrated on quite a general (not necessarily closed) $A\subset\mathbb R^n$. We also provide various alternative characterizations of the minimizer $λ_{A,f}$, analyze the continuity of both $λ_{A,f}$ and the modified Robin constant for monotone families of sets, and give a description of the support of $λ_{A,f}$. The significant improvement of the theory in question thereby achieved is due to a new approach based on the close interaction between the strong and the vague topologies, as well as on the theory of inner balayage, developed recently by the author.
