论文标题
当地紧凑型空间上外部田野的最小能量问题
Minimum energy problems with external fields on locally compact spaces
论文作者
论文摘要
本文在局部紧凑空间$ x $的外部田地存在的情况下处理最小能量问题，相对于功能内核$κ$满足能量和一致性原则。对于非常一般的（不一定要降低半连续性）外部字段$ f $，我们为存在$λ_{a，f} $最小化Gauss功能\ [\intκ（x，y）\，d（μ\otimesμ）（x，y）+2 \ int f \ $ y的$λ_{a，f} $ coplational offictiatiation \ intk（x，y）\ $ complose $λ_{a，f} $ coplant $λ_{a，f} $ youst $ co \ $ undime，我们建立了足够和/或必要的条件。 $μ（x）= 1 $，集中在相当一般的（不一定是关闭或有限的）$ a \ subset x $上，从而回答了M. Ohtsuka提出的问题（J.Sci。HiroshimaUniv。，1961年）。此类结果是为Riesz内核$ | x-y |^{α-n} $，$ 0 <α<n $，在$ \ mathbb r^n $，$ n \ geqslant2 $上的，并通过一些示例说明的。此外，我们提供了最小化器$λ_{a，f} $的各种替代特征，作为副产品，我们在$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ k \ k \ subset a $中分析了$λ_{a，f} $的强连续性。获得的结果是正确的，对于古典和现代潜在理论中许多有趣的内核来说是新的。
The paper deals with minimum energy problems in the presence of external fields on a locally compact space $X$ with respect to a function kernel $κ$ satisfying the energy and consistency principles. For quite a general (not necessarily lower semicontinuous) external field $f$, we establish sufficient and/or necessary conditions for the existence of $λ_{A,f}$ minimizing the Gauss functional \[\intκ(x,y)\,d(μ\otimesμ)(x,y)+2\int f\,dμ\] over all positive Radon measures $μ$ with $μ(X)=1$, concentrated on quite a general (not necessarily closed or bounded) $A\subset X$, thereby giving an answer to a question raised by M. Ohtsuka (J. Sci. Hiroshima Univ., 1961). Such results are specified for the Riesz kernels $|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$, and are illustrated by some examples. Furthermore, we provide various alternative characterizations of the minimizer $λ_{A,f}$, and as a by-product we analyze the strong and vague continuity of $λ_{A,f}$ under the exhaustion of $A$ by compact $K\subset A$. The results obtained hold true and are new for many interesting kernels in classical and modern potential theory.
