论文标题
在埃文斯和Choquet的定理上
On Evans' and Choquet's theorems on polar sets
论文作者
论文摘要
通过G.C.的经典结果埃文斯（Evans）和G. $g_δ$ - set。 Evans定理的已知应用使Perron-Wiener-Brelot方法仅使用谐波上和下功能，将广义Dirichlet问题的解决方案用于开放集。 在本说明中，可以表明，通过基本的“度量”注意事项并且没有使用任何潜在理论，可以为满足本地三角属性的一般内核$ g $获得这样的结果。特定情况是$ g（x，y）= | x-y |^{α-d} $，$ r^d $，$ 2 <α<d $，解决了一个长期存在的开放问题。
By classical results of G.C. Evans and G. Choquet on "good kernels $G$ in potential theory", for every polar $K_σ$-set $P$, there exists a finite measure $μ$ on $P$ such that $Gμ=\infty$ on $P$, and a set $P$ admits a finite measure $μ$ on $P$ such that $\{Gμ=\infty\}=P$ if and only if $P$ is a polar $G_δ$-set. A known application of Evans' theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic upper and lower functions. In this note it is shown that, by elementary "metric" considerations and without using any potential theory, such results can be obtained for general kernels $G$ satisfying a local triangle property. The particular case, $G(x,y)=|x-y|^{α-d}$ on $R^d$, $2<α<d$, solves a long-standing open problem.
