测度的扫除：尖点附近的行为
Balayage of measures: behavior near a cusp
AI总结：
本文研究正测度在具有向外尖点的区域边界上的扫除测度，在测度密度满足幂次渐近条件下给出尖点附近的首阶项，并推广到多角点与尖点情形，还应用于二维库仑气体。
摘要：
设 $μ$ 为支撑在区域 $Ω$ 上的正测度。我们考虑扫除测度 $ν:=\mathrm{Bal}(μ,\partial Ω)$ 在边界点 $z_{0}\in \partial Ω$ 附近的行为，其中 $Ω$ 在 $z_{0}$ 处具有向外的尖点。假设该尖点的切触阶数和切触系数分别为 $d>0$ 和 $a>0$，并且当 $z\to z_0$ 时 $dμ(z) \asymp |z-z_{0}|^{2b-2}d^{2}z$ 对某个 $b > 0$ 成立，我们得到了 $ν$ 在 $z_{0}$ 附近的首阶项。该首阶项具有普适性，即它只依赖于 $d$、$a$ 和 $b$。我们还处理了区域在同一点具有多个角点和尖点的情况。最后，我们得到了两个密切圆之间尖结点区域上均匀测度的扫除的显式表达式，并将该结果应用于二维库仑气体。
英文摘要：
Let $μ$ be a positive measure supported on a domain $Ω$. We consider the behavior of the balayage measure $ν:=\mathrm{Bal}(μ,\partial Ω)$ near a point $z_{0}\in \partial Ω$ at which $Ω$ has an outward-pointing cusp. Assuming that the order and coefficient of tangency of the cusp are $d>0$ and $a>0$, respectively, and that $dμ(z) \asymp |z-z_{0}|^{2b-2}d^{2}z$ as $z\to z_0$ for some $b > 0$, we obtain the leading order term of $ν$ near $z_{0}$. This leading term is universal in the sense that it only depends on $d$, $a$, and $b$. We also treat the case when the domain has multiple corners and cusps at the same point. Finally, we obtain an explicit expression for the balayage of the uniform measure on the tacnodal region between two osculating circles, and we give an application of this result to two-dimensional Coulomb gases.
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