Balayage of measures: behavior near a corner

Abstract

We consider the balayage of a measure $\mu$ defined on a domain $\Omega$ onto its boundary $\partial \Omega$. Assuming that $\Omega$ has a corner of opening $\pi \alpha$ at a point $z_0\in \partial \Omega$ for some $0<\alpha \le 2$ and that $d\mu (z)\asymp |z-z_0{|}^{2b-2}{d}^{2}z$ as $z\to z_0$ for some $b>0$, we obtain the precise rate of vanishing of the balayage of $\mu$ near $z_0$. The rate of vanishing is universal in the sense that it only depends on $\alpha$ and $b$. We also treat the case when the domain has multiple corners at the same point. Moreover, when $2b\le \frac{1}{\alpha }$, we provide explicit constants for the upper and lower bounds.

arXiv commentResults are improved; 21 pages, 6 figures