Balayage of measures: behavior near a cusp

TL;DR

This paper analyzes the behavior of balayage measures near cusps on domain boundaries, deriving universal leading order terms and explicit formulas, with applications to Coulomb gases.

Contribution

It provides the first detailed asymptotic description of balayage measures near cusps and multiple corners, including explicit formulas for complex geometries.

Findings

Derived universal leading order behavior of balayage near cusps.

Obtained explicit formulas for balayage of uniform measures on tacnodal regions.

Applied results to analyze two-dimensional Coulomb gases.

Abstract

Let $\mu$ be a positive measure supported on a domain $\Omega$. We consider the behavior of the balayage measure $\nu:=\mathrm{Bal}(\mu,\partial \Omega)$ near a point $z_{0}\in \partial \Omega$ at which $\Omega$ 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 $d\mu(z) \asymp |z-z_{0}|^{2b-2}d^{2}z$ as $z\to z_0$ for some $b > 0$, we obtain the leading order term of $\nu$ 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.