Balayage of Measures : Behavior Near a Cusp - Charlier, Christophe; Lenells, Jonatan (Potential Analysis, 2025)
Let μ be a positive measure supported on a planar domain Ω. We consider the behavior of the balayage measure ν:=Bal(μ,∂Ω) near a point z0∈∂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)≍|z-z0|2b-2d2z as z→z0 for some b>0 (here d2z is the Lebesgue measure on C), we obtain the leading order term of ν near z0. 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.
Keywords: Balayage measure; Boundary behavior; Coulomb gas; Cusp; Harmonic measure; Tacnode