Balayage of measures: behavior near a corner

2024/03/05 by Charlier, Christophe, Lenells, Jonatan
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2403.02964

Abstract

We consider the balayage of a measure μ defined on a domain Ω onto its boundary ∂ Ω. Assuming that Ω has a corner of opening πα at a point z₀ ∈ ∂ Ω for some 0 < α≤ 2 and that dμ(z) \asymp |z-z₀|^2b-2d²z as z→ z₀ for some b > 0, we obtain the precise rate of vanishing of the balayage of μ near z₀. The rate of vanishing is universal in the sense that it only depends on α and b. We also treat the case when the domain has multiple corners at the same point. Moreover, when 2b≤ \frac1α, we provide explicit constants for the upper and lower bounds.

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