Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Balayage of Measures : Behavior Near a Cusp

Charlier, Christophe LU and Lenells, Jonatan LU (2025) In Potential Analysis
Abstract

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... (More)

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.

(Less)
Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
Balayage measure, Boundary behavior, Coulomb gas, Cusp, Harmonic measure, Tacnode
in
Potential Analysis
article number
021602
publisher
Springer
external identifiers
  • scopus:105001636549
ISSN
0926-2601
DOI
10.1007/s11118-025-10212-5
language
English
LU publication?
yes
id
d0e7b046-6757-485e-ba0a-2fa7a988e21b
date added to LUP
2025-09-08 11:21:50
date last changed
2025-10-14 10:11:11
@article{d0e7b046-6757-485e-ba0a-2fa7a988e21b,
  abstract     = {{<p>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&gt;0 and a&gt;0, respectively, and that dμ(z)≍|z-z0|2b-2d2z as z→z0 for some b&gt;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.</p>}},
  author       = {{Charlier, Christophe and Lenells, Jonatan}},
  issn         = {{0926-2601}},
  keywords     = {{Balayage measure; Boundary behavior; Coulomb gas; Cusp; Harmonic measure; Tacnode}},
  language     = {{eng}},
  publisher    = {{Springer}},
  series       = {{Potential Analysis}},
  title        = {{Balayage of Measures : Behavior Near a Cusp}},
  url          = {{http://dx.doi.org/10.1007/s11118-025-10212-5}},
  doi          = {{10.1007/s11118-025-10212-5}},
  year         = {{2025}},
}