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Uniqueness theorems for weighted harmonic functions in the upper half-plane

Olofsson, Anders LU and Wittsten, Jens LU (2023) In Journal d'Analyse Mathematique
Abstract

We consider a class of weighted harmonic functions in the open upper half-plane known as α-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case (α ≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case (α = 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp... (More)

We consider a class of weighted harmonic functions in the open upper half-plane known as α-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case (α ≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case (α = 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin.

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Contribution to journal
publication status
epub
subject
in
Journal d'Analyse Mathematique
publisher
Magnes Press
external identifiers
  • scopus:85169924123
ISSN
0021-7670
DOI
10.1007/s11854-023-0298-8
language
English
LU publication?
yes
id
8bd864b0-1d5e-4dd7-afb8-fdb84d26455b
date added to LUP
2023-11-06 10:28:45
date last changed
2023-11-06 10:28:45
@article{8bd864b0-1d5e-4dd7-afb8-fdb84d26455b,
  abstract     = {{<p>We consider a class of weighted harmonic functions in the open upper half-plane known as α-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case (α ≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case (α = 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin.</p>}},
  author       = {{Olofsson, Anders and Wittsten, Jens}},
  issn         = {{0021-7670}},
  language     = {{eng}},
  publisher    = {{Magnes Press}},
  series       = {{Journal d'Analyse Mathematique}},
  title        = {{Uniqueness theorems for weighted harmonic functions in the upper half-plane}},
  url          = {{http://dx.doi.org/10.1007/s11854-023-0298-8}},
  doi          = {{10.1007/s11854-023-0298-8}},
  year         = {{2023}},
}