PreprintCC BY 4.0arXiv
Harmonic mappings on Grushin planes, part II
Abstract
We study harmonic mappings between the source domain in the α-Grushin plane and the target domain in the β-Grushin plane for α not necessarily equal to β and α∈ [0,1). Our harmonic mappings arise as a Euler--Lagrange system of equations related to the Dirichlet energy and are studied in weak and strong forms. The key results enclose the second order regularity of harmonic mappings, the Bochner identity, the weak Harnack estimates for coordinate functions of a harmonic mapping, as well as the Caccioppoli estimates and the Liouville type theorem. Furthermore, we discuss several examples of (Grushin) harmonic mappings and, in particular, relate them to holomorphic mappings in the plane.
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BibTeX
@misc{adamowicz2026harmonic,
title = {{Harmonic mappings on Grushin planes, part II}},
author = {Tomasz Adamowicz and Krzysztof Chełmiński and Marcin Walicki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25801},
eprint = {2609.25801},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25801},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Harmonic mappings on Grushin planes, part II
AU - Tomasz Adamowicz
AU - Krzysztof Chełmiński
AU - Marcin Walicki
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25801
UR - https://arxiv.org/abs/2609.25801
AB - We study harmonic mappings between the source domain in the α-Grushin plane and the target domain in the β-Grushin plane for α not necessarily equal to β and α∈ [0,1). Our harmonic mappings arise as a Euler--Lagrange system of equations related to the Dirichlet energy and are studied in weak and strong forms. The key results enclose the second order regularity of harmonic mappings, the Bochner identity, the weak Harnack estimates for coordinate functions of a harmonic mapping, as well as the Caccioppoli estimates and the Liouville type theorem. Furthermore, we discuss several examples of (Grushin) harmonic mappings and, in particular, relate them to holomorphic mappings in the plane.
ER -