PreprintCC BY 4.0arXiv
Classification of solutions to the Liouville equation with a nonlinear Robin boundary condition on the unit disk
Abstract
In this paper, we study the nonlinear boundary value problem −Δu=e2u,& in 𝔻, (∂ u)/∂ν+λ=eu ,& on 𝕊1, where 𝔻 is the unit disk, λ is a constant and ν denotes the outer unit normal on 𝕊1. For 0<λ≤2, we establish a complete classification of smooth solutions. For 2<λ≤3, we prove a dichotomy between radial solutions and nonradial solutions. We develop a Hardy-Wronskian boundary rigidity method that transforms the nonlinear boundary value problem into a spectral rigidity problem for normalized holomorphic frames. This method exploits the complex-analytic structure of the Liouville equation with a Robin boundary condition and provides a new rigidity framework for related two-dimensional elliptic boundary value problems.
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BibTeX
@misc{dou2026classification,
title = {{Classification of solutions to the Liouville equation with a nonlinear Robin boundary condition on the unit disk}},
author = {Jingbo Dou and Yunyun Hu and Keqing Peng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12847},
eprint = {2609.12847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12847},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of solutions to the Liouville equation with a nonlinear Robin boundary condition on the unit disk
AU - Jingbo Dou
AU - Yunyun Hu
AU - Keqing Peng
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12847
UR - https://arxiv.org/abs/2609.12847
AB - In this paper, we study the nonlinear boundary value problem −Δu=e^2u,& in 𝔻, (∂ u)/∂ν+λ=e^u ,& on 𝕊^1, where 𝔻 is the unit disk, λ is a constant and ν denotes the outer unit normal on 𝕊^1. For 0<λ≤2, we establish a complete classification of smooth solutions. For 2<λ≤3, we prove a dichotomy between radial solutions and nonradial solutions. We develop a Hardy-Wronskian boundary rigidity method that transforms the nonlinear boundary value problem into a spectral rigidity problem for normalized holomorphic frames. This method exploits the complex-analytic structure of the Liouville equation with a Robin boundary condition and provides a new rigidity framework for related two-dimensional elliptic boundary value problems.
ER -