单位圆盘上带非线性Robin边界条件的Liouville方程解的分类

中文摘要 AI 辅助

本文研究非线性边值问题 \begin{equation*} \begin{cases} -\Delta u=e^{2u},& \mbox{in } {\mathbb{D}},\\\\ \frac{\partial u}{\partial\nu}+\lambda=e^u,& \mbox{on } {\mathbb{S}^{1}}, \end{cases} \end{equation*} 其中 $\mathbb D$ 是单位圆盘，$\lambda$ 是常数，$\nu$ 表示 $\mathbb S^1$ 上的外单位法向量。对于 $0<\lambda\le2$，我们建立了光滑解的完全分类。对于 $2<\lambda\leq3$，我们证明了径向解与非径向解之间的二分性。我们发展了一种Hardy-Wronskian边界刚性方法，将非线性边值问题转化为归一化全纯框架的谱刚性问题。该方法利用了带Robin边界条件的Liouville方程的复解析结构，并为相关的二维椭圆边值问题提供了新的刚性框架。

英文摘要

In this paper, we study the nonlinear boundary value problem \begin{equation*} \begin{cases} -Δu=e^{2u},& \mbox{in } {\mathbb{D}},\\ \frac{\partial u}{\partialν}+λ=e^u ,& \mbox{on } {\mathbb{S}^{1}}, \end{cases} \end{equation*} where $\mathbb D$ is the unit disk, $λ$ is a constant and $ν$ denotes the outer unit normal on $\mathbb S^1$. For $0<λ\le2$, we establish a complete classification of smooth solutions. For $2<λ\leq3$, 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.

发表机构

• School of Mathematics and Statistics, Shaanxi Normal University（陕西师范大学数学与统计学院）

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