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 \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*} where $\mathbb D$ is the unit disk, $\lambda$ is a constant and $\nu$ denotes the outer unit normal on $\mathbb S^1$. For $0<\lambda\le2$, we establish a complete classification of smooth solutions. For $2<\lambda\leq3$, 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.