AI 中文总结
研究光滑一致严格凸域上三个指数狄利克雷问题，通过结合域变形等多种理论和方法，证明\(w = -\text{arcosh}(e^{-u/2})\)在基础实变量中严格凸，确定了相关问题的共同凸性结构。
AI 中文摘要
我们确定了光滑一致严格凸域上三个指数狄利克雷问题的共同凸性结构：刘维尔方程\(\Delta u = e^u\)、实方程\(\sigma_2(D^2u)=e^{2u}\)及其复方程\(\sigma_2(u_{i\bar j})=e^{2u}\)。每种情况下\(u\)在域内小于\(0\)且在边界上为\(0\)。我们证明\(w = -\text{arcosh}(e^{-u/2})\)在基础实变量中是严格凸的。论证结合了域变形、常秩理论、逆凸性估计、径向球模型、边界严格凸性和局部\(C^2\)稳定性。
英文摘要
We identify a common convexity structure for three exponential Dirichlet problems on smooth uniformly strictly convex domains: the Liouville equation $Δu=e^u$, the real equation $σ_2(D^2u)=e^{2u}$, and its complex counterpart $σ_2(u_{i\bar j})=e^{2u}$. In each case $u<0$ in the domain and $u=0$ on the boundary. We prove that \[ w=-\operatorname{arcosh}(e^{-u/2}) \] is strictly convex in the underlying real variables. The argument combines domain deformation, constant-rank theory, inverse-convexity estimates, radial ball models, boundary strict convexity, and local $C^2$ stability.
