Author

Shuning Xu

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Preprint Aug 2026

Strict Convexity and Sharp Power Concavity for a Graphical $\sigma_2$-Curvature Equation

We prove strict convexity of the square-root transformation $v=-\sqrt{-u}$ for admissible solutions of a graphical $\sigma_2$-curvature Dirichlet problem on smooth uniformly convex domains. A key ingredient is a constant-rank theorem for $D^2v$, proved by a direct Ma-Xu type argument in dimension three and by the Bian-Guan microscopic convexity principle together with inverse-convexity methods in arbitrary dimensions. Combined with boundary strict convexity and a domain-deformation argument, the constant-rank theorem yields $D^2v>0$ throughout the domain.

Shuning Xu · 0 citations

Preprint Jul 2026

Strict Convexity for Solution of Liouville-Type Dirichlet Problems

We identify a common convexity structure for three exponential Dirichlet problems on smooth uniformly strictly convex domains: the Liouville equation $\Delta u=e^u$, the real equation $\sigma_2(D^2u)=e^{2u}$, and its complex counterpart $\sigma_2(u_{i\bar j})=e^{2u}$. 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.

Jiahuan Li, Shuning Xu · 0 citations