Strict Convexity for Solution of Liouville-Type Dirichlet Problems

Jiahuan Li; Shuning Xu

analysis of partial differential equations

Strict Convexity for Solution of Liouville-Type Dirichlet Problems

arXiv:2607.12849

summary

The paper proves that a transformed solution of three exponential Dirichlet problems on smooth uniformly strictly convex domains is strictly convex, using techniques like domain deformation and constant‑rank theory.

Abstract

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

Topics & keywords

#convexity #liouville equation #dirichlet problem #sigma_2 equation #convex domainsexponential Dirichlet problemstrict convexityarcosh transformationconstant-rank theoryinverse-convexity estimatesC^2 stability