Strict Convexity for Solution of Liouville-Type Dirichlet Problems

2026/07/14 by Jiahuan Li, Shuning Xu
Mathematics · #math.AP

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Abstract

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 σ₂(D²u)=e^2u, and its complex counterpart σ₂(u_{i j})=e^2u. In each case u<0 in the domain and u=0 on the boundary. We prove that w=-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 C² stability.

Citations

• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

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