AI 中文总结
针对光滑一致凸区域上的图σ₂曲率狄利克雷问题，通过Ma-Xu型论证、Bian-Guan微观凸性原理等方法证明其容许解对应变换的严格凸性，得到区域内二阶导数正定的结论。
AI 中文摘要
我们证明了在光滑一致凸区域上的图σ₂曲率狄利克雷问题的容许解对应的平方根变换v=-√(-u)具有严格凸性。关键要素是D²v的常秩定理，该定理通过三维下直接的Ma-Xu型论证、任意维下结合Bian-Guan微观凸性原理与逆凸性方法得到。结合边界严格凸性与区域变形论证，常秩定理推出区域内D²v>0处处成立。
英文摘要
We prove strict convexity of the square-root transformation $v=-\sqrt{-u}$ for admissible solutions of a graphical $σ_2$-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.
Comments28 pages
