AI 中文总结
该研究建立了特定参数下Hessian商算子的凹性不等式与对应Jacobi不等式，结合已有框架得到Hessian商方程凸解的内部Hessian估计，并证明了具有二次增长的整体凸解必为二次多项式。
AI 中文摘要
我们建立了$k-l\in\{1,2\}$情形下Hessian商算子$\frac{σ_k}{σ_l}$的凹性不等式，并推导了相应的Jacobi不等式。将其与Lu和Tsai提出的框架相结合，我们得到了$\frac{σ_k(D^2u)}{σ_l(D^2u)}=f$凸解的内部Hessian估计。作为应用，我们证明了$\mathbb R^n$中任何具有二次增长的整体凸解必为二次多项式。
英文摘要
We establish a concavity inequality for the Hessian quotient operators $\frac{σ_k}{σ_l}$ in the cases $k-l\in\{1,2\}$, and then derive the corresponding Jacobi inequality. Combining this with the framework developed by Lu and Tsai, we obtain an interior Hessian estimate for convex solutions of $\frac{σ_k(D^2u)}{σ_l(D^2u)}=f$. As an application, we prove that any entire convex solution in $\mathbb R^n$ with quadratic growth must be a quadratic polynomial.
Comments15 pages
