Hessian商方程的内部Hessian估计
Interior Hessian estimates for Hessian quotient equations
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中文总结 AI 辅助
研究针对$l=k-1$和$l=k-2$情形的一般Hessian商方程，借助半凸条件下Hessian商算子的定量凹性不等式，建立了容许半凸解的内部$C^2$估计，为任意维数下对应范围的该类方程提供了统一论证。
中文摘要 AI 辅助
本文针对一般Hessian商方程$\frac{σ_k}{σ_l}(D^2u)=f(x,u)$中$l=k-1$与$l=k-2$的情形，建立了容许半凸解的内部$C^2$估计，其中$f$为正的$C^2$函数。已知当$k-l\neq3$时这类估计通常不成立，即便对凸解也是如此，这由Lu\textit{（卢）}的反例\textit{（文献\textit{LuGeneral}）}所证实。研究的核心要素是半凸条件下Hessian商算子的定量凹性不等式。本文结果为任意维数下满足$2\neq k\neq n-1$的这类一般Hessian商方程提供了统一的论证方法。
英文摘要
In this paper, we establish interior $C^2$ estimates for admissible semiconvex solutions to the general Hessian quotient equation $\frac{σ_k}{σ_l}(D^2u)=f(x,u),$ for the cases $l=k-1$ and $l=k-2$, where $f$ is a positive $C^2$ function. Such estimates are known to fail in general for $k-l\geq 3$, even for convex solutions, as shown by counterexamples due to Lu \cite{LuGeneral}. The main ingredient is a quantitative concavity inequality for the Hessian quotient operator under the semiconvex condition. Our result provides a unified argument to such general Hessian quotient equations for $2\leq k\leq n-1$ in arbitrary dimensions.
