AI 中文总结
本文针对特定条件下的黑塞商方程，建立逐点加倍论证来获取凸解的黑塞估计，不依赖勒让德变换等，还给出反例说明结构假设的必要性，最后将论证扩展到一般黑塞商方程。
AI 中文摘要
在本文中，我们建立了一个加倍论证，以在\(1/f\)关于\(Du\)变量为凹的条件下，获得\(k = n - 1\)和\(k = n - 2\)时黑塞商方程\(\frac{\sigma_n}{\sigma_k}(D^2u)=f(x,u,Du)\)的凸解的黑塞估计。特别地，我们的方法是逐点的，且不使用勒让德变换或基于积分的局部最大值原理。我们给出一个反例，表明若不对\(f\)在\(Du\)变量上施加结构假设，内部估计可能不成立。最后，在对\(f\)在\(Du\)变量上施加类似结构条件以及Lu - Tsai 2026引入的算子的额外结构凹性假设下，我们将加倍论证扩展到一般黑塞商方程\(\frac{\sigma_l}{\sigma_k}(D^2u)=f(x,u,Du)\)，其中\(k\in\{l - 1, l - 2\}\)。
英文摘要
In this paper, we establish a doubling argument to obtain Hessian estimates for convex solutions to the Hessian quotient equation $\frac{σ_n}{σ_k}(D^2u) = f(x,u,Du)$ for $k=n-1$ and $k=n-2$ under the condition that $\log f$ is convex in the $Du$ variable. In particular, our approach is pointwise and does not make use of the Legendre transform or integral-based local maximum principles. We provide a counterexample demonstrating that interior estimates can fail if no structural assumption is imposed on $f$ in the $Du$ variable. Finally, we extend our doubling argument to general Hessian quotient equations $\frac{σ_l}{σ_k}(D^2u) = f(x,u,Du)$ for $k \in \{l-1, l-2\}$, under a similar structural condition imposed on $f$ in the $Du$ variable, alongside an additional structural concavity assumption on the operator introduced by Lu-Tsai 2026, which has very recently been established in independent works.
Comments41 pages; updated references; slightly relaxed the assumption on the gradient variable; added a remark on the recent establishment of the concavity assumption
