Author

Zhisu Li

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Preprint Aug 2026

Exterior Dirichlet Problems for Hessian Quotient Equations of Mixed Type

We study the exterior Dirichlet problem for the mixed Hessian quotient equation \[ \frac{\sigma_k(\eta(D^2 u))}{\sigma_l(\eta(D^2 u))} = 1, \] where $\eta(M) = (\operatorname{tr} M)I - M$. We establish existence and uniqueness of smooth admissible solutions with prescribed quadratic asymptotics at infinity, and obtain full derivative decay of the remainder. The proof relies on a three-stage subsolution construction.

yu Lei, Zhisu Li · 0 citations

Preprint Aug 2026

A concavity inequality and interior $C^2$ estimate for Hessian quotient equations

We establish a concavity inequality for the Hessian quotient operators $\frac{\sigma_k}{\sigma_l}$ 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 $\frac{\sigma_k(D^2u)}{\sigma_l(D^2u)}=f$. As an application, we prove that any entire convex solution in $\mathbb R^n$ with quadratic growth must be a quadratic polynomial.

Zhisu Li, Ke Wu · 6 citations · ⚡5