Overview
The research focuses on establishing interior $C^2$ estimates for a specific class of partial differential equations known as Hessian quotient equations. These equations are characterized by the form $\frac{\sigma_k}{\sigma_l}(D^2u)=f(x,u)$. The study specifically addresses admissible semiconvex solutions within the context of two particular relationships between the indices $k$ and $l$: when $l=k-1$ and when $l=k-2$. The function $f(x,u)$ is defined as a positive $C^2$ function.
Research Context
Hessian quotient equations represent a significant area within the study of partial differential equations. Prior work, specifically counterexamples presented by Lu in \cite{LuGeneral}, indicated that interior $C^2$ estimates generally fail for these equations when the difference between the indices, $k-l$, is greater than or equal to 3. This failure was noted even for solutions exhibiting convexity. The current research addresses specific cases where such estimates are achievable under certain solution conditions.
Approach
The methodology employed to achieve the interior $C^2$ estimates relies on a quantitative concavity inequality. This inequality is applied to the Hessian quotient operator under the specific condition that the solutions are semiconvex. The application of this inequality serves as a fundamental component in deriving the reported estimates.
Findings
- Interior $C^2$ estimates were established for admissible semiconvex solutions to the general Hessian quotient equation $\frac{\sigma_k}{\sigma_l}(D^2u)=f(x,u)$.
- These estimates are specifically applicable to the cases where the index $l$ is $k-1$ or $k-2$.
- The function $f(x,u)$ is required to be a positive $C^2$ function for these estimates to hold.
- The study confirms that these estimates are not generally known to hold for $k-l \geq 3$, referencing prior counterexamples by Lu.
- A key component enabling these findings is a quantitative concavity inequality for the Hessian quotient operator, applicable under the semiconvex condition.
- The results offer a unified argument for these general Hessian quotient equations for $2 \leq k \leq n-1$ in arbitrary dimensions.
Why This Matters
This work provides a unified framework for understanding $C^2$ regularity properties for a class of nonlinear partial differential equations under specific conditions. By establishing these estimates, the research addresses a gap where general estimates were previously understood to fail for certain parameter ranges, contributing to the foundational understanding of these complex equations across various dimensions.