2026/07/24 by Cheuk Yan Fung
#math.AP
In this paper, we establish a doubling argument to obtain Hessian estimates for convex solutions to the Hessian quotient equation (σn)/(σk)(D2u) = f(x,u,Du) for k=n-1 and k=n-2 under the condition that 1/f is concave 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 (σl)/(σk)(D2u) = f(x,u,Du) for k ∈ \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.