Pogorelov type interior 𝐶2 estimate for Hessian quotient equation and its application

Abstract In this paper, we derive a Pogorelov type interior C 2 {C^{2}} estimate for the Hessian quotient equation σ n σ k ⁢ ( D 2 ⁢ u ) = f {\frac{\sigma_{n}}{\sigma_{k}}(D^{2}u)=f} . As an application, we show that convex viscosity solutions are regular for k ≤ n - 3 {k\leq n-3} if u ∈ C 1 , α {u\in C^{1,\alpha}} with α > 1 - 2 n - k {\alpha>1-\frac{2}{n-k}} or u ∈ W 2 , p {u\in W^{2,p}} with p ≥ ( n - 1 ) ⁢ ( n - k ) 2 {p\geq\frac{(n-1)(n-k)}{2}} . Both exponents are sharp in view of the example in [S. Lu, Interior C 2 C^{2} estimate for Hessian quotient equation in general dimension, Ann. PDE 11 2025, 2, Paper No. 17].