The local regularity for strong solutions of the Hessian quotient equation (Q1770985)

The authors consider the local \(C^{1,1}\) estimate and the regularity of the strong solutions for the Hessian quotient equation \[ {S_k(D^2 u)\over S_e(D^2 u)}= c,\quad\text{a. e. }x\in\Omega,\;0\leq\ell< k\leq n, \] where \(\Omega\) is a domain in \(\mathbb{R}^n\), \(C\) is a positive constant, \(D^2u\) denotes the Hessian of a function \(u\) in \(\Omega\), and \(S_j(D^2u)\) is denoted to the \(j\)th elementary symmetric function of the eigenvalues \(\lambda= (\lambda_1,\dots, \lambda_n)\) of \(D^2u\). To this end, the authors use the Reilly formula and the Alexander maximum principle. They also extend the regularity result in the special Lagrangian case to the Hessian quotient case.