The local regularity for strong solutions of the Hessian quotient equation
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.
scientific article; zbMATH DE number 2153755