The local regularity for strong solutions of the Hessian quotient equation

By means of the Reilly formula and the Alexandrov maximum principle, we obtain the local C1,1 estimates of the W2,p strong solutions to the Hessian quotient equations for p sufficiently large, and then prove that these solutions are smooth. There are counterexamples to show that the integral exponent p is optimal in some cases. We modify partially the known result in the Hessian case, and extend the regularity result in the special Lagrangian case to the Hessian quotient case.