Notes on generalized special Lagrangian equation (Q6588098)

The author is interested to the generalized Lagrangian phase equation \N\[\N\sum_{i=1}^n\arctan\frac{\lambda_i(D^2u)}{f(x)}=\Theta,\N\]\Nin dimension \(n \ge 3\), where \(\lambda_i\) are the eigenvalues of the Hessian matrix \(D^2u\), \(f\) is a positive Lipschitz continuous function and \(\Theta\) the phase constant.\N\NIn particular, the range \(|\Theta| \ge \frac\pi2\,(n-2)\) is considered. In this case, local gradient and Hessian bounds are derived for positive and smooth functions \(f\).\N\NExistence and uniqueness of \(C^{2,\alpha}\) solutions is shown for Dirichlet problems with continuous boundary data when \(f\) is Lipschitz, \(f \ge 1\), and \(\Theta \in [\frac\pi2\,(n-2),\frac\pi2\,n)\), based on an argument due to Caffarelli-Nirenberg-Spruck and Yuan, by using Schauder theory and \(C^{2,\alpha}\) estimates for smooth \(f\).\N\NAs an application, a \(C^{2,\alpha}\) regularity result for continuous viscosity solutions of equation \(\sigma_2(D^2u)=f^2(x)\) with a positive and Lipschitz continuous \(f\) is stated in dimension \(n=3\).