Concavity of the Lagrangian Phase Operator and Applications

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if $\Omega$ is a compact domain in $\mathbb{R}^{n}$ or $\mathbb{C}^n$, then there exists a solution to the Dirichlet problem with right-hand side $h(x)$ satisfying $|h(x)| > (n-2)\frac{\pi}{2}$ and boundary data $\phi$ if and only if there exists a subsolution.