On the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase

In this paper, we solve the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase. Lagrangian phase equation arises naturally from the study of Lagrangian submanifold. To solve the Dirichlet problem, we adopt the continuity method by obtaining the full a priori estimates. A crucial ingredient is the interior $$ C^2 $$ estimate. Our result is sharp in the sense that there exist singular solutions in the subcritical phase case.