The Lagrangian phase operator arises in the study of calibrated geometries and the deformed Hermitian-Yang-Mills equation in complex geometry. We study a local version of these geometric problems, and solve the Dirichlet problem for the Lagrangian phase operator with supercritical phase given the existence of a subsolution. They key step is to find hidden concavity properties in order to obtain a priori estimates. This is joint work with T. Collins and X. Wu.