Facial behaviour of analytic functions on the bidisk

We prove that if $\varphi$ is an analytic function bounded by 1 on the bidisk and $\tau$ is a point in a face of the bidisk at which $\varphi$ satisfies Caratheodory’s condition then both $\varphi$ and the angular gradient $\mathrm{\nabla }\varphi$ exist and are constant on the face. Moreover, the class of all $\varphi$ with prescribed $\varphi (\tau )$ and $\mathrm{\nabla }\varphi (\tau )$ can be parametrized in terms of a function in the two-variable Pick class. As an application we solve an interpolation problem with nodes that lie on faces of the bidisk.