An example of constructing the Bellman function for extremal problems in BMO

An example of solution of a boundary value problem for a homogeneous Monge–Ampère equation is given, which produces a Bellman function for an extremal problem on the space BMO. The paper contains a step-by-step instruction for calculation of this function. The cases of a rather complicated foliation are considered. This illustrates the technique elaborated in a paper by Ivanishvili, Stoyanov, Vasyunin, and Zatitsky.