  Monge--Amp\`ere equation plays an important part in Analysis. For example, it
is instrumental in mass transport problems. On the other hand, the Bellman
function technique appeared recently as a way to consider certain Harmonic
Analysis problems as the problems of Stochastic Optimal Control. This brings us
to Bellman PDE, which in stochastic setting is often a Monge--Amp\`ere equation
or its close relative. We explore the way of solving Monge--Amp\`ere equation
by a sort of method of characteristics to find the Bellman function of certain
classical Harmonic Analysis problems, and, therefore, of finding full structure
of sharp constants and extremal sequences for those problems.
