  The goal of this note is to have a systematic approach to generating
isoperimetric inequalities from two concrete type of PDEs. We call these PDEs
Bellman type because a totally analogous equations happen to rule many sharp
estimates for singular integrals in harmonic analysis, and such estimates were
obtained with the use of Hamilton--Jacobi--Bellman PDE. We show how classical
inequalities of Brascamp--Lieb, Prekopa--Leindler, Ehrhard are particular case
of this scheme, which allows us to augment the stock of such inequalities. We
approach the isoperimetric inequalities as a maximum (minimum) principle for
special types of functions. These functions are compositions of "Bellman
function" and an appropriate flow built on test functions. Then the existence
of maximum (minimum) principle for such compositions can be reduced to the
requirement that Bellman function satisfies a concrete class of nonlinear PDE
(written down below). We are left to solve this nonlinear PDE (sometimes a
possible task) to enjoy isoperimetric inequalities. The nonlinear PDE that we
will describe in this article can be reduced sometimes to solving Laplacian
eigenvalue problem, $\bar{\partial}$-equation of certain type or just the
linear heat equation.
