Bellman partial differential equation and the hill property for classical isoperimetric problems

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.