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

@article{Ivanisvili2015BellmanPD, title={Bellman partial differential equation and the hill property for classical isoperimetric problems}, author={Paata Ivanisvili and Alexander Volberg}, journal={arXiv: Analysis of PDEs}, year={2015} }

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…

## 11 Citations

Hessian of Bellman functions and uniqueness of the Brascamp–Lieb inequality

- MathematicsJ. Lond. Math. Soc.
- 2015

Under some assumptions on the vectors a1,..,an 2 R k and the function B and thefunction B : R n ! R the authors find the sharp estimate of the expression Rk B(u1(a1 · x),...,un(an · x))dx in terms of R uj(y)dy,j = 1,...,n.

The Bellman Function Technique in Harmonic Analysis

- Computer Science
- 2020

The Bellman function, a powerful tool originating in control theory, can be used successfully in a large class of difficult harmonic analysis problems and has produced some notable results over the…

Isoperimetric Functional Inequalities via the Maximum Principle: The Exterior Differential Systems Approach

- Mathematics
- 2018

Our goal in this note is to give a unified approach to the solutions of a class of isoperimetric problems by relating them to the exterior differential systems studied by R. Bryant and P. Griffiths.

On the tightness of Gaussian concentration for convex functions

- MathematicsJournal d'Analyse Mathématique
- 2019

The concentration of measure phenomenon in Gauss' space states that every $L$-Lipschitz map $f$ on $\mathbb R^n$ satisfies \[ \gamma_{n} \left(\{ x : | f(x) - M_{f} | \geqslant t \} \right) \leqslant…

The Borell–Ehrhard game

- Mathematics
- 2016

A precise description of the convexity of Gaussian measures is provided by sharp Brunn–Minkowski type inequalities due to Ehrhard and Borell. We show that these are manifestations of a game-theoretic…

A Gaussian small deviation inequality for convex functions

- Mathematics
- 2016

Let $Z$ be an $n$-dimensional Gaussian vector and let $f: \mathbb R^n \to \mathbb R$ be a convex function. We show that: $$\mathbb P \left( f(Z) \leq \mathbb E f(Z) -t\sqrt{ {\rm Var} f(Z)} \right)…

A boundary value problem and the Ehrhard inequality

- MathematicsStudia Mathematica
- 2019

Let $I, J\subset \mathbb{R}$ be closed intervals, and let $H$ be $C^{3}$ smooth real valued function on $I\times J$ with nonvanishing $H_{x}$ and $H_{y}$. Take any fixed positive numbers $a,b$, and…

Improving Beckner's bound via Hermite functions

- Mathematics
- 2016

We obtain an improvement of the Beckner's inequality $\| f\|^{2}_{2} -\|f\|^{2}_{p} \leq (2-p) \| \nabla f\|_{2}^{2}$ valid for $p \in [1,2]$ and the Gaussian measure. Our improvement is essential…

An Interpolation Proof of Ehrhard’s Inequality

- Mathematics
- 2020

We prove Ehrhard’s inequality using interpolation along the Ornstein–Uhlenbeck semi-group. We also provide an improved Jensen inequality for Gaussian variables that might be of independent interest.

## References

SHOWING 1-10 OF 47 REFERENCES

Monge--Amp\`ere equation and Bellman optimization of Carleson Embedding Theorems

- Mathematics, Computer Science
- 2008

This work explores 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.

The Brascamp–Lieb Inequalities: Finiteness, Structure and Extremals

- Mathematics
- 2005

Abstract.We consider the Brascamp–Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the…

An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies

- Mathematics
- 2007

The BrunnMinkowski theorem and related geometric and functional inequalities

- Mathematics
- 2006

The Brunn�Minkowski inequality gives a lower bound of the Lebesgue measure of a
sum-set in terms of the measures of the individual sets. It has played a crucial role in the theory
of convex bodies.…

Bellman function for extremal problems in BMO

- Mathematics
- 2012

In this paper we develop the method of nding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John{Nirenberg inequality and L p…

Sharp estimates of integral functionals on classes of functions with small mean oscillation

- Mathematics
- 2014

Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator

- Mathematics
- 1996

Abstract. We establish, by simple semigroup arguments, a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with…

Best Constants in Young's Inequality, Its Converse, and Its Generalization to More than Three Functions

- Mathematics
- 1976

Hessian of Bellman functions and uniqueness of the Brascamp–Lieb inequality

- MathematicsJ. Lond. Math. Soc.
- 2015

Under some assumptions on the vectors a1,..,an 2 R k and the function B and thefunction B : R n ! R the authors find the sharp estimate of the expression Rk B(u1(a1 · x),...,un(an · x))dx in terms of R uj(y)dy,j = 1,...,n.

Analysis and Geometry of Markov Diffusion Operators

- Mathematics
- 2013

Introduction.- Part I Markov semigroups, basics and examples: 1.Markov semigroups.- 2.Model examples.- 3.General setting.- Part II Three model functional inequalities: 4.Poincare inequalities.-…