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Bellman function and Monge–Amp` ere equation

Vasily Vasyunin

St.-Petersburg department of the Steklov Mathematical Institute

May 19, 2017

Probability and Analysis B¸ edlewo, Poland, May 15–19, 2017

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 1 / 21

St.-Petersburg group

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 2 / 21

St.-Petersburg group

Paata Ivanisvili

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 2 / 21

St.-Petersburg group

Paata Ivanisvili Dmitriy Stolyarov

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 2 / 21

St.-Petersburg group

Paata Ivanisvili Dmitriy Stolyarov Pavel Zatitskii

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 2 / 21

St.-Petersburg group

Paata Ivanisvili Dmitriy Stolyarov Pavel Zatitskii and partially Nikolay Osipov

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 2 / 21

St.-Petersburg group

Paata Ivanisvili Dmitriy Stolyarov Pavel Zatitskii and partially Nikolay Osipov Bellman function for extremal problems in BMO, Transactions of the American Mathematical Society, 368 (2016), 3415–3468.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 2 / 21

St.-Petersburg group

Paata Ivanisvili Dmitriy Stolyarov Pavel Zatitskii and partially Nikolay Osipov Bellman function for extremal problems in BMO, Transactions of the American Mathematical Society, 368 (2016), 3415–3468. Bellman function for extremal problems in BMO II: evolution, accepted in Memoirs of the American Mathematical Society, 153 p. (http://arxiv.org/abs/1510.01010)

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 2 / 21

Notation

ϕI = 1 |I|

• I

ϕ(t) dt is the average of ϕ over I.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 3 / 21

Notation

ϕI = 1 |I|

• I

ϕ(t) dt is the average of ϕ over I. Domain: Let ωε be a family of open strictly convex domains in R2 satisfying the following conditions

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 3 / 21

Notation

ϕI = 1 |I|

• I

ϕ(t) dt is the average of ϕ over I. Domain: Let ωε be a family of open strictly convex domains in R2 satisfying the following conditions clos ωε1 ⊂ ωε2 for ε1 < ε2;

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 3 / 21

Notation

ϕI = 1 |I|

• I

ϕ(t) dt is the average of ϕ over I. Domain: Let ωε be a family of open strictly convex domains in R2 satisfying the following conditions clos ωε1 ⊂ ωε2 for ε1 < ε2; any ray from ωε1 can be shifted to lay inside ωε2.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 3 / 21

Notation

ϕI = 1 |I|

• I

ϕ(t) dt is the average of ϕ over I. Domain: Let ωε be a family of open strictly convex domains in R2 satisfying the following conditions clos ωε1 ⊂ ωε2 for ε1 < ε2; any ray from ωε1 can be shifted to lay inside ωε2.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 3 / 21

Notation

ϕI = 1 |I|

• I

ϕ(t) dt is the average of ϕ over I. Domain: Let ωε be a family of open strictly convex domains in R2 satisfying the following conditions clos ωε1 ⊂ ωε2 for ε1 < ε2; any ray from ωε1 can be shifted to lay inside ωε2. We consider domains Ωε = ωε \ ω0.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 3 / 21

Notation

ϕI = 1 |I|

• I

ϕ(t) dt is the average of ϕ over I. Domain: Let ωε be a family of open strictly convex domains in R2 satisfying the following conditions clos ωε1 ⊂ ωε2 for ε1 < ε2; any ray from ωε1 can be shifted to lay inside ωε2. We consider domains Ωε = ωε \ ω0. Parametrization of the fixed boundary of Ωε: ∂ω0 = {g(t) = (g1(t), g2(t)): t ∈ J0 ⊂ R} .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 3 / 21

Class of test functions

Definition: Let J ⊂ R be an interval and ϕ: J → J0 be a summable

• function. We say that the function ϕ belongs to the class AΩ if

g ◦ ϕI ∈ Ω for every subinterval I ⊂ J.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 4 / 21

Class of test functions

Definition: Let J ⊂ R be an interval and ϕ: J → J0 be a summable

• function. We say that the function ϕ belongs to the class AΩ if

g ◦ ϕI ∈ Ω for every subinterval I ⊂ J. Examples: ωε = {(x1, x2): x2 > x2

1 + ε2}, g(s) = (s, s2), s ∈ (−∞, ∞),

ϕ ∈ BMO, ϕ ≤ ε ⇐ ⇒ ϕ ∈ AΩε;

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 4 / 21

Class of test functions

Definition: Let J ⊂ R be an interval and ϕ: J → J0 be a summable

• function. We say that the function ϕ belongs to the class AΩ if

g ◦ ϕI ∈ Ω for every subinterval I ⊂ J. Examples: ωε = {(x1, x2): x2 > x2

1 + ε2}, g(s) = (s, s2), s ∈ (−∞, ∞),

ϕ ∈ BMO, ϕ ≤ ε ⇐ ⇒ ϕ ∈ AΩε; ωε = {(x1, x2): xp−1

2

x1 > ε + 1}, g(s) = (s, s−

1 p−1 ), s ∈ (0, ∞),

ϕ ∈ Ap, [ϕ]Ap ≤ δ ⇐ ⇒ ϕ ∈ AΩδ−1 .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 4 / 21

Extremal problems and their Bellman functions

Extremal problem. For a given function f : J0 → R maximize (or minimize) the value of the following integral functional f (ϕ)J

• ver the set AΩε.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 5 / 21

Extremal problems and their Bellman functions

Extremal problem. For a given function f : J0 → R maximize (or minimize) the value of the following integral functional f (ϕ)J

• ver the set AΩε.

Definition of the corresponding Bellman function. B(x; ε) def = sup

ϕ { f ◦ ϕJ }

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 5 / 21

Extremal problems and their Bellman functions

Extremal problem. For a given function f : J0 → R maximize (or minimize) the value of the following integral functional f (ϕ)J

• ver the set AΩε.

Definition of the corresponding Bellman function. B(x; ε) def = sup

ϕ { f ◦ ϕJ }

Supremum is taken over the set of all functions ϕ such that ϕ ∈ AΩε,

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 5 / 21

Extremal problems and their Bellman functions

Extremal problem. For a given function f : J0 → R maximize (or minimize) the value of the following integral functional f (ϕ)J

• ver the set AΩε.

Definition of the corresponding Bellman function. B(x; ε) def = sup

ϕ { f ◦ ϕJ }

Supremum is taken over the set of all functions ϕ such that ϕ ∈ AΩε, g ◦ ϕJ = x .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 5 / 21

Various choices of f

Integral form of the John–Nirenberg inequality: f (s) = es, B(x; ε) = sup

ϕ {eϕJ } .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 6 / 21

Various choices of f

Integral form of the John–Nirenberg inequality: f (s) = es, B(x; ε) = sup

ϕ {eϕJ } .

Classical weak form of the John–Nirenberg inequality: f (s) = χ(−∞,−λ)∪(λ,∞)(s), B(x; ε, λ) = sup

ϕ

1 |J| |{t ∈ J : |ϕ(t)| ≥ λ}|

• .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 6 / 21

Various choices of f

Integral form of the John–Nirenberg inequality: f (s) = es, B(x; ε) = sup

ϕ {eϕJ } .

Classical weak form of the John–Nirenberg inequality: f (s) = χ(−∞,−λ)∪(λ,∞)(s), B(x; ε, λ) = sup

ϕ

1 |J| |{t ∈ J : |ϕ(t)| ≥ λ}|

• .

Lp-estimates, in particular, equivalence of different BMO-norms: f (s) = |s|p, B(x; p, ε) = sup

ϕ {|ϕ|pJ } .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 6 / 21

Various choices of f (continued)

Reverse H¨

• lder inequality for Ap-weights:

Ωδ = {(x1, x2): 1 ≤ xp−1

2

x1 ≤ δ}, f (s) = sq, B(x; δ) = sup

ϕ {ϕqJ : ϕ ∈ Ap, [ϕ]Ap ≤ δ} .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 7 / 21

Various choices of f (continued)

Reverse H¨

• lder inequality for Ap-weights:

Ωδ = {(x1, x2): 1 ≤ xp−1

2

x1 ≤ δ}, f (s) = sq, B(x; δ) = sup

ϕ {ϕqJ : ϕ ∈ Ap, [ϕ]Ap ≤ δ} .

Equivalence of the “norms” in RH1 and A∞: Ωδ = {(x1, x2): log x1 ≤ x2 ≤ log x1 + δ}, f (s) = s log s, B(x; δ) = sup

ϕ {ϕ log ϕJ : ϕ ∈ A∞, [ϕ]A∞ ≤ eδ} .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 7 / 21

Properties of the Bellman function

1 The Bellman function does not depend of the interval J, where test

functions ϕ are defined.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 8 / 21

Properties of the Bellman function

1 The Bellman function does not depend of the interval J, where test

functions ϕ are defined.

2 Domain of B is Ωε. Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 8 / 21

Properties of the Bellman function

1 The Bellman function does not depend of the interval J, where test

functions ϕ are defined.

2 Domain of B is Ωε. 3 Boundary values on the fixed boundary:

B(g(s); ε) = f (s).

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 8 / 21

Local concavity

Theorem (Stolyarov, Zatitskii; 2014)

Bellman function is the minimal locally concave function with the given boundary conditions on the fixed boundary.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 9 / 21

Monge–Amp` ere equation

d2B dx2 =  

∂2B ∂x2

1

∂2B ∂x1∂x2 ∂2B ∂x1∂x2 ∂2B ∂x2

2

  ≤ 0 .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 10 / 21

Monge–Amp` ere equation

d2B dx2 =  

∂2B ∂x2

1

∂2B ∂x1∂x2 ∂2B ∂x1∂x2 ∂2B ∂x2

2

  ≤ 0 . Concavity has to be degenerate along some direction.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 10 / 21

Monge–Amp` ere equation

d2B dx2 =  

∂2B ∂x2

1

∂2B ∂x1∂x2 ∂2B ∂x1∂x2 ∂2B ∂x2

2

  ≤ 0 . Concavity has to be degenerate along some direction. The Hessian matrix d2B

dx2 has to be degenerate.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 10 / 21

Monge–Amp` ere equation

d2B dx2 =  

∂2B ∂x2

1

∂2B ∂x1∂x2 ∂2B ∂x1∂x2 ∂2B ∂x2

2

  ≤ 0 . Concavity has to be degenerate along some direction. The Hessian matrix d2B

dx2 has to be degenerate.

Monge–Amp` ere equation ∂2B ∂x2

1

· ∂2B ∂x2

2

= ∂2B ∂x1∂x2 2 .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 10 / 21

Monge–Amp` ere equation

d2B dx2 =  

∂2B ∂x2

1

∂2B ∂x1∂x2 ∂2B ∂x1∂x2 ∂2B ∂x2

2

  ≤ 0 . Concavity has to be degenerate along some direction. The Hessian matrix d2B

dx2 has to be degenerate.

Monge–Amp` ere equation ∂2B ∂x2

1

· ∂2B ∂x2

2

= ∂2B ∂x1∂x2 2 . The Bellman function is a solution of the boundary value problems for this equation: B(g(s)) = f (s) .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 10 / 21

Properties of the solutions

Properties of the solutions of the Monge–Amp` ere equation Integral curves of the vector field generated by the kernel vectors of the Hessian matrix are segments of straight lines.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 11 / 21

Properties of the solutions

Properties of the solutions of the Monge–Amp` ere equation Integral curves of the vector field generated by the kernel vectors of the Hessian matrix are segments of straight lines. Solutions are linear along these extremal lines.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 11 / 21

Properties of the solutions

Properties of the solutions of the Monge–Amp` ere equation Integral curves of the vector field generated by the kernel vectors of the Hessian matrix are segments of straight lines. Solutions are linear along these extremal lines. All partial derivatives of the solution are constant on any extremal line.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 11 / 21

Properties of the solutions

Properties of the solutions of the Monge–Amp` ere equation Integral curves of the vector field generated by the kernel vectors of the Hessian matrix are segments of straight lines. Solutions are linear along these extremal lines. All partial derivatives of the solution are constant on any extremal line. Properties of the Monge–Amp` ere foliation If two extremal lines intersect at a point, then B is linear in a neighborhood of this point.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 11 / 21

Properties of the solutions

Properties of the solutions of the Monge–Amp` ere equation Integral curves of the vector field generated by the kernel vectors of the Hessian matrix are segments of straight lines. Solutions are linear along these extremal lines. All partial derivatives of the solution are constant on any extremal line. Properties of the Monge–Amp` ere foliation If two extremal lines intersect at a point, then B is linear in a neighborhood of this point. If an extremal line intersects the free boundary ∂ωε then it touches it tangentially.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 11 / 21

Left tangent foliation

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Left tangent foliation

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

Right tangent foliation

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 13 / 21

When the foliation is right and when it is left?

Any smooth functional f on BMO with f ′′′ >0 produces the left foliation.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 14 / 21

When the foliation is right and when it is left?

Any smooth functional f on BMO with f ′′′ >0 produces the left foliation. Any smooth functional f on BMO with f ′′′<0 produces the right foliation.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 14 / 21

When the foliation is right and when it is left?

Any smooth functional f on BMO with f ′′′ >0 produces the left foliation. Any smooth functional f on BMO with f ′′′<0 produces the right foliation. For more general domains we have to consider the curvature of the graph

• f the boundary condition instead of third derivative.

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 14 / 21

When the foliation is right and when it is left?

Any smooth functional f on BMO with f ′′′ >0 produces the left foliation. Any smooth functional f on BMO with f ′′′<0 produces the right foliation. For more general domains we have to consider the curvature of the graph

• f the boundary condition instead of third derivative.

The boundary of the graph of B is the curve γ(t) =

• g1(t), g2(t), f (t)
• ,

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 14 / 21

When the foliation is right and when it is left?

Any smooth functional f on BMO with f ′′′ >0 produces the left foliation. Any smooth functional f on BMO with f ′′′<0 produces the right foliation. For more general domains we have to consider the curvature of the graph

• f the boundary condition instead of third derivative.

The boundary of the graph of B is the curve γ(t) =

• g1(t), g2(t), f (t)
• ,

and in general case we have to look on the sign of the curvature of this curve, i.e., on the sign of the determinant

• g′

1(t)

g′

2(t)

f ′(t) g′′

1 (t)

g′′

2 (t)

f ′′(t) g′′′

1 (t)

g′′′

2 (t)

f ′′′(t)

• =
• γ′(t)

γ′′(t) γ′′′(t)

• .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 14 / 21

A cup

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c)

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

A cup

ω0 ωε

q

g(c) curvature>0 curvature<0

q

g(a)

q

g(b)

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

The ends of a chord

To determine which points are connected by a chord, we need to solve the following equation:

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 16 / 21

The ends of a chord

To determine which points are connected by a chord, we need to solve the following equation:

• γ′(a)

γ′(b) γ(b) − γ(a)

• =

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 16 / 21

The ends of a chord

To determine which points are connected by a chord, we need to solve the following equation:

• γ′(a)

γ′(b) γ(b) − γ(a)

• =
• g′

1(a)

g′

2(a)

f ′(a) g′

1(b)

g′

2(b)

f ′(b) g1(b) − g1(a) g2(b) − g2(a) f (b) − f (a)

• = 0 .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 16 / 21

The ends of a chord

To determine which points are connected by a chord, we need to solve the following equation:

• γ′(a)

γ′(b) γ(b) − γ(a)

• =
• g′

1(a)

g′

2(a)

f ′(a) g′

1(b)

g′

2(b)

f ′(b) g1(b) − g1(a) g2(b) − g2(a) f (b) − f (a)

• = 0 .

In the case of BMO (parabolic strip: g(s) = (s, s2)) this “cap equation” has the following form: f ′[a,b] = f ′(a) + f ′(b) 2 .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 16 / 21

The ends of a chord

To determine which points are connected by a chord, we need to solve the following equation:

• γ′(a)

γ′(b) γ(b) − γ(a)

• =
• g′

1(a)

g′

2(a)

f ′(a) g′

1(b)

g′

2(b)

f ′(b) g1(b) − g1(a) g2(b) − g2(a) f (b) − f (a)

• = 0 .

In the case of BMO (parabolic strip: g(s) = (s, s2)) this “cap equation” has the following form: f ′[a,b] = f ′(a) + f ′(b) 2 . f (b) − f (a) b − a =

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 16 / 21

An angle

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

An angle

u

q

curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

A trolleybus

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c)

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A trolleybus

q

g(c) curvature>0 curvature<0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

A birdie

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c)

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0 curvature<0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

A birdie

q

g(c) curvature>0 curvature<0 curvature<0 curvature>0

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 19 / 21

Classification of subdomains

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 20 / 21

Classification of subdomains

1 Tangent domain Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 20 / 21

Classification of subdomains

1 Tangent domain 2 Chordal domain Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 20 / 21

Classification of subdomains

1 Tangent domain 2 Chordal domain 3 Domain of linearity Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 20 / 21

Classification of subdomains

1 Tangent domain 2 Chordal domain 3 Domain of linearity 1

Domain with one point on the lower boundary (angle)

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 20 / 21

Classification of subdomains

1 Tangent domain 2 Chordal domain 3 Domain of linearity 1

Domain with one point on the lower boundary (angle)

2

Domain with two points on the lower boundary (trolleybus, birdie)

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 20 / 21

Classification of subdomains

1 Tangent domain 2 Chordal domain 3 Domain of linearity 1

Domain with one point on the lower boundary (angle)

2

Domain with two points on the lower boundary (trolleybus, birdie)

3

Domain with more than two points on the lower boundary (multicup, multitrolleybus, multibirdie)

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 20 / 21

Evolution

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 21 / 21

Evolution

1 angle + cup = trolleybus Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 21 / 21

Evolution

1 angle + cup = trolleybus 2 trolleybus + angle = birdie Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 21 / 21

Evolution

1 angle + cup = trolleybus 2 trolleybus + angle = birdie 3 trolleybus + cup = multicup Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 21 / 21

Evolution

1 angle + cup = trolleybus 2 trolleybus + angle = birdie 3 trolleybus + cup = multicup 4 multicup + angle = multitrolleybus Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 21 / 21

Evolution

1 angle + cup = trolleybus 2 trolleybus + angle = birdie 3 trolleybus + cup = multicup 4 multicup + angle = multitrolleybus 5 multitrolleybus + angle = multibirdie Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 21 / 21

Evolution

1 angle + cup = trolleybus 2 trolleybus + angle = birdie 3 trolleybus + cup = multicup 4 multicup + angle = multitrolleybus 5 multitrolleybus + angle = multibirdie 6 . . .

and so on . . .

Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 21 / 21