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Bellman function and MongeAmp` ere equation Vasily Vasyunin St.-Petersburg department of the Steklov Mathematical Institute May 19, 2017 Probability and Analysis B edlewo, Poland, May 1519, 2017 Vasily Vasyunin (Steklov Institute)

  1. Various choices of f (continued) Reverse H¨ older inequality for A p -weights: Ω δ = { ( x 1 , x 2 ): 1 ≤ x p − 1 x 1 ≤ δ } , 2 f ( s ) = s q , ϕ {� ϕ q � J : ϕ ∈ A p , [ ϕ ] A p ≤ δ } . B ( x ; δ ) = sup Equivalence of the “norms” in RH 1 and A ∞ : Ω δ = { ( x 1 , x 2 ): log x 1 ≤ x 2 ≤ log x 1 + δ } , ϕ {� ϕ log ϕ � J : ϕ ∈ A ∞ , [ ϕ ] A ∞ ≤ e δ } . f ( s ) = s log s , B ( x ; δ ) = sup Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 7 / 21

  2. 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

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

  4. 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 Ω ε . 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

  5. 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

  6. Monge–Amp` ere equation  ∂ 2 B ∂ 2 B  d 2 B ∂ x 2 ∂ x 1 ∂ x 2  ≤ 0 . 1 dx 2 =  ∂ 2 B ∂ 2 B ∂ x 2 ∂ x 1 ∂ x 2 2 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 10 / 21

  7. Monge–Amp` ere equation  ∂ 2 B ∂ 2 B  d 2 B ∂ x 2 ∂ x 1 ∂ x 2  ≤ 0 . 1 dx 2 =  ∂ 2 B ∂ 2 B ∂ x 2 ∂ x 1 ∂ x 2 2 Concavity has to be degenerate along some direction. Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 10 / 21

  8. Monge–Amp` ere equation  ∂ 2 B ∂ 2 B  d 2 B ∂ x 2 ∂ x 1 ∂ x 2  ≤ 0 . 1 dx 2 =  ∂ 2 B ∂ 2 B ∂ x 2 ∂ x 1 ∂ x 2 2 Concavity has to be degenerate along some direction. The Hessian matrix d 2 B dx 2 has to be degenerate. Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 10 / 21

  9. Monge–Amp` ere equation  ∂ 2 B ∂ 2 B  d 2 B ∂ x 2 ∂ x 1 ∂ x 2  ≤ 0 . 1 dx 2 =  ∂ 2 B ∂ 2 B ∂ x 2 ∂ x 1 ∂ x 2 2 Concavity has to be degenerate along some direction. The Hessian matrix d 2 B dx 2 has to be degenerate. Monge–Amp` ere equation � ∂ 2 B � 2 ∂ 2 B · ∂ 2 B = . ∂ x 2 ∂ x 2 ∂ x 1 ∂ x 2 1 2 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 10 / 21

  10. Monge–Amp` ere equation  ∂ 2 B ∂ 2 B  d 2 B ∂ x 2 ∂ x 1 ∂ x 2  ≤ 0 . 1 dx 2 =  ∂ 2 B ∂ 2 B ∂ x 2 ∂ x 1 ∂ x 2 2 Concavity has to be degenerate along some direction. The Hessian matrix d 2 B dx 2 has to be degenerate. Monge–Amp` ere equation � ∂ 2 B � 2 ∂ 2 B · ∂ 2 B = . ∂ x 2 ∂ x 2 ∂ x 1 ∂ x 2 1 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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 ∂ω ε then it touches it tangentially. Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 11 / 21

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

  17. Left tangent foliation ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  18. Left tangent foliation ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  19. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  20. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  21. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  22. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  23. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  24. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  25. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  26. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  27. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

  28. Left tangent foliation ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 12 / 21

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

  30. 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

  31. 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

  32. 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 of the boundary condition instead of third derivative. Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 14 / 21

  33. 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 of the boundary condition instead of third derivative. � � The boundary of the graph of B is the curve γ ( t ) = g 1 ( t ) , g 2 ( t ) , f ( t ) , Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 14 / 21

  34. 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 of the boundary condition instead of third derivative. � � The boundary of the graph of B is the curve γ ( t ) = g 1 ( t ) , g 2 ( 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 ′ g ′ f ′ ( t ) � � γ ′ ( t ) � 1 ( t ) 2 ( t ) � � � � � � � � g ′′ g ′′ f ′′ ( t ) γ ′′ ( t ) 1 ( t ) 2 ( t ) = . � � � � � � � � g ′′′ 1 ( t ) g ′′′ 2 ( t ) f ′′′ ( t ) γ ′′′ ( t ) � � � � Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 14 / 21

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

  36. A cup ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  37. A cup ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  38. A cup ω ε ω 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  39. A cup ω ε ω 0 q g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  40. A cup ω ε ω 0 curvature > 0 q g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  41. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  42. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  43. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  44. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  45. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  46. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  47. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  48. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  49. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  50. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  51. A cup ω ε ω 0 curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  52. A cup ω ε ω 0 q g ( b ) q g ( a ) curvature > 0 q curvature < 0 g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 15 / 21

  53. 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

  54. 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

  55. The ends of a chord To determine which points are connected by a chord, we need to solve the following equation: � � � � γ ′ ( a ) g ′ 1 ( a ) g ′ 2 ( a ) f ′ ( a ) � � � � � � � � γ ′ ( b ) g ′ g ′ f ′ ( b ) = 1 ( b ) 2 ( b ) = 0 . � � � � � � � � γ ( b ) − γ ( a ) g 1 ( b ) − g 1 ( a ) g 2 ( b ) − g 2 ( a ) f ( b ) − f ( a ) � � � � Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 16 / 21

  56. The ends of a chord To determine which points are connected by a chord, we need to solve the following equation: � � � � γ ′ ( a ) g ′ 1 ( a ) g ′ 2 ( a ) f ′ ( a ) � � � � � � � � γ ′ ( b ) g ′ g ′ f ′ ( b ) = 1 ( b ) 2 ( b ) = 0 . � � � � � � � � γ ( b ) − γ ( a ) g 1 ( b ) − g 1 ( a ) g 2 ( b ) − g 2 ( a ) f ( b ) − f ( a ) � � � � In the case of BMO (parabolic strip: g ( s ) = ( s , s 2 )) 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

  57. The ends of a chord To determine which points are connected by a chord, we need to solve the following equation: � � � � γ ′ ( a ) g ′ 1 ( a ) g ′ 2 ( a ) f ′ ( a ) � � � � � � � � γ ′ ( b ) g ′ g ′ f ′ ( b ) = 1 ( b ) 2 ( b ) = 0 . � � � � � � � � γ ( b ) − γ ( a ) g 1 ( b ) − g 1 ( a ) g 2 ( b ) − g 2 ( a ) f ( b ) − f ( a ) � � � � In the case of BMO (parabolic strip: g ( s ) = ( s , s 2 )) this “ cap equation ” has the following form: � f ′ � [ a , b ] = f ′ ( a ) + f ′ ( b ) f ( b ) − f ( a ) = . b − a 2 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 16 / 21

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

  59. An angle q u Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  60. An angle q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  61. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  62. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  63. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  64. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  65. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  66. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  67. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  68. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  69. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

  70. An angle curvature > 0 q u curvature < 0 Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 17 / 21

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

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

  73. A trolleybus q g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

  74. A trolleybus curvature > 0 q g ( c ) Vasily Vasyunin (Steklov Institute) Bellman function and Monge–Amp` ere equation May 19, 2017 18 / 21

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