COLLOQUIUM HEIDELBERG

Description

& $ % COLLOQUIUM, HEIDELBERG, 18/06/2009 HIDDEN CONVEXITY IN SOME NONLINEAR PDEs FROM GEOMETRY AND PHYSICS YANN BRENIER CNRS, FR 2800, Université de Ni e-Sophia-Antipolis, Web site: e.fr/ brenier June 15, 2009 1

• fun tions


• monge-ampère equation


• tions ?


• legendre-fen hel


• test fun tion


• d2?


• pde well-posed


• transport theory sin

Sujets

• Monge?Ampère equation

Informations

• Publié par: pefav
• Publié le: 01 juin 2009
• Nombre de lectures: 8

Extrait

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& %
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α β
d
R Φ
2
β(DΦ(x))det(D Φ(x)) = α(x)
2
D Φ(x)
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2
D Φ(x)
d d
x∈R →DΦ(x)∈R
y =DΦ(x)
Z Z Z
2
f(y)β(y)dy = f(DΦ(x)))β(DΦ(x))det(D Φ(x))dx = f(DΦ(x)))α(x)dx
f
Z Z
f(y)β(y)dy = f(DΦ(x)))α(x)dx, ∀f
β(y)dy α(x)dx
d d
x∈R →DΦ(x)∈R
& %
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2
β(DΦ(x))det(D Φ(x)) = α(x)
CONVEX
Z Z
∗
Φ ⇒ Φ(x)α(x)dx+ Φ (y)β(y)dy
∗
Φ Φ
∗
Φ (y) = sup x·y−Φ(x).
d
x∈R
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Z Z Z
∗ ∗
Ψ(x)α(x)dx+ Ψ (y)β(y)dy = (Ψ(x)+Ψ (DΦ(x)))α(x)dx
DΦ α β
Z
≥ x·DΦ(x)α(x)dx
∗
Ψ (y) = sup x·y−Ψ(x).
d
x∈R
Z
∗
= (Φ(x)+Φ (DΦ(x)))α(x)dx
∗
Φ (y) = sup x·y−Φ(x)
∗
y =DΦ(x) Φ (DΦ(x)) =x·DΦ(x)−Φ(x)
Z Z
∗
= Φ(x)α(x)dx+ Φ (y)β(y)dy
Φ
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α β
Z Z
2 2
|x| α(x)dx < +∞, |y| β(y)dy < +∞,
unique x→DΦ(x)
x→DΦ(x)
α β
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d
Ω B R
1
1
1−1/d 1/d
|Ω| |B | ≤ |∂Ω|
1
d
DΦ
1 1
α(x) = , x∈Ω , β(y) = , y∈B .
1
|Ω| |B |
1
2
(Ω,α)→ (B ,β) , β(DΦ(x))det(D Φ(x)) = α(x)
1
|B |
1
2
i.e. det(D Φ(x)) = , x∈Ω.
|Ω|
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DΦ
Z Z Z
|∂Ω| = dσ(x)≥ DΦ(x)·n(x)dσ(x) = ΔΦ(x)dx
∂Ω ∂Ω Ω
Z
2 1/d
≥d (det(D Φ(x)) dx
Ω
1/d
(detA) ≤1/d Trace(A)
A
1−1/d 1/d
=d|Ω| |B |
1
|B |
2 1
det(D Φ(x)) = , x∈Ω.
|Ω|
1
1−1/d 1/d
|Ω| |B | ≤ |∂Ω|
1
d
only Ω
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d
D R
2
d g
t
−1
◦g +∇p =0,
t
t
2
dt
t→ g SDiff(D)
t
D p
t
D
2
L
SDiff(D)
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