• Aucun résultat trouvé

Entropy methods and sharp functional inequalities: new results

N/A
N/A
Protected

Academic year: 2022

Partager "Entropy methods and sharp functional inequalities: new results"

Copied!
87
0
0

Texte intégral

(1)

Entropy methods and sharp functional inequalities:

new results

Jean Dolbeault

http://www.ceremade.dauphine.fr/ ∼ dolbeaul Ceremade, Universit´e Paris-Dauphine

September 8, 2015

Nonlinear PDEs

Brussels, September 7-11, 2015

(2)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Outline

Entropy and the fast diffusion equation: from functional inequalities and characterization of optimal rates to best matching Barenblatt functions and improved inequalities Fast diffusion equations on manifolds and sharp functional inequalities: rigidity results, the carr´e du champ or Bakry-Emery method, and the use of nonlinear diffusion equations

An equivalent point of view: optimal Keller-Lieb-Thirring estimates on manifolds

Symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities; an introduction to the lecture of Michael Loss

J. Dolbeault Entropy methods and sharp functional inequalities

(3)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Entropy and the fast diffusion equation

A summary —

⊲ Relative entropy, linearization, functional inequalities,

improvements, improved rates of convergence, delays

(4)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

The fast diffusion equation

The fast diffusion equation corresponds to m < 1 u t = ∆u m x ∈ R d , t > 0

Self-similar (Barenblatt) functions attract all solutions as t → + ∞ [Friedmann, Kamin]

⊲ Entropy methods allow to measure the speed of convergence of any solution to U in norms which are adapted to the equation

⊲ Entropy methods provide explicit constants

The Bakry-Emery method [Carrillo, Toscani], [Juengel, Markowich, Toscani], [Carrillo, Juengel, Markowich, Toscani, Unterreiter], [Carrillo, V´ azquez]

The variational approach and Gagliardo-Nirenberg inequalities:

[del Pino, JD]

Mass transportation and gradient flow issues: [Otto et al.]

Large time asymptotics and the spectral approach: [Blanchet, Bonforte, JD, Grillo, V´ azquez], [Denzler, Koch, McCann], [Seis]

Refined relative entropy methods

J. Dolbeault Entropy methods and sharp functional inequalities

(5)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Time-dependent rescaling, free energy

Time-dependent rescaling: Take u(τ, y ) = R −d (τ) v (t , y /R(τ)) where

dR

dτ = R d(1 −m) 1 , R(0) = 1 , t = log R The function v solves a Fokker-Planck type equation

∂v

∂t = ∆v m + ∇ · (x v) , v | τ=0 = u 0

[Ralston, Newman, 1984] Lyapunov functional:

Generalized entropy or Free energy F [v ] :=

Z

R

d

v m m − 1 + 1

2 | x | 2 v

dx − F 0

Entropy production is measured by the Generalized Fisher information

d F [v ] = −I [v ] , I [v ] :=

Z v

∇ v m

+ x

2

dx

(6)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Relative entropy and entropy production

Stationary solution: choose C such that k v k L

1

= k u k L

1

= M > 0 v (x ) :=

C + 1 − m 2 m | x | 2

− 1/(1 −m) +

Relative entropy: Fix F 0 so that F [v ] = 0 Entropy – entropy production inequality

Theorem

d ≥ 3, m ∈ [ d d 1 , + ∞ ), m > 1 2 , m 6 = 1 I [v] ≥ 2 F [v ] Corollary

A solution v with initial data u 0 ∈ L 1 + ( R d ) such that | x | 2 u 0 ∈ L 1 ( R d ), u 0 m ∈ L 1 ( R d ) satisfies F [v(t , · )] ≤ F [u 0 ] e 2 t

J. Dolbeault Entropy methods and sharp functional inequalities

(7)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

An equivalent formulation: Gagliardo-Nirenberg inequalities

F [v] = Z

R

d

v m m − 1 + 1

2 | x | 2 v

dx −F 0 ≤ 1 2 Z

R

d

v

∇ v m v + x

2

dx = 1 2 I [v]

Rewrite it with p = 2m− 1 1 , v = w 2p , v m = w p+1 as 1

2 2m

2m − 1 2 Z

R

d

|∇ w | 2 dx + 1

1 − m − d Z

R

d

| w | 1+p dx − K ≥ 0 Theorem

[Del Pino, J.D.] With 1 < p ≤ d d 2 (fast diffusion case) and d ≥ 3

k w k L

2p

( R

d

) ≤ C p,d GN k∇ w k θ L

2

( R

d

) k w k 1 L

p+1

θ ( R

d

)

(8)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Improved asymptotic rates

[Denzler, McCann], [Denzler, Koch, McCann], [Seis]

[Blanchet, Bonforte, J.D., Grillo, V´ azquez], [Bonforte, J.D., Grillo, V´ azquez], [J.D., Toscani]

mc=d−2d 0

m1=d−1d m2=d+1d+2

e m2:=d+4d+6

m 1

2 4

Case 1 Case 2 Case 3

γ(m)

(d= 5)

e m1:=d+2d

J. Dolbeault Entropy methods and sharp functional inequalities

(9)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Fast diffusion equations:

some recent results

1

improved inequalities and scalings

2

scalings and a concavity property

3

improved rates and best matching

(10)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Improved inequalities and scalings

J. Dolbeault Entropy methods and sharp functional inequalities

(11)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Gagliardo-Nirenberg inequalities and the FDE

k∇ w k ϑ L

2

( R

d

) k w k 1 L

q+1

ϑ ( R

d

) ≥ C GN k w k L

2q

( R

d

)

With the right choice of the constants, the functional J[w ] := 1 4 (q 2 − 1) R

R

d

|∇ w | 2 dx+β R

R

d

| w | q+1 dx −K C α GN R

R

d

| w | 2q dx

2qα

is nonnegative and J[w ] ≥ J[w ] = 0

Theorem

[Dolbeault-Toscani] For some nonnegative, convex, increasing ϕ J[w ] ≥ ϕ

β R

R

d

| w | q+1 dx − R

R

d

| w | q+1 dx for any w ∈ L q+1 ( R d ) such that R

R

d

|∇ w | 2 dx < ∞ and R

R

d

| w | 2q | x | 2 dx = R

R

d

w 2q | x | 2 dx

Consequence for decay rates of relative R´enyi entropies: faster rates of

(12)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Scalings and a concavity property

⊲ R´enyi entropies, the entropy approach without rescaling: [Savar´e, Toscani]

⊲ faster rates of convergence: [Carrillo, Toscani], [JD, Toscani]

J. Dolbeault Entropy methods and sharp functional inequalities

(13)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

The fast diffusion equation in original variables

Consider the nonlinear diffusion equation in R d , d ≥ 1

∂u

∂t = ∆u m

with initial datum u(x, t = 0) = u 0 (x) ≥ 0 such that R

R

d

u 0 dx = 1 and R

R

d

| x | 2 u 0 dx < + ∞ . The large time behavior of the solutions is governed by the source-type Barenblatt solutions

U ⋆ (t, x) := 1 κ t 1/µ d B ⋆

x κ t 1/µ

where

µ := 2 + d (m − 1) , κ :=

2 µ m m − 1

1/µ

and B is the Barenblatt profile B (x) :=

 C ⋆ − | x | 2 1/(m− 1)

+ if m > 1

2 1/(m− 1)

(14)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

The entropy

The entropy is defined by E :=

Z

R

d

u m dx and the Fisher information by

I :=

Z

R

d

u |∇ p | 2 dx with p = m m − 1 u m 1

p is the pressure variable. If u solves the fast diffusion equation, then E = (1 − m) I

To compute I , we will use the fact that

∂p

∂t = (m − 1) p ∆p + |∇ p | 2 F := E σ with σ = µ

d (1 − m) = 1+ 2 1 − m

1

d + m − 1

= 2 d

1 1 − m − 1 has a linear growth asymptotically as t → + ∞

J. Dolbeault Entropy methods and sharp functional inequalities

(15)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

The concavity property

Theorem

[Toscani-Savar´e] Assume that m ≥ 1 − 1 d if d > 1 and m > 0 if d = 1.

Then F (t) is increasing, (1 − m) F ′′ (t) ≤ 0 and

t→ lim + ∞

1

t F(t ) = (1 − m) σ lim

t→ + ∞ E σ 1 I = (1 − m) σ E σ 1 I ⋆

[Dolbeault-Toscani] The inequality

σ 1 F = E σ 1 I ≥ E σ 1 I ⋆

is equivalent to the Gagliardo-Nirenberg inequality k∇ w k θ L

2

( R

d

) k w k 1 L

q+1

θ ( R

d

) ≥ C GN k w k L

2q

( R

d

)

if 1 − 1 ≤ m < 1. Hint: u m 1/2 = w , q = 1

(16)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

The proof

Lemma

If u solves ∂u ∂t = ∆u m with 1 d ≤ m < 1, then I = d

dt Z

R

d

u |∇ p | 2 dx = − 2 Z

R

d

u m

k D 2 p k 2 + (m − 1) (∆p) 2 dx

k D 2 p k 2 = 1

d (∆p) 2 +

D 2 p − 1 d ∆p Id

2

1

σ (1 − m) E 2 σ (E σ ) ′′ = (1 − m) (σ − 1) Z

R

d

u |∇ p | 2 dx 2

− 2 1

d + m − 1 Z

R

d

u m dx Z

R

d

u m (∆p) 2 dx

− 2 Z

R

d

u m dx Z

R

d

u m

D 2 p − 1 d ∆p Id

2

dx

J. Dolbeault Entropy methods and sharp functional inequalities

(17)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Best matching

(18)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Relative entropy and best matching

Consider the family of the Barenblatt profiles B σ (x) := σ

d2

C ⋆ + σ 1 | x | 2

m−11

∀ x ∈ R d

The Barenblatt profile B σ plays the role of a local Gibbs state if C ⋆ is chosen so that R

R

d

B σ dx = R

R

d

v dx The relative entropy is defined by

F σ [v] := 1 m − 1

Z

R

d

v m − B σ m − m B σ m− 1 (v − B σ ) dx To minimize F σ [v] with respect to σ is equivalent to fix σ such that

σ Z

R

d

| x | 2 B 1 dx = Z

R

d

| x | 2 B σ dx = Z

R

d

| x | 2 v dx

J. Dolbeault Entropy methods and sharp functional inequalities

(19)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

A Csisz´ar-Kullback(-Pinsker) inequality

Let m ∈ ( d+2 d , 1) and consider the relative entropy F σ [u] := 1

m − 1 Z

R

d

u m − B σ m − m B σ m− 1 (u − B σ ) dx

Theorem

[J.D., Toscani] Assume that u is a nonnegative function in L 1 ( R d ) such that u m and x 7→ | x | 2 u are both integrable on R d . If k u k L

1

( R

d

) = M and R

R

d

| x | 2 u dx = R

R

d

| x | 2 B σ dx, then F σ [u]

σ

d2

(1 −m) ≥ m 8 R

R

d

B 1 m dx

C ⋆ k u − B σ k L

1

( R

d

) + 1 σ

Z

R

d

| x | 2 | u − B σ | dx

2

(20)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Temperature (fast diffusion case)

The second moment functional (temperature) is defined by Θ(t ) := 1

d Z

R

d

| x | 2 u(t, x) dx and such that

Θ = 2 E

0 t

τ(0)

τ(t) τ

= lim

s→+∞

τ (s) s Θ

µ/2

(s) s (s τ(0))Θ

s (s

+

τ(t))Θ

µ/2

µ/2

s (s

+

τ

µ/2

s

Fast diffusion case

+

J. Dolbeault Entropy methods and sharp functional inequalities

(21)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

Temperature (porous medium case)

Let U ⋆ s be the best matching Barenblatt function, in the sense of relative entropy F [u | U ⋆ s ], among all Barenblatt functions ( U ⋆ s ) s>0 .

0

s τ(t) τ(0)

τ

= lim

s→+∞

τ(s) t

s (s

+

τ

(0)) Θ

µ/2

s Θ

µ/2

(s)

s (s

+

τ (t))

Θ

µ/2

s (s

+

τ

) Θ

µ/2

Porous medium case

(22)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Improved inequalities and scalings Scalings and a concavity property Best matching

A result on delays

Theorem

Assume that m ≥ 1 − d 1 and m 6 = 1. The best matching Barenblatt function of a solution u is (t, x) 7→ U (t + τ(t), x) and the function t 7→ τ(t ) is nondecreasing if m > 1 and nonincreasing if 1 − d 1 ≤ m < 1 With G := Θ 1

η2

, η = d (1 − m) = 2 − µ, the R´enyi entropy power functional H := Θ

η2

E is such that

G = µ H with H := Θ

η2

E H

1 − m = Θ 1

η2

Θ I − d E 2

= d E 2

Θ

η2

+1 (q − 1) with q := Θ I d E 2 ≥ 1 d E 2 = 1

d

− Z

R

d

x · ∇ (u m ) dx 2

= 1 d

Z

R

d

x · u ∇ p dx 2

≤ 1 d

Z

R

d

u | x | 2 dx Z

R

d

u |∇ p | 2 dx = Θ I

J. Dolbeault Entropy methods and sharp functional inequalities

(23)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Fast diffusion equations on manifolds and sharp functional inequalities

1

The sphere

2

The line

3

Compact Riemannian manifolds

4

The Moser-Trudinger-Onofri inequality on

Riemannian manifolds

(24)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Interpolation inequalities on the sphere

Joint work with M.J. Esteban, M. Kowalczyk and M. Loss

J. Dolbeault Entropy methods and sharp functional inequalities

(25)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

A family of interpolation inequalities on the sphere

The following interpolation inequality holds on the sphere p − 2

d Z

S

d

|∇ u | 2 d v g + Z

S

d

| u | 2 d v g ≥ Z

S

d

| u | p d v g

2/p

∀ u ∈ H 1 ( S d , dv g ) for any p ∈ (2, 2 ] with 2 = d 2d 2 if d ≥ 3

for any p ∈ (2, ∞ ) if d = 2

Here dv g is the uniform probability measure: v g ( S d ) = 1

1 is the optimal constant, equality achieved by constants

p = 2 corresponds to Sobolev’s inequality...

(26)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Stereographic projection

J. Dolbeault Entropy methods and sharp functional inequalities

(27)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Sobolev’s inequality

The stereographic projection of S d ⊂ R d × R ∋ (ρ φ, z) onto R d : to ρ 2 + z 2 = 1, z ∈ [ − 1, 1], ρ ≥ 0, φ ∈ S d− 1 we associate x ∈ R d such that r = | x | , φ = | x x |

z = r 2 − 1

r 2 + 1 = 1 − 2

r 2 + 1 , ρ = 2 r r 2 + 1

and transform any function u on S d into a function v on R d using u(y ) = ρ r

d−22

v(x) = r

2

2 +1

d−22

v(x) = (1 − z )

d−22

v(x) p = 2 , S d = 1 4 d (d − 2) | S d | 2/d : Euclidean Sobolev inequality

Z

R

d

|∇ v | 2 dx ≥ S d

Z

R

d

| v |

d−2d2

dx

d−2d

∀ v ∈ D 1,2 ( R d )

(28)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Schwarz symmetrization and the ultraspherical setting

(ξ 0 , ξ 1 , . . . ξ d ) ∈ S d , ξ d = z, P d

i=0 | ξ i | 2 = 1 [Smets-Willem]

Lemma

Up to a rotation, any minimizer of Q depends only on ξ d = z

• Let dσ(θ) := (sin Z θ)

dd−1

d θ, Z d := √ π Γ(

d 2 ) Γ( d+1 2 )

: ∀ v ∈ H 1 ([0, π], dσ) p − 2

d Z π

0 | v (θ) | 2 dσ + Z π

0 | v (θ) | 2 d σ ≥ Z π

0 | v (θ) | p

2p

• Change of variables z = cos θ, v(θ) = f (z ) p − 2

d Z 1

− 1 | f | 2 ν dν d + Z 1

− 1 | f | 2 dν d ≥ Z 1

− 1 | f | p dν d

2 p

where ν d (z ) dz = dν d (z) := Z d 1 ν

d2

1 dz , ν(z ) := 1 − z 2

J. Dolbeault Entropy methods and sharp functional inequalities

(29)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The ultraspherical operator

With dν d = Z d 1 ν

d2

1 dz, ν(z) := 1 − z 2 , consider the space L 2 (( − 1, 1), dν d ) with scalar product

h f 1 , f 2 i = Z 1

− 1

f 1 f 2 dν d , k f k p = Z 1

− 1

f p dν d

p1

The self-adjoint ultraspherical operator is

L f := (1 − z 2 ) f ′′ − d z f = ν f ′′ + d 2 ν f which satisfies h f 1 , L f 2 i = − R 1

− 1 f 1 f 2 ν dν d

Proposition

Let p ∈ [1, 2) ∪ (2, 2 ], d ≥ 1

−h f , L f i = Z 1

| f | 2 ν dν d ≥ d k f k 2 p − k f k 2 2

∀ f ∈ H 1 ([ − 1, 1], d ν d )

(30)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Flows on the sphere

Heat flow and the Bakry-Emery method

Fast diffusion (porous media) flow and the choice of the exponents

Joint work with M.J. Esteban, M. Kowalczyk and M. Loss

J. Dolbeault Entropy methods and sharp functional inequalities

(31)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Heat flow and the Bakry-Emery method

With g = f p , i.e. f = g α with α = 1/p

(Ineq.) −h f , L f i = −h g α , L g α i =: I [g ] ≥ d k g k 2 1 α − k g 2 α k 1

p − 2 =: F [g ] Heat flow

∂g

∂t = L g d

dt k g k 1 = 0 , d

dt k g 2 α k 1 = − 2 (p − 2) h f , L f i = 2 (p − 2) Z 1

− 1 | f | 2 ν dν d

which finally gives d

dt F [g (t , · )] = − d p − 2

d

dt k g 2 α k 1 = − 2 d I [g (t, · )]

Ineq. ⇐⇒ d

F [g (t , · )] ≤ − 2 d F [g (t, · )] ⇐ = d

I [g (t, · )] ≤ − 2 d I [g (t , · )]

(32)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The equation for g = f p can be rewritten in terms of f as

∂f

∂t = L f + (p − 1) | f | 2 f ν

− 1 2

d dt

Z 1

− 1 | f | 2 ν dν d = 1 2

d

dt h f , L f i = hL f , L f i + (p − 1) h | f | 2 f ν, L f i d

dt I [g (t, · )] + 2 d I [g (t, · )] = d dt

Z 1

− 1 | f | 2 ν dν d + 2 d Z 1

− 1 | f | 2 ν dν d

= − 2 Z 1

− 1

| f ′′ | 2 + (p − 1) d d + 2

| f | 4

f 2 − 2 (p − 1) d − 1 d + 2

| f | 2 f ′′

f

ν 2 dν d

is nonpositive if

| f ′′ | 2 + (p − 1) d d + 2

| f | 4

f 2 − 2 (p − 1) d − 1 d + 2

| f | 2 f ′′

f is pointwise nonnegative, which is granted if

(p − 1) d − 1 d + 2

2

≤ (p − 1) d

d + 2 ⇐⇒ p ≤ 2 d 2 + 1

(d − 1) 2 = 2 # < 2 d d − 2 = 2

J. Dolbeault Entropy methods and sharp functional inequalities

(33)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

... up to the critical exponent: a proof in two slides

d dz , L

u = ( L u) − L u = − 2 z u ′′ − d u Z 1

− 1

( L u) 2 dν d = Z 1

− 1 | u ′′ | 2 ν 2 d ν d + d Z 1

− 1 | u | 2 ν dν d

Z 1

− 1

( L u) | u | 2

u ν dν d = d d + 2

Z 1

− 1

| u | 4

u 2 ν 2 dν d − 2 d − 1 d + 2

Z 1

− 1

| u | 2 u ′′

u ν 2 dν d

On ( − 1, 1), let us consider the porous medium (fast diffusion) flow u t = u 2

L u + κ | u | 2 u ν

If κ = β (p − 2) + 1, the L p norm is conserved d

dt Z 1

u βp dν d = β p (κ − β (p − 2) − 1) Z 1

u β(p 2) | u | 2 ν dν d = 0

(34)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

f = u β , k f k 2 L

2

( S

d

) + p d 2

k f k 2 L

2

( S

d

) − k f k 2 L

p

( S

d

)

≥ 0 ?

A :=

Z 1

− 1 | u ′′ | 2 ν 2 dν d − 2 d − 1

d + 2 (κ + β − 1) Z 1

− 1

u ′′ | u | 2 u ν 2 dν d

+

κ (β − 1) + d

d + 2 (κ + β − 1) Z 1

− 1

| u | 4 u 2 ν 2 dν d

A is nonnegative for some β if 8 d 2

(d + 2) 2 (p − 1) (2 − p) ≥ 0

A is a sum of squares if p ∈ (2, 2 ) for an arbitrary choice of β in a certain interval (depending on p and d)

A = Z 1

− 1

u ′′ − p + 2 6 − p

| u | 2 u

2

ν 2 dν d ≥ 0 if p = 2 and β = 4 6 − p

J. Dolbeault Entropy methods and sharp functional inequalities

(35)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The rigidity point of view

Which computation have we done ? u t = u 2

L u + κ |u u

|

2

ν

− L u − (β − 1) | u | 2

u ν + λ

p − 2 u = λ p − 2 u κ Multiply by L u and integrate

...

Z 1

− 1 L u u κ dν d = − κ Z 1

− 1

u κ | u | 2 u dν d

Multiply by κ | u u

|

2

and integrate ... = + κ

Z 1

− 1

u κ | u | 2 u dν d

The two terms cancel and we are left only with the two-homogenous

(36)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Improvements of the inequalities (subcritical range)

as long as the exponent is either in the range (1, 2) or in the range (2, 2 ), on can establish improved inequalities

An improvement automatically gives an explicit stability result of the optimal functions in the (non-improved) inequality

By duality, this provides a stability result for Keller-Lieb-Tirring inequalities

J. Dolbeault Entropy methods and sharp functional inequalities

(37)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

What does “improvement” mean ?

An improved inequality is

d Φ (e) ≤ i ∀ u ∈ H 1 ( S d ) s.t. k u k 2 L

2

( S

d

) = 1 for some function Φ such that Φ(0) = 0, Φ (0) = 1, Φ > 0 and Φ(s) > s for any s. With Ψ(s) := s − Φ 1 (s)

i − d e ≥ d (Ψ ◦ Φ)(e) ∀ u ∈ H 1 ( S d ) s.t. k u k 2 L

2

( S

d

) = 1 Lemma (Generalized Csisz´ar-Kullback inequalities)

k∇ u k 2 L

2

( S

d

) − d p − 2

h k u k 2 L

p

( S

d

) − k u k 2 L

2

( S

d

)

i

≥ d k u k 2 L

2

( S

d

) (Ψ ◦ Φ)

C k u k

2 (1−r) Ls(Sd)

k u k

2L2 (Sd)

k u r − u ¯ r k 2 L

q

( S

d

)

∀ u ∈ H 1 ( S d )

s(p) := max { 2, p } and p ∈ (1, 2): q(p) := 2/p, r (p) := p; p ∈ (2, 4):

(38)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Linear flow: improved Bakry-Emery method

Cf. [Arnold, JD]

w t = L w + κ | w | 2 w ν With 2 := (d− 2 d

2

+1 1)

2

γ 1 :=

d − 1 d + 2

2

(p − 1) (2 # − p) if d > 1 , γ 1 := p − 1

3 if d = 1 If p ∈ [1, 2) ∪ (2, 2 ] and w is a solution, then

d

dt (i − d e) ≤ − γ 1

Z 1

− 1

| w | 4

w 2 dν d ≤ − γ 1 | e | 2 1 − (p − 2) e Recalling that e = − i, we get a differential inequality

e ′′ + d e ≥ γ 1 | e | 2 1 − (p − 2) e After integration: d Φ(e(0)) ≤ i(0)

J. Dolbeault Entropy methods and sharp functional inequalities

(39)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Nonlinear flow: the H¨older estimate of J. Demange

w t = w 2

L w + κ | w | 2 w

For all p ∈ [1, 2 ], κ = β (p − 2) + 1, dt d R 1

− 1 w βp dν d = 0

1

2

dt d R 1

− 1

| (w β ) | 2 ν + p d 2 w − w

dν d ≥ γ R 1

− 1

|w

|

4

w

2

ν 2 dν d

Lemma (Demange)

For all w ∈ H 1 ( − 1, 1), dν d

, such that R 1

− 1 w βp dν d = 1 Z 1

− 1

| w | 4

w 2 ν 2 dν d ≥ 1 β 2

R 1

− 1 | (w β ) | 2 ν d ν d R 1

− 1 | w | 2 ν dν d

R 1

− 1 w dν d

δ

.... but there are conditions on β

(40)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Admissible ( p, β ) for d = 5

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1 2 3 4 5 6

J. Dolbeault Entropy methods and sharp functional inequalities

(41)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The line

A first example of a non-compact manifold

Joint work with M.J. Esteban, A. Laptev and M. Loss

(42)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

One-dimensional Gagliardo-Nirenberg-Sobolev inequalities

k f k L

p

( R ) ≤ C GN (p) k f k θ L

2

( R ) k f k 1 L

2

( θ R ) if p ∈ (2, ∞ ) k f k L

2

( R ) ≤ C GN (p) k f k η L

2

( R ) k f k 1 L

p

( η R ) if p ∈ (1, 2) with θ = p 2 p 2 and η = 2 2+p p

The threshold case corresponding to the limit as p → 2 is the logarithmic Sobolev inequality

R

R u 2 log

u

2

k u k

2L2 (R)

dx ≤ 1 2 k u k 2 L

2

( R ) log

2 π e

ku

k

2L2 (R)

k u k

2L2 (R)

If p > 2, u ⋆ (x) = (cosh x)

p−22

solves

− (p − 2) 2 u ′′ + 4 u − 2 p | u | p− 2 u = 0 If p ∈ (1, 2) consider u (x) = (cos x)

2−p2

, x ∈ ( − π/2, π/2)

J. Dolbeault Entropy methods and sharp functional inequalities

(43)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Flow

Let us define on H 1 ( R ) the functional F [v] := k v k 2 L

2

( R ) + 4

(p − 2) 2 k v k 2 L

2

( R ) − C k v k 2 L

p

( R ) s.t. F [u ⋆ ] = 0 With z (x) := tanh x , consider the flow

v t = v 1

p2

√ 1 − z 2

v ′′ + 2 p

p − 2 z v + p 2

| v | 2

v + 2

p − 2 v

Theorem (Dolbeault-Esteban-Laptev-Loss) Let p ∈ (2, ∞ ). Then

d

dt F [v(t)] ≤ 0 and lim

t →∞ F [v(t)] = 0

d

dt F [v (t )] = 0 ⇐⇒ v 0 (x) = u ⋆ (x − x 0 )

(44)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The inequality ( p > 2) and the ultraspherical operator

The problem on the line is equivalent to the critical problem for the ultraspherical operator

Z

R

| v | 2 dx + 4 (p − 2) 2

Z

R

| v | 2 dx ≥ C Z

R

| v | p dx

2p

With

z (x) = tanh x , v ⋆ = (1 − z 2 )

p−21

and v (x) = v ⋆ (x ) f (z(x)) equality is achieved for f = 1 and, if we let ν(z ) := 1 − z 2 , then

Z 1

− 1 | f | 2 ν d ν d + 2 p (p − 2) 2

Z 1

− 1 | f | 2 dν d ≥ 2 p (p − 2) 2

Z 1

− 1 | f | p dν d

2 p

where dν p denotes the probability measure dν p (z ) := ζ 1

p

ν

p−22

dz d = p− 2 p 2 ⇐⇒ p = d− 2d 2

J. Dolbeault Entropy methods and sharp functional inequalities

(45)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Change of variables = stereographic projection + Emden-Fowler

(46)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Compact Riemannian manifolds

no sign is required on the Ricci tensor and an improved integral criterion is established

the flow explores the energy landscape... and shows the non-optimality of the improved criterion

J. Dolbeault Entropy methods and sharp functional inequalities

(47)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Riemannian manifolds with positive curvature

( M , g ) is a smooth closed compact connected Riemannian manifold dimension d, no boundary, ∆ g is the Laplace-Beltrami operator vol( M ) = 1, R is the Ricci tensor, λ 1 = λ 1 ( − ∆ g )

ρ := inf

M inf

ξ ∈ S

d−1

R (ξ , ξ) Theorem (Licois-V´eron, Bakry-Ledoux)

Assume d ≥ 2 and ρ > 0. If λ ≤ (1 − θ) λ 1 + θ d ρ

d − 1 where θ = (d − 1) 2 (p − 1) d (d + 2) + p − 1 > 0 then for any p ∈ (2, 2 ), the equation

− ∆ g v + λ

p − 2 v − v p− 1

= 0

(48)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Riemannian manifolds: first improvement

Theorem (Dolbeault-Esteban-Loss) For any p ∈ (1, 2) ∪ (2, 2 )

0 < λ < λ ⋆ = inf

u∈ H

2

(M)

Z

M

h (1 − θ) (∆ g u) 2 + θ d

d − 1 R ( ∇ u, ∇ u) i d v g

R

M |∇ u | 2 d v g

there is a unique positive solution in C 2 ( M ): u ≡ 1 lim p → 1

+

θ(p) = 0 = ⇒ lim p → 1

+

λ ⋆ (p) = λ 1 if ρ is bounded λ ⋆ = λ 1 = d ρ/(d − 1) = d if M = S d since ρ = d − 1

(1 − θ) λ 1 + θ d ρ

d − 1 ≤ λ ⋆ ≤ λ 1

J. Dolbeault Entropy methods and sharp functional inequalities

(49)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Riemannian manifolds: second improvement

H g u denotes Hessian of u and θ = (d − 1) 2 (p − 1) d (d + 2) + p − 1 Q g u := H g u − g

d ∆ g u − (d − 1) (p − 1) θ (d + 3 − p)

∇ u ⊗ ∇ u

u − g

d

|∇ u | 2 u

Λ ⋆ := inf

u ∈ H

2

( M ) \{ 0 }

(1 − θ) Z

M

(∆ g u) 2 d v g + θ d d − 1

Z

M

h k Q g u k 2 + R ( ∇ u, ∇ u) i Z

M |∇ u | 2 d v g

Theorem (Dolbeault-Esteban-Loss)

Assume that Λ ⋆ > 0. For any p ∈ (1, 2) ∪ (2, 2 ), the equation has a

unique positive solution in C 2 ( M ) if λ ∈ (0, Λ ⋆ ): u ≡ 1

(50)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Optimal interpolation inequality

For any p ∈ (1, 2) ∪ (2, 2 ) or p = 2 if d ≥ 3 k∇ v k 2 L

2

(M) ≥ λ

p − 2

h k v k 2 L

p

(M) − k v k 2 L

2

(M)

i ∀ v ∈ H 1 ( M ) Theorem (Dolbeault-Esteban-Loss)

Assume Λ ⋆ > 0. The above inequality holds for some λ = Λ ∈ [Λ ⋆ , λ 1 ] If Λ ⋆ < λ 1 , then the optimal constant Λ is such that

Λ ⋆ < Λ ≤ λ 1

If p = 1, then Λ = λ 1

Using u = 1 + ε ϕ as a test function where ϕ we get λ ≤ λ 1

A minimum of

v 7→ k∇ v k 2 L

2

(M) − p λ 2 h

k v k 2 L

p

(M) − k v k 2 L

2

(M)

i under the constraint k v k L

p

( M ) = 1 is negative if λ > λ 1

J. Dolbeault Entropy methods and sharp functional inequalities

(51)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The flow

The key tool is the flow u t = u 2

∆ g u + κ |∇ u | 2 u

, κ = 1 + β (p − 2) If v = u β , then dt d k v k L

p

( M ) = 0 and the functional

F [u] :=

Z

M |∇ (u β ) | 2 d v g + λ p − 2

"

Z

M

u 2 β d v g − Z

M

u β p d v g

2/p #

is monotone decaying

(52)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Elementary observations (1/2)

Let d ≥ 2, u ∈ C 2 ( M ), and consider the trace free Hessian L g u := H g u − g

d ∆ g u Lemma

Z

M

(∆ g u) 2 d v g = d d − 1

Z

M

k L g u k 2 d v g + d d − 1

Z

M

R ( ∇ u, ∇ u) d v g

Based on the Bochner-Lichnerovicz-Weitzenb¨ ock formula 1

2 ∆ |∇ u | 2 = k H g u k 2 + ∇ (∆ g u) · ∇ u + R ( ∇ u, ∇ u)

J. Dolbeault Entropy methods and sharp functional inequalities

(53)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Elementary observations (2/2)

Lemma

Z

M

∆ g u |∇ u | 2 u d v g

= d

d + 2 Z

M

|∇ u | 4

u 2 d v g − 2 d d + 2

Z

M

[L g u] :

∇ u ⊗ ∇ u u

d v g

Lemma

Z

M

(∆ g u) 2 d v g ≥ λ 1

Z

M

|∇ u | 2 d v g ∀ u ∈ H 2 ( M )

and λ 1 is the optimal constant in the above inequality

(54)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The key estimates

G [u] := R

M

h θ (∆ g u) 2 + (κ + β − 1) ∆ g u |∇u| u

2

+ κ (β − 1) |∇u| u

24

i d v g

Lemma 1 2 β 2

d

dt F [u] = − (1 − θ) Z

M

(∆ g u) 2 d v g − G [u] + λ Z

M

|∇ u | 2 d v g

Q θ g u := L g u − 1 θ d d+2 1 (κ + β − 1) h

∇ u ⊗∇ u

u − g d |∇u| u

2

i Lemma

G [u] = θ d d − 1

Z

M k Q θ g u k 2 d v g + Z

M

R ( ∇ u, ∇ u) d v g

− µ Z

M

|∇ u | 4 u 2 d v g

with µ := 1 θ

d − 1 d + 2

2

(κ + β − 1) 2 − κ (β − 1) − (κ + β − 1) d d + 2

J. Dolbeault Entropy methods and sharp functional inequalities

(55)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The end of the proof

Assume that d ≥ 2. If θ = 1, then µ is nonpositive if β (p) ≤ β ≤ β + (p) ∀ p ∈ (1, 2 ) where β ± := b ± 2 a b

2

a with a = 2 − p + h

(d− 1) (p− 1) d+2

i 2

and b = d+3 d+2 p Notice that β (p) < β + (p) if p ∈ (1, 2 ) and β (2 ) = β + (2 )

θ = (d − 1) 2 (p − 1)

d (d + 2) + p − 1 and β = d + 2 d + 3 − p Proposition

Let d ≥ 2, p ∈ (1, 2) ∪ (2, 2 ) (p 6 = 5 or d 6 = 2) 1

2 β 2 d

dt F [u] ≤ (λ − Λ ⋆ ) Z

|∇ u | 2 d v g

(56)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The Moser-Trudinger-Onofri inequality on Riemannian manifolds

Joint work with G. Jankowiak and M.J. Esteban

Extension to compact Riemannian manifolds of dimension 2...

J. Dolbeault Entropy methods and sharp functional inequalities

(57)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

We shall also denote by R the Ricci tensor, by H g u the Hessian of u and by

L g u := H g u − g d ∆ g u

the trace free Hessian. Let us denote by M g u the trace free tensor M g u := ∇ u ⊗ ∇ u − g

d |∇ u | 2 We define

λ ⋆ := inf

u ∈ H

2

( M ) \{ 0 }

Z

M

h

k L g u − 1 2 M g u k 2 + R ( ∇ u, ∇ u) i

e u/2 d v g

Z

M |∇ u | 2 e u/2 d v g

(58)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Theorem

Assume that d = 2 and λ ⋆ > 0. If u is a smooth solution to

− 1

2 ∆ g u + λ = e u then u is a constant function if λ ∈ (0, λ ⋆ ) The Moser-Trudinger-Onofri inequality on M

1

4 k∇ u k 2 L

2

( M ) + λ Z

M

u d v g ≥ λ log Z

M

e u d v g

∀ u ∈ H 1 ( M ) for some constant λ > 0. Let us denote by λ 1 the first positive eigenvalue of − ∆ g

Corollary

If d = 2, then the MTO inequality holds with λ = Λ := min { 4 π, λ ⋆ } . Moreover, if Λ is strictly smaller than λ 1 /2, then the optimal constant in the MTO inequality is strictly larger than Λ

J. Dolbeault Entropy methods and sharp functional inequalities

(59)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

The flow

∂f

∂t = ∆ g (e −f /2 ) − 1 2 |∇ f | 2 e −f /2 G λ [f ] :=

Z

M k L g f − 1 2 M g f k 2 e f /2 d v g + Z

M

R ( ∇ f , ∇ f ) e f /2 d v g

− λ Z

M

|∇ f | 2 e f /2 d v g Then for any λ ≤ λ ⋆ we have

d

dt F λ [f (t, · )] = Z

M − 1 2 ∆ g f + λ

∆ g (e f /2 ) − 1 2 |∇ f | 2 e f /2 d v g

= − G λ [f (t , · )]

Since F λ is nonnegative and lim t→∞ F λ [f (t, · )] = 0, we obtain that F λ [u] ≥

Z ∞

G λ [f (t , · )] dt

(60)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

The sphere The line

Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality

Weighted Moser-Trudinger-Onofri inequalities on the two-dimensional Euclidean space

On the Euclidean space R 2 , given a general probability measure µ does the inequality

1 16 π

Z

R

2

|∇ u | 2 dx ≥ λ

log Z

R

d

e u

− Z

R

d

u dµ

hold for some λ > 0 ? Let

Λ ⋆ := inf

x ∈ R

2

− ∆ log µ 8 π µ Theorem

Assume that µ is a radially symmetric function. Then any radially symmetric solution to the EL equation is a constant if λ < Λ ⋆ and the inequality holds with λ = Λ ⋆ if equality is achieved among radial functions

J. Dolbeault Entropy methods and sharp functional inequalities

(61)

Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds

Caffarelli-Kohn-Nirenberg inequalities

The symmetry issue

A brief introduction to the lecture of Michael Loss

⊲ Nonlinear flows (fast diffusion equation) can be used as a tool for

Références

Documents relatifs

By using inequalities involving several means, Neuman [4] presented the fol- lowing inequality chain generalizing the Adamovic-Mitrinovic inequality... (5) Proof Indeed, Let us

Keywords: Gagliardo–Nirenberg–Sobolev inequalities; Improved inequalities; Manifold of optimal functions; Entropy–entropy production method; Fast diffusion equation;

Abstract — The purpose of this paper is to explain the phenomenon of symmetry breaking for optimal functions in functional inequalities by the numerical computations of some well

On the Caffarelli-Kohn-Nirenberg inequalities: sharp con- stants, existence (and nonexistence), and symmetry of extremal functions.. On the best constant for a weighted

B Gagliardo-Nirenberg inequalities: optimal constants and rates B Asymptotic rates of convergence, Hardy-Poincaré inequality B The Rényi entropy powers approach.. Symmetry and

We introduce the good functional analysis framework to study this equation on a bounded domain and prove the existence of weak solutions defined globally in time for general

Given its prevalence in many systems, understanding how multichannel feeding affects food-web stability, especially in comparison to the more commonly used models that have only

We study the asymptotic behavior of random simply generated noncrossing planar trees in the space of compact subsets of the unit disk, equipped with the Hausdorff distance..