Sharp functional inequalities and nonlinear diffusions

diffusions

Jean Dolbeault

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

July 7, 2015

CMO Workshop 15w5049 on Kinetic and Related Equations,

Oaxaca, July 6-10, 2015

Introduction

1

Caffarelli-Kohn-Nirenberg inequalities

2

Entropy and the fast diffusion equation

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Caffarelli-Kohn-Nirenberg inequalities

The symmetry issue —

In collaboration with M.J. Esteban and M. Loss

⊲ Nonlinear flows (fast diffusion equation) can be used as a tool for the investigation of sharp functional inequalities

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Caffarelli-Kohn-Nirenberg inequalities and the symmetry breaking issue

Let D a,b := n

v ∈ L ^p R ^d , | x | ^−b dx

: | x | ^−a |∇ v | ∈ L ² R ^d , dx o Z

R

| v | ^p

| x | ^{b p} dx 2/p

≤ C a,b

Z

R

|∇ v | ²

| x | ^{2 a} dx ∀ v ∈ D a,b

hold under the conditions that a ≤ b ≤ a + 1 if d ≥ 3, a < b ≤ a + 1 if d = 2, a + 1/2 < b ≤ a + 1 if d = 1, and a < a c := (d − 2)/2

p = 2 d

d − 2 + 2 (b − a)

⊲ With v ⋆ (x ) =

1 + | x | ^{(p − 2) (a}

^{− a)} ₋

and C ^⋆ _a,b = k | x | ^−b v ⋆ k ² p

k | x | ^{− a} ∇ v ⋆ k ² 2

do we have C a,b = C ^⋆ _a,b (symmetry)

or C a,b > C ^⋆ _a,b (symmetry breaking) ?

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

CKN: range of the parameters

Figure: d = 3 Z

R

| v | ^p

| x | ^{b p} dx 2/p

≤ C a,b

Z

R

|∇ v | ²

| x | ^{2 a} dx

a ≤ b ≤ a + 1 if d ≥ 3

a < b ≤ a + 1 if d = 2, a + 1/2 < b ≤ a + 1 if d = 1 and a < a c := (d − 2)/2

p = 2 d

d − 2 + 2 (b − a)

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Symmetry vs. symmetry breaking

[J.D., Esteban, Loss, Tarantello, 2009] There is a curve which separates the symmetry region from the symmetry breaking region, which is parametrized by a function p 7→ a + b

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Moving planes and symmetrization techniques

[Chou, Chu], [Horiuchi]

[Betta, Brock, Marcaldo, Posteraro]

+ Perturbation results: [CS Lin, ZQ Wang], [Smets, Willem], [JD, Esteban, Tarantello 2007], [J.D., Esteban, Loss, Tarantello, 2009]

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Linear instability of radial minimizers:

the Felli-Schneider curve

[Catrina, Wang], [Felli, Schneider] The functional C ^⋆ _a,b

Z

R

|∇ v | ²

| x | ^{2 a} dx − Z

R

| v | ^p

| x | ^{b p} dx 2/p

is linearly instable at v = v ⋆

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Direct spectral estimates

[J.D., Esteban, Loss, 2011]: sharp interpolation on the sphere and a Keller-Lieb-Thirring spectral estimate on the line

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Numerical results

20 40 60 80 100

10 20 30 40 50

Λ(µ) symmetric non-symmetric asymptotic

bifurcation µ α

Parametric plot of the branch of optimal functions for p = 2.8, d = 5.

Non-symmetric solutions bifurcate from symmetric ones at a bifurcation point computed by V. Felli and M. Schneider. The branch behaves for large values of Λ as predicted by F. Catrina and Z.-Q. Wang

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Other evidences

Further numerical results [J.D., Esteban, 2012]

Formal commutation of the non-symmetric branch near the bifurcation point [J.D., Esteban, 2013]

Asymptotic energy estimates [J.D., Esteban, 2013]

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Symmetry vs. symmetry breaking:

the sharp result

A result based on entropies and nonlinear flows

[J.D., Esteban, Loss, 2015]: http://arxiv.org/abs/1506.03664

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The symmetry result

The Felli & Schneider curve is defined by b FS (a) := d (a c − a)

2 p

(a c − a) ² + d − 1 + a − a c

Theorem

Let d ≥ 2 and p < 2 ^∗ . If either a ∈ [0, a c ) and b > 0, or a < 0 and b ≥ b FS (a), then the optimal functions for the Caffarelli-Kohn-Nirenberg inequalities are radially symmetric

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The Emden-Fowler transformation and the cylinder

⊲ With an Emden-Fowler transformation, Caffarelli-Kohn-Nirenberg inequalities on the Euclidean space are equivalent to

Gagliardo-Nirenberg inequalities on a cylinder

v (r , ω) = r ^{a − a}

ϕ(s, ω) with r = | x | , s = − log r and ω = x r With this transformation, the Caffarelli-Kohn-Nirenberg inequalities can be rewritten as

k ∂ s ϕ k ² L

( C ) + k∇ ω ϕ k ² L

( C ) + Λ k ϕ k ² L

( C ) ≥ µ(Λ) k ϕ k ² L

( C ) ∀ ϕ ∈ H ¹ ( C )

where Λ := (a c − a) ² , C = R × S ^{d− 1} and the optimal constant µ(Λ) is µ(Λ) = 1

C a,b

with a = a c ± √

Λ and b = d p ± √

Λ

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Entropy and the fast diffusion equation (2000-2014)

A short review —

⊲ Relative entropy, linearization, functional inequalities, improvements, improved rates of convergence, delays

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The fast diffusion equation

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

Self-similar (Barenblatt) function: U (t) = O(t ^{− d/(2 − d(1 − m))} ) as t → + ∞

[Friedmann, Kamin, 1980] k u(t, · ) − U (t, · ) k L

= o(t ^{− d/(2 − d(1 − m))} )

⊲ 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]

Refined relative entropy methods

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

v ^m m − 1 + 1

2 | x | ² v

dx − F 0

Entropy production is measured by the Generalized Fisher information

d

dt F [v ] = −I [v ] , I [v ] :=

Z

R

v

∇ v ^m v + x

2

dx

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Relative entropy and entropy production

Stationary solution: choose C such that k v _∞ k L

= k u k L

= M > 0 v _∞ (x ) := C + ¹ _2m ^−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 Sharp functional inequalities and nonlinear diffusions

An equivalent formulation: Gagliardo-Nirenberg inequalities

F [v] = Z

R

v ^m m − 1 + 1

2 | x | ² v

dx −F ⁰ ≤ 1 2 Z

R

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

|∇ w | ² dx + 1

1 − m − d Z

R

| w | ^1+p dx − K ≥ 0 for some γ, K = K 0

R

R

v dx = R

R

w ^2p dx γ

w = w _∞ = v _∞ ^1/2p is optimal Theorem

[Del Pino, J.D.] With 1 < p ≤ _{d −} ^d ₂ (fast diffusion case) and d ≥ 3 k w k L

( R

) ≤ C p,d ^GN k∇ w k ^θ L

( R

) k w k ¹ _L ⁻

^θ ₍ R

)

C p,d ^GN =

y(p− 1)

2πd

2y − d 2y

_Γ(y)

Γ(y −

)

, θ = _p(d+2 ^d(p− _{− (d} ¹⁾ _{− 2)p)} , y = ^p+1 _{p − 1}

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Entropy methods and linearization: sharp asymptotic rates of convergence

Generalized Barenblatt profiles: V D (x) := D + | x | ²

(H1) V D

≤ v 0 ≤ V D

for some D 0 > D 1 > 0

(H2) if d ≥ 3 and m ≤ m _∗ := ^d− _d− ⁴ ₂ , (v 0 − V D ) is integrable for a suitable D ∈ [D 1 , D 0 ]

Theorem

[Blanchet, Bonforte, J.D., Grillo, V´ azquez] Under Assumptions (H1)-(H2), if m < 1 and m 6 = m _∗ , the entropy decays according to

F [v (t, · )] ≤ C e ^{− 2 (1 − m) Λ}

^t ∀ t ≥ 0

where Λ α,d > 0 is the best constant in the Hardy–Poincar´e inequality Λ α,d

Z

R

| f | ² dµ α − 1 ≤ Z

R

|∇ f | ² dµ α ∀ f ∈ H ¹ (d µ α ) with α := 1/(m − 1) < 0, dµ α := h α dx, h α (x) := (1 + | x | ² ) ^α

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Plots ( d = 5)

λ01=−4α−2d

λ10=−2α λ₁₁=−6α−2 (d+ 2) λ02=−8α−4 (d+ 2)

λ₂₀=−4α λ₃₀ λ21λ

12 λ₀₃

λcontα,d:=¹4(d+ 2α−2)²

α=−d α=−(d+ 2)

α=−d+2 2

α=−^d−22 α=−^d+6₂

α 0 Essential spectrum ofLα,d

α=−√d−1−^d2 α=−√d−1−^d+42 α=− −^d+22

√2d (d= 5) Spectrum ofLα,d

mc=^d−2d

m1=^d−1d m2=^d+1_d+2

e m₁=_d+2^d

e m₂=^d+4_d+6

m Spectrum of (1−m)L1/(m−1),d

(d= 5)

Essential spectrum of (1

1

−m)L1/(m−1),d

2 4 6

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Improved asymptotic rates

[Bonforte, J.D., Grillo, V´ azquez] Assume that m ∈ (m 1 , 1), d ≥ 3.

Under Assumption (H1), if v is a solution of the fast diffusion equation with initial datum v 0 such that R

R

x v 0 dx = 0, then the asymptotic convergence holds with an improved rate corresponding to the improved spectral gap.

m1=d+2^d

m1=d−1 d

m2=^d+4d+6 m2=^d+1d+2

4

2

m 1 mc=d−2

d (d= 5)

γ(m)

0

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Higher order matching asymptotics

[J.D., G. Toscani] For some m ∈ (m c , 1) with m c := (d − 2)/d, we consider on R ^d the fast diffusion equation

∂u

∂τ + ∇ · u ∇ u ^{m − 1}

= 0

Without choosing R, we may define the function v such that u(τ, y + x 0 ) = R ^{− d} v (t, x) , R = R(τ) , t = ¹ ₂ log R , x = y

R Then v has to be a solution of

∂v

∂t + ∇ · h v

σ

^(m−m

⁾ ∇ v ^{m− 1} − 2 x i

= 0 t > 0 , x ∈ R ^d with (as long as we make no assumption on R)

2 σ ⁻

^(m−m

⁾ = R ^{1 −d (1 −m)} dR dτ

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Refined relative entropy and best matching Barenblatt functions

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

C _M + ¹ _σ | x | ²

∀ x ∈ R ^d (1) Note that σ is a function of t : as long as ^dσ _dt 6 = 0, the Barenblatt profile B σ is not a solution (it plays the role of a local Gibbs state) but we may still consider the relative entropy

F ^σ [v] := 1 m − 1

Z

R

v ^m − B _σ ^m − m B _σ ^{m − 1} (v − B σ ) dx The time derivative of this relative entropy is

d

dt F σ(t) [v(t , · )] = dσ dt

d dσ F ^σ [v]

| σ=σ(t)

| {z } choose it = 0

⇐⇒ Minimize F σ [v ] w.r.t. σ ⇐⇒ R

R

| x | ² B σ dx = R

R

| x | ² v dx

+ m

m − 1 Z

R

v ^{m− 1} − B _σ(t) ^{m − 1} ∂v

∂t dx

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Fast diffusion equations:

New points of view (2014-2015)

1

improved inequalities and scalings

2

scalings and a concavity property

3

improved rates and best matching

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Improved inequalities and scalings

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Gagliardo-Nirenberg inequalities and the FDE

k∇ w k ^ϑ L

( R

) k w k ¹ L ⁻

^ϑ ( R

) ≥ C GN k w k L

( R

)

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

R

|∇ w | ² dx+β R

R

| w | ^q+1 dx −K C ^α _GN R

R

| w | ^2q dx

is nonnegative and J[w ] ≥ J[w _∗ ] = 0

Theorem

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

β R

R

| w _∗ | ^q+1 dx − R

R

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

R

|∇ w | ² dx < ∞ and R

R

| w | ^2q | x | ² dx = R

R

w _∗ ^2q | x | ² dx

Consequence for decay rates of relative R´enyi entropies: faster rates of convergence in intermediate asymptotics for ^∂u _∂t = ∆u ^p

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 Sharp functional inequalities and nonlinear diffusions

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

u 0 dx = 1 and R

R

| 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 C ⋆ + | x | ² 1/(m− 1)

if m < 1

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The entropy

The entropy is defined by E :=

Z

R

u ^m dx and the Fisher information by

I :=

Z

R

u |∇ p | ² dx with p = m m − 1 u ^{m − 1} 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 Sharp functional inequalities and nonlinear diffusions

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

E ^{σ − 1} I ≥ E ^σ _⋆ ^{− 1} I ⋆

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

( R

) k w k ¹ L ⁻

^θ ( R

) ≥ C GN k w k L

( R

)

if 1 − ¹ _d ≤ m < 1. Hint: u ^{m − 1/2} = _kwk ^w

L2q(Rd)

, q = _{2 m−} ¹ ₁

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The proof

Lemma

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

dt Z

R

u |∇ p | ² dx = − 2 Z

R

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

u |∇ p | ² dx 2

− 2 1

d + m − 1 Z

R

u ^m dx Z

R

u ^m (∆p) ² dx

− 2 Z

R

u ^m dx Z

R

u ^m

D ² p − 1 d ∆p Id

2

dx

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Best matching

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Relative entropy and best matching

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

C ⋆ + _σ ¹ | x | ²

∀ x ∈ R ^d

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

R

B σ dx = R

R

v dx The relative entropy is defined by

F σ [v] := 1 m − 1

Z

R

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

| x | ² B 1 dx = Z

R

| x | ² B σ dx = Z

R

| x | ² v dx

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

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

( R

) = M and R

R

| x | ² u dx = R

R

| x | ² B σ dx, then F σ [u]

σ

^{(1 −m)} ≥ m 8 R

R

B ₁ ^m dx

C ⋆ k u − B σ k ^L

( R

) + 1 σ

Z

R

| x | ² | u − B σ | dx 2

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Temperature (fast diffusion case)

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

d Z

R

| x | ² u(t, x) dx and such that

Θ ^′ = 2 E

0 t

τ(0)

τ(t) τ

= lim

τ (s) s Θ

(s) s (s τ(0))Θ

s (s

τ(t))Θ

µ/2

s (s

τ

)Θ

s

Fast diffusion case

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Temperature (porous medium case) and delay

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 . We define s as a function of t and consider the delay given by

τ(t ) :=

Θ(t ) Θ ⋆

− t

0

s τ(t) τ(0)

τ

= lim

τ(s) t

s (s

τ

(0)) Θ

s Θ

(s)

s (s

τ (t))

Θ

s (s

τ

) Θ

Porous medium case

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 −}

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

E is such that

G ^′ = µ H with H := Θ ⁻

E H ^′

1 − m = Θ ^{− 1 −}

Θ I − d E ²

= d E ²

Θ

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

d

− Z

R

x · ∇ (u ^m ) dx 2

= 1 d

Z

R

x · u ∇ p dx 2

≤ 1 d

Z

R

u | x | ² dx Z

R

u |∇ p | ² dx = Θ I

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Fast diffusion equations on manifolds and sharp functional inequalities

1

The sphere

2

The line

3

Compact Riemannian manifolds

4

The cylinder: Caffarelli-Kohn-Nirenberg inequalities

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Interpolation inequalities on the sphere

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

A family of interpolation inequalities on the sphere

The following interpolation inequality holds on the sphere p − 2

d Z

S

|∇ u | ² d v g + Z

S

| u | ² d v g ≥ Z

S

| 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...

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Stereographic projection

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

v(x) = ^r

₂ ⁺¹

v(x) = (1 − z ) ⁻

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

Z

R

|∇ v | ² dx ≥ S d

Z

R

| v |

dx

∀ v ∈ D ^1,2 ( R ^d )

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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)

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 dσ

• 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

where ν d (z ) dz = dν d (z) := Z _d ^{− 1} ν

^{− 1} dz , ν(z ) := 1 − z ²

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The ultraspherical operator

With dν d = Z _d ^{− 1} ν

^{− 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

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

− 1 | f ^′ | ² ν dν d ≥ d k f k ² p − k f k ² 2

p − 2 ∀ f ∈ H ¹ ([ − 1, 1], d ν d )

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 Sharp functional inequalities and nonlinear diffusions

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

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

dt I [g (t, · )] ≤ − 2 d I [g (t , · )]

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 Sharp functional inequalities and nonlinear diffusions

... 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 − 2β}

L u + κ | u ^′ | ² u ν

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

dt Z 1

− 1

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

− 1

u ^{β(p − 2)} | u ^′ | ² ν dν d = 0

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

f = u ^β , k f ^′ k ² L

( S

) + _{p −} ^d ₂

k f k ² L

( S

) − k f k ² L

( S

)

≥ 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 Sharp functional inequalities and nonlinear diffusions

The rigidity point of view

Which computation have we done ? u t = u ^{2 − 2β}

L u + κ ^|u

′

|

u 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 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 terms

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 Sharp functional inequalities and nonlinear diffusions

What does “improvement” mean ?

An improved inequality is

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

( S

) = 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

( S

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

k∇ u k ² L

( S

) − d p − 2

h k u k ² L

( S

) − k u k ² L

( S

)

i

≥ d k u k ² L

( S

) (Ψ ◦ Φ)

C ^{k u k}

2 (1−r) Ls(Sd)

k u k

k u ^r − u ¯ ^r k ² L

( S

)

∀ u ∈ H ¹ ( S ^d ) s(p) := max { 2, p } and p ∈ (1, 2): q(p) := 2/p, r (p) := p; p ∈ (2, 4):

q = p/2, r = 2; p ≥ 4: q = p/(p − 2), r = p − 2

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Linear flow: improved Bakry-Emery method

Cf. [Arnold, JD]

w t = L w + κ | w ^′ | ² w ν With 2 ^♯ := _(d− ^{2 d}

⁺¹ ₁₎

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 Sharp functional inequalities and nonlinear diffusions

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

w t = w ^{2 − 2β}

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

− _2β ¹

_dt ^d R 1

− 1

| (w ^β ) ^′ | ² ν + _{p −} ^d ₂ w ^2β − w ^2β

dν d ≥ γ R 1

− 1

|w

|

w

ν ² dν d

Lemma

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 ^2β dν d

δ

.... but there are conditions on β

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 Sharp functional inequalities and nonlinear diffusions

The line

A first example of a non-compact manifold

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

One-dimensional Gagliardo-Nirenberg-Sobolev inequalities

k f k L

( R ) ≤ C GN (p) k f ^′ k ^θ L

( R ) k f k ¹ L ⁻

( ^θ R ) if p ∈ (2, ∞ ) k f k L

( R ) ≤ C GN (p) k f ^′ k ^η L

( R ) k f k ¹ L ⁻

( ^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

k u k

dx ≤ ¹ 2 k u k ² L

( R ) log

2 π e

ku

k

k u k

If p > 2, u ⋆ (x) = (cosh x) ⁻

solves

− (p − 2) ² u ^′′ + 4 u − 2 p | u | ^{p− 2} u = 0 If p ∈ (1, 2) consider u _∗ (x) = (cos x)

, x ∈ ( − π/2, π/2)

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Flow

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

( R ) + 4

(p − 2) ² k v k ² L

( R ) − C k v k ² L

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

v t = v ^{1 −}

√ 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 ) Similar results for p ∈ (1, 2)

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

With

z (x) = tanh x , v ⋆ = (1 − z ² )

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

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

p

ν

dz d = _p− ^{2 p} ₂ ⇐⇒ p = _d− ^2d ₂

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Change of variables = stereographic projection + Emden-Fowler

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 Sharp functional inequalities and nonlinear diffusions

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

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 has a unique positive solution v ∈ C ² ( M ): v ≡ 1

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Riemannian manifolds: first improvement

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

0 < λ < λ ⋆ = inf

u∈ H

(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 Sharp functional inequalities and nonlinear diffusions

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

( 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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Optimal interpolation inequality

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

(M) ≥ λ

p − 2 h

k v k ² L

(M) − k v k ² L

(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

(M) − _{p −} ^λ ₂ h

k v k ² L

(M) − k v k ² L

(M)

i

under the constraint k v k ^L

( M ) = 1 is negative if λ > λ 1

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The flow

The key tools the flow u t = u ^{2 − 2β}

∆ g u + κ |∇ u | ² u

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

(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

J. Demange, Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature, J. Funct. Anal., 254 (2008), pp. 593–611. Also see C. Villani, Optimal Transport, Old and New

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 Sharp functional inequalities and nonlinear diffusions

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The key estimates

G [u] := R

M

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

+ κ (β − 1) ^|∇u| _u

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

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 Sharp functional inequalities and nonlinear diffusions

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

^{− 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

M

|∇ u | ² d v g

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 Sharp functional inequalities and nonlinear diffusions

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

( 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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

( 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 Sharp functional inequalities and nonlinear diffusions

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 _∞

0 G λ [f (t , · )] dt

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

|∇ u | ² dx ≥ λ

log Z

R

e ^u dµ

− Z

R

u dµ

hold for some λ > 0 ? Let

Λ ⋆ := inf

x ∈ R

− ∆ 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 Sharp functional inequalities and nonlinear diffusions

Caffarelli-Kohn-Nirenberg inequalities

A sketch of the proof —

In collaboration with M.J. Esteban and M. Loss

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

a b

0

− 1

1

b = a + 1

b = a b = b

(a)

b = b

(a) Symmetry region

Symmetry breaking region

a

=

The Felli-Schneider region, or symmetry breaking region, appears in dark grey and is defined by a < 0, a ≤ b < b

(a). We prove that symmetry holds in the light grey region defined by b ≥ b

(a) when a < 0 and for any b ∈ [a, a + 1] if a ∈ [0, a

)

J. Dolbeault Sharp functional inequalities and nonlinear diffusions