Sharp functional inequalities and nonlinear diﬀusions

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diffusions

Jean Dolbeault

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

June 5, 2015

Conference onRecent trends in geometric analysis, Carry-Le-Rouet, June 1-5, 2015

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Fast diffusion equations:

New points of view

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

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Improved inequalities and scalings

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The logarithmic Sobolev inequality

dµ=µdx,µ(x) = (2π)^−d/2e^−|x|²^/2, onR^d withd≥1 Gaussian logarithmic Sobolev inequality

Z

Rd|∇u|²dµ≥ 1 2

Z

Rd|u|²log|u|²dµ for any functionu∈H¹(R^d,dµ)such thatR

Rd|u|²dµ= 1 ϕ(t) :=d

4

exp 2t

d

−1−2t d

∀t ∈R

[Stam, 1959],[Weissler, 1978],[Bakry, Ledoux (2006)],[Fathi et al.

(2014)],[Dolbeault, Toscani (2014)]

Proposition

Z

Rd|∇u|²dµ−1 2

Z

Rd|u|² log|u|²dµ≥ϕ Z

Rd|u|²log|u|²dµ

∀u∈H¹(R^d,dµ) s.t.

Z

Rd|u|²dµ= 1 and Z

Rd|x|²|u|²dµ=d

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Consequences for the heat equation

Ornstein-Uhlenbeck equation (or backward Kolmogorov equation)

∂f

∂t = ∆f −x· ∇f

with initial datumf0∈L¹₊(R^d,(1 +|x|²)dµand define theentropy as E[f] :=

Z

Rd

f logf dµ , d

dtE[f] =−4 Z

Rd|∇√

f|²dµ≤ −2E[f] thus proving that E[f(t,·)]≤ E[f0]e⁻^2t. Moreover,

d dt

Z

Rd

f|x|²dµ= 2 Z

Rd

f (d− |x|²)dµ Theorem

Assume that E[f0]is finite and R

Rdf0|x|²dµ=d R

Rdf0dµ. Then E[f(t,·)]≤ −d

2 logh 1−

1−e⁻²^d^E^[f⁰^] e⁻^2ti

∀t ≥0

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Gagliardo-Nirenberg inequalities and the FDE

k∇wk^ϑL²(Rd)kwk¹L⁻^q+1^ϑ(Rd)≥CGNkwkL^2q(Rd)

With the right choice of the constants, the functional J[w] := ¹₄(q²−1)R

Rd|∇w|²dx+βR

Rd|w|^q+1dx−KC^α_GN R

Rd|w|^2qdx_2q^α is nonnegative and J[w]≥J[w_∗] = 0

Theorem

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

β R

Rd|w_∗|^q+1dx−R

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

Rd|∇w|²dx<∞and R

Rd|w|^2q|x|²dx =R

Rdw_∗^2q|x|²dx

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

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Scalings and a concavity property

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The fast diffusion equation in original variables

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

∂u

∂t = ∆u^p

with initial datumu(x,t = 0) =u0(x)≥0such thatR

Rdu0dx= 1and R

Rd|x|²u0dx<+∞. The large time behavior of the solutions is governed by the source-type Barenblatt solutions

U^⋆(t,x) := 1

κt^1/µdB^⋆ x κt^1/µ

where

µ:= 2 +d(p−1), κ:=

2µp p−1

1/µ

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







C⋆− |x|²1/(p−1)

+ ifp>1 C⋆+|x|²1/(p−1)

ifp<1

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The entropy

Theentropy is defined by E :=

Z

Rd

u^pdx and theFisher information by

I :=

Z

Rd

u|∇v|²dx with v = p p−1u^p⁻¹ Ifu solves the fast diffusion equation, then

E^′= (1−p) I To computeI^′, we will use the fact that

∂v

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

d(1−p) = 1+ 2 1−p

1

d +p−1

= 2 d

1 1−p−1 has a linear growth asymptotically ast →+∞

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The concavity property

Theorem

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

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

t→lim+∞

1

t F(t) = (1−p)σ lim

t→+∞E^σ⁻¹I = (1−p)σE^σ_⋆⁻¹I⋆

[Dolbeault-Toscani]The inequality

E^σ⁻¹I≥E^σ_⋆⁻¹I⋆

is equivalent to the Gagliardo-Nirenberg inequality k∇wk^θL²(Rd)kwk¹L⁻^q+1^θ(Rd)≥CGNkwkL^2q(Rd)

if1−¹_d ≤p<1. Hint: u^p⁻^1/2= _kwk^w

L2q(Rd),q=₂_p−¹₁

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The proof

Lemma

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

dt Z

Rd

u|∇v|²dx =−2 Z

Rd

u^p

kD²vk²+ (p−1) (∆v)² dx

kD²vk²= 1

d(∆v)²+

D²v− 1 d∆vId

2

1

σ(1−p)E²⁻^σ (E^σ)^′′= (1−p) (σ−1) Z

Rd

u|∇v|²dx 2

−2 1

d +p−1 Z

Rd

u^pdx Z

Rd

u^p(∆v)²dx

−2 Z

Rd

u^pdx Z

Rd

u^p

D²v− 1 d ∆vId

2

dx

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Best matching

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Relative entropy and best matching

Consider the family of the Barenblatt profiles Bσ(x) :=σ⁻^d² C⋆+_σ¹|x|²_p−1¹

∀x ∈R^d

The Barenblatt profile Bσ plays the role of alocal Gibbs stateifC⋆ is chosen so thatR

RdBσdx=R

Rdv dx The relative entropy is defined by

Fσ[v] := 1 p−1

Z

Rd

v^p−B_σ^p−p B_σ^p−¹(v−Bσ) dx To minimize F^σ[v]with respect to σis equivalent to fixσsuch that

σ Z

Rd|x|²B1dx= Z

Rd|x|²Bσdx= Z

Rd|x|²v dx

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A Csisz´ar-Kullback(-Pinsker) inequality

Letp∈(_d+2^d ,1)and consider the relative entropy Fσ[u] := 1

p−1 Z

Rd

u^p−B_σ^p−p B_σ^p−¹(u−Bσ) dx

Theorem

[J.D., Toscani] Assume that u is a nonnegative function inL¹(R^d)such that u^pand x 7→ |x|²u are both integrable onR^d. IfkukL¹(Rd)=M and R

Rd|x|²u dx =R

Rd|x|²Bσdx, then Fσ[u]

σ^d²⁽¹^−p) ≥ p 8R

RdB₁^pdx

C⋆ku−Bσk^L¹(Rd)+1 σ

Z

Rd|x|²|u−Bσ|dx 2

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Temperature (fast diffusion case)

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

d Z

Rd|x|²u(t,x)dx and such that

Θ^′= 2 E

0 t

τ(0)

τ(t) τ∞= lims→+∞τ(s) s Θ^µ/2(s) s (s τ(0))Θ

s (s+τ(t))Θ µ/2

µ/2 s (s+τ∞)Θ^µ/2 s

Fast diffusion case +

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Temperature (porous medium case) and delay

LetU⋆^s be thebest matching Barenblattfunction, in the sense of relative entropyF[u| U⋆^s], among all Barenblatt functions(U⋆^s)s>0. We define s as a function oft and consider thedelay given by

τ(t) :=

Θ(t) Θ⋆

^µ₂

−t

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

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A result on delays

Theorem

Assume that p≥1−¹_d and p6= 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 p >1 and nonincreasing if1−_d¹ ≤p<1 With G := Θ¹⁻^η²,η=d(1−p) = 2−µ, theR´enyi entropy power functional H := Θ⁻^η² Eis such that

G^′=µH with H := Θ⁻^η² E H^′

1−p = Θ⁻¹⁻^η² Θ I−dE²

= dE²

Θ^η²⁺¹(q−1) with q := Θ I dE² ≥1 dE²= 1

d

− Z

Rd

x· ∇(u^p)dx 2

= 1 d

Z

Rd

x·u∇v dx 2

≤ 1 d

Z

Rd

u|x|²dx Z

Rd

u|∇v|²dx= Θ I

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An estimate of the delay

Theorem

If p >1−¹_d and p 6= 1, then the delay satisfies

t→lim+∞|τ(t)−τ(0)| ≥ |1−p|Θ(0)¹⁻^d²⁽¹^−p) 2 H⋆

H⋆−H(0)2

Θ(0) I(0)−dE(0)²

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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

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The Cylinder: symmetry breaking in CKN inequalities

Interpolation inequalities on the sphere

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

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A family of interpolation inequalities on the sphere

The following interpolation inequality holds on the sphere p−2

d Z

Sd|∇u|²d vg+ Z

Sd|u|²d vg ≥ Z

Sd|u|^pd vg

2/p

∀u∈H¹(S^d,dvg) for anyp∈(2,2^∗]with2^∗= _d^2d₋₂ ifd ≥3

for anyp∈(2,∞)ifd= 2

Here dvg is the uniform probability measure: vg(S^d) = 1

1 is the optimal constant, equality achieved by constants p= 2^∗ corresponds to Sobolev’s inequality...

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Stereographic projection

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Sobolev inequality

The stereographic projection of S^d ⊂R^d×R∋(ρ φ,z)ontoR^d: to ρ²+z²= 1,z ∈[−1,1],ρ≥0,φ∈S^d−¹ we associatex∈R^d such that r =|x|,φ= _|^x_x_|

z = r²−1

r²+ 1 = 1− 2

r²+ 1 , ρ= 2r r²+ 1

and transform any function uonS^d into a functionv onR^d using u(y) = _ρ^r^d−₂²

v(x) = ^r²₂⁺¹^d−₂²

v(x) = (1−z)⁻^d−²²v(x) p= 2^∗,Sd =¹₄d(d−2)|S^d|^2/d: Euclidean Sobolev inequality

Z

Rd|∇v|²dx≥Sd

Z

Rd|v|^d−²^d² dx ^d−2_d

∀v∈ D^1,2(R^d)

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Extended inequality

Z

Sd|∇u|²d vg ≥ d p−2

"

Z

Sd|u|^pd vg

2/p

− Z

Sd|u|²d vg

#

∀u∈H¹(S^d,dµ) is valid

for anyp∈(1,2)∪(2,∞)ifd= 1,2 for anyp∈(1,2)∪(2,2^∗]ifd ≥3 Casep= 2: Logarithmic Sobolev inequality Z

Sd|∇u|²d vg≥ d 2

Z

Sd|u|²log

|u|² R

Sd|u|²d vg

d vg ∀u∈H¹(S^d,dµ) Casep= 1: Poincar´e inequality

Z

Sd|∇u|²d vg ≥d Z

Sd|u−u¯|²d vg with u¯:=

Z

Sd

u d vg ∀u∈H¹(S^d,dµ)

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Optimality: a perturbation argument

For anyp∈(1,2^∗]ifd≥3, anyp>1 ifd = 1or2, it is remarkable that

Q[u] := (p−2)k∇uk²L²(Sd)

kuk²_Lp(Sd)− kuk²_L²₍Sd)

≥ inf

u∈H¹(Sd,dµ)Q[u] = 1 d is achieved in the limiting case

Q[1 +εv]∼ k∇vk²L²(Sd)

kvk²L²(Sd)

as ε→0 when v is an eigenfunction associated with the first nonzero eigenvalue of∆g, thus proving the optimality

p<2: a proof by semi-groups using Nelson’s hypercontractivity lemma. p>2: no simple proof based on spectral analysis is available:

[Beckner], an approach based on Lieb’s duality, the Funk-Hecke formula and some (non-trivial) computations

elliptic methods /Γ2formalism of Bakry-Emery / nonlinear flows

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Schwarz symmetrization and the ultraspherical setting

(ξ0, ξ1, . . . ξd)∈S^d,ξd =z, Pd

i=0|ξi|²= 1[Smets-Willem]

Lemma

Up to a rotation, any minimizer ofQdepends only onξd =z

• Letdσ(θ) := ^(sin_Z^θ)_d^d−¹dθ,Zd :=√ π ^Γ(

d 2 ) Γ(d+1 2 )

: ∀v ∈H¹([0, π],dσ) p−2

d Z π

0 |v^′(θ)|²dσ+ Z π

0 |v(θ)|²dσ≥ Z π

0 |v(θ)|^pdσ ²_p

• Change of variablesz = cosθ,v(θ) =f(z) p−2

d Z 1

−1|f^′|²νdνd+ Z 1

−1|f|²dνd ≥ Z 1

−1|f|^pdνd

2 p

where νd(z)dz =dνd(z) :=Z_d⁻¹ν^d²⁻¹dz,ν(z) := 1−z²

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The ultraspherical operator

With dνd=Z_d⁻¹ν^d²⁻¹dz, ν(z) := 1−z², consider the space L²((−1,1),dνd)with scalar product

hf1,f2i= Z 1

−1

f1f2dνd, kfkp= Z 1

−1

f^p dνd

p¹

The self-adjointultraspherical operator is

Lf := (1−z²)f^′′−d z f^′=νf^′′+d 2 ν^′f^′ which satisfieshf1,Lf2i=−R1

−1f₁^′f₂^′ν dνd

Proposition

Let p∈[1,2)∪(2,2^∗], d ≥1

−hf,Lfi= Z 1

−1|f^′|²νdνd≥d kfk²p− kfk²2

p−2 ∀f ∈H¹([−1,1],dνd)

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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

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Heat flow and the Bakry-Emery method

With g =f^p,i.e. f =g^αwithα= 1/p

(Ineq.) −hf,Lfi=−hg^α,Lg^αi=:I[g]≥d kgk²1^α− kg²^αk1

p−2 =:F[g] Heat flow

∂g

∂t =Lg d

dt kgk1= 0, d

dt kg²^αk1=−2 (p−2)hf,Lfi= 2 (p−2) Z 1

−1|f^′|²ν dνd

which finally gives d

dtF[g(t,·)] =− d p−2

d

dtkg²^αk1=−2dI[g(t,·)]

Ineq. ⇐⇒ d

dtF[g(t,·)]≤ −2dF[g(t,·)] ⇐= d

dtI[g(t,·)]≤ −2dI[g(t,·)]

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The equation forg =f^p can be rewritten in terms off as

∂f

∂t =Lf + (p−1)|f^′|² f ν

−1 2

d dt

Z 1

−1|f^′|²ν dνd =1 2

d

dthf,Lfi=hLf,Lfi+ (p−1)h|f^′|² f ν,Lfi d

dtI[g(t,·)] + 2dI[g(t,·)]= d dt

Z 1

−1|f^′|²νdνd+ 2d Z 1

−1|f^′|²νdνd

=−2 Z 1

−1

|f^′′|²+ (p−1) d d+ 2

|f^′|⁴

f² −2 (p−1)d−1 d+ 2

|f^′|²f^′′

f

ν²dνd

is nonpositive if

|f^′′|²+ (p−1) d d+ 2

|f^′|⁴

f² − 2 (p−1)d−1 d+ 2

|f^′|²f^′′

f is pointwise nonnegative, which is granted if

(p−1)d−1 d+ 2

2

≤(p−1) d

d+ 2 ⇐⇒ p≤ 2d²+ 1

(d−1)² = 2^#< 2d d−2 = 2^∗

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

d dz,L

u= (Lu)^′− Lu^′ =−2z u^′′−d u^′ Z 1

−1

(Lu)²dνd = Z 1

−1|u^′′|²ν²dνd+d Z 1

−1|u^′|²ν dνd

Z 1

−1

(Lu)|u^′|²

u νdνd = d d+ 2

Z 1

−1

|u^′|⁴

u² ν²dνd−2d−1 d+ 2

Z 1

−1

|u^′|²u^′′

u ν²dνd

On (−1,1), let us consider the porous medium (fast diffusion)flow ut=u²⁻^2β

Lu+κ|u^′|² u ν

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

dt Z 1

−1

u^βpdνd =βp(κ−β(p−2)−1) Z 1

−1

u^β(p⁻²⁾|u^′|²ν dνd= 0

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f =u^β,kf^′k²L²(Sd)+_p₋^d₂

kfk²L²(Sd)− kfk²L^p(Sd)

≥0 ?

A:=

Z 1

−1|u^′′|²ν²dνd−2d−1

d+ 2(κ+β−1) Z 1

−1

u^′′|u^′|² u ν²dνd

+

κ(β−1) + d

d+ 2(κ+β−1) Z 1

−1

|u^′|⁴ u² ν²dνd

Ais nonnegative for someβ if 8d²

(d+ 2)²(p−1) (2^∗−p)≥0

Ais a sum of squares ifp∈(2,2^∗)for an arbitrary choice ofβ in a certain interval (depending on pand

A= Z 1

−1

u^′′−p+ 2 6−p

|u^′|² u

2

ν²dνd≥0 ifp= 2^∗andβ = 4 6−p

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The rigidity point of view

Which computation have we done ? ut=u²⁻^2β

Lu+κ^|u_u^′^|²ν

− Lu−(β−1)|u^′|²

u ν+ λ

p−2u= λ p−2u^κ Multiply by Luand integrate

...

Z 1

−1Lu u^κdνd=−κ Z 1

−1

u^κ|u^′|² u dνd

Multiply by κ^|^u_u^′^|² and integrate ...= +κ

Z 1

−1

u^κ|u^′|² u dνd

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

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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 establishimproved 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

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What does “improvement” mean ?

An improvedinequality is

dΦ (e)≤i ∀u∈H¹(S^d) s.t. kuk²L²(Sd)= 1 for some functionΦsuch thatΦ(0) = 0,Φ^′(0) = 1,Φ^′>0 and Φ(s)>s for anys. WithΨ(s) :=s−Φ⁻¹(s)

i−de≥d(Ψ◦Φ)(e) ∀u∈H¹(S^d) s.t. kuk²L²(Sd)= 1 Lemma (Generalized Csisz´ar-Kullback inequalities)

k∇uk²L²(Sd)− d p−2

hkuk²L^p(Sd)− kuk²L²(Sd)

i

≥dkuk²L²(Sd)(Ψ◦Φ)

C^k^u^k

2 (1−r) Ls(Sd)

kuk²_{L2 (}Sd) ku^r−u¯^rk²L^q(Sd)

∀u∈H¹(S^d) s(p) := max{2,p} andp∈(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

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Linear flow: improved Bakry-Emery method

Cf. [Arnold, JD]

wt =Lw+κ|w^′|² w ν With 2^♯:= _(d−²^d²⁺¹₁₎2

γ1:=

d−1 d+ 2

2

(p−1) (2^#−p) if d>1, γ1:= p−1

3 if d= 1 Ifp∈[1,2)∪(2,2^♯] andw is a solution, then

d

dt (i−de)≤ −γ1

Z 1

−1

|w^′|⁴

w² dνd ≤ −γ1 |e^′|² 1−(p−2) e Recalling that e^′=−i, we get a differential inequality

e^′′+ de^′ ≥γ1 |e^′|² 1− (p−2) e After integration: dΦ(e(0))≤i(0)

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

wt =w²⁻^2β

Lw+κ|w^′|² w

For allp∈[1,2^∗],κ=β(p−2) + 1, _dt^d R1

−1w^βpdνd = 0

−_2β¹² _dt^d R1

−1

|(w^β)^′|²ν+_p₋^d₂ w^2β−w^2β

dνd ≥γR1

−1

|w^′|⁴ w² ν²dνd

Lemma

For all w ∈H¹ (−1,1),dνd

, such thatR1

−1w^βpdνd = 1 Z 1

−1

|w^′|⁴

w² ν²dνd ≥ 1 β²

R1

−1|(w^β)^′|²νdνdR1

−1|w^′|²ν dνd

R1

−1w^2βdνd

δ

.... but there are conditions on β

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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

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The line

A first example of a non-compact manifold

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

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One-dimensional Gagliardo-Nirenberg-Sobolev inequalities

kfkL^p(R)≤CGN(p)kf^′k^θL²(R)kfk¹L⁻²(^θR) if p∈(2,∞) kfkL²(R) ≤CGN(p)kf^′k^ηL²(R)kfk¹L⁻^p(^ηR) if p∈(1,2) withθ= ^p₂⁻_p² andη=²_2+p⁻^p

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

R

Ru²log

u² kuk²_{L2 (}R)

dx≤ ¹2kuk²L²(R) log

2 πe

ku^′k²_{L2 (}R)

kuk²_{L2 (}R)

Ifp>2,u⋆(x) = (coshx)⁻^p−2² solves

−(p−2)²u^′′+ 4u−2p|u|^p−²u= 0 Ifp∈(1,2)consider u_∗(x) = (cosx)^2−p² ,x∈(−π/2, π/2)

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Flow

Let us define on H¹(R)the functional F[v] :=kv^′k²L²(R)+ 4

(p−2)²kvk²L²(R)−Ckvk²L^p(R) s.t.F[u⋆] = 0 With z(x) := tanhx, consider theflow

vt= v¹⁻^p²

√1−z²

v^′′+ 2p

p−2z v^′+ p 2

|v^′|²

v + 2

p−2v

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

d

dtF[v(t)]≤0 and lim

t→∞F[v(t)] = 0

d

dtF[v(t)] = 0 ⇐⇒ v0(x) =u⋆(x−x0) Similar results for p∈(1,2)

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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^′|²dx+ 4 (p−2)²

Z

R

|v|²dx≥C Z

R

|v|^pdx ²_p With

z(x) = tanhx, v⋆= (1−z²)^p−2¹ and v(x) =v⋆(x)f(z(x)) equality is achieved forf = 1and, if we letν(z) := 1−z², then

Z 1

−1|f^′|²νdνd+ 2p (p−2)²

Z 1

−1|f|²dνd ≥ 2p (p−2)²

Z 1

−1|f|^pdνd

2 p

where dνp denotes the probability measuredνp(z) :=_ζ¹

pν^p−2² dz d= _p−²^p₂ ⇐⇒ p=_d−^2d₂

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Change of variables = stereographic projection + Emden-Fowler

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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

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Riemannian manifolds with positive curvature

(M,g)is a smooth closed compact connected Riemannian manifold dimensiond, no boundary,∆g is the Laplace-Beltrami operator vol(M) = 1,R is the Ricci tensor,λ1=λ1(−∆g)

ρ:= inf

M inf

ξ∈Sd−1

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

Assume d ≥2 andρ >0. If λ≤(1−θ)λ1+θ dρ

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

−∆gv+ λ

p−2 v−v^p−¹

= 0 has a unique positive solution v ∈C²(M): v ≡1

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Riemannian manifolds: first improvement

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

0< λ < λ⋆= inf

u∈H²(M)

Z

M

h(1−θ) (∆gu)²+ θd

d−1R(∇u,∇u)i d vg

R

M|∇u|²d vg

there is a unique positive solution in C²(M): u≡1 limp→1+θ(p) = 0 =⇒limp→1+λ⋆(p) =λ1 ifρis bounded λ⋆=λ1=dρ/(d−1) =d ifM=S^d sinceρ=d−1

(1−θ)λ1+θ dρ

d−1 ≤λ⋆≤λ1

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Riemannian manifolds: second improvement

Hgu denotes Hessian ofuandθ= (d−1)²(p−1) d(d+ 2) +p−1 Qgu:=Hgu−g

d ∆gu−(d−1) (p−1) θ(d+ 3−p)

∇u⊗ ∇u

u −g

d

|∇u|² u

Λ⋆:= inf

u∈H²(M)\{0}

(1−θ) Z

M

(∆gu)²d vg+ θd d−1

Z

M

hkQguk²+R(∇u,∇u)i Z

M|∇u|²d vg

Theorem (Dolbeault-Esteban-Loss)

Assume that Λ⋆>0. For any p∈(1,2)∪(2,2^∗), the equation has a unique positive solution in C²(M)ifλ∈(0,Λ⋆): u≡1

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Optimal interpolation inequality

For any p∈(1,2)∪(2,2^∗)orp= 2^∗ ifd≥3 k∇vk²L²(M)≥ λ

p−2

hkvk²L^p(M)− kvk²L²(M)

i ∀v ∈H¹(M) Theorem (Dolbeault-Esteban-Loss)

AssumeΛ⋆>0. The above inequality holds for someλ= Λ∈[Λ⋆, λ1] IfΛ⋆< λ1, then the optimal constantΛ is such that

Λ⋆<Λ≤λ1

If p = 1, thenΛ =λ1

Using u= 1 +ε ϕas a test function whereϕwe getλ≤λ1

A minimum of

v 7→ k∇vk²L²(M)−_p₋^λ₂ h

kvk²L^p(M)− kvk²L²(M)

i under the constraintkvk^L^p(M)= 1is negative ifλ > λ1

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The flow

The key tools the flow ut =u²⁻^2β

∆gu+κ|∇u|² u

, κ= 1 +β(p−2) Ifv =u^β, then _dt^dkvkL^p(M) = 0and the functional

F[u] :=

Z

M|∇(u^β)|²d vg+ λ p−2

"

Z

M

u²^βd vg− Z

M

u^β^pd vg

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

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Elementary observations (1/2)

Letd ≥2,u∈C²(M), and consider the trace free Hessian Lgu:=Hgu−g

d ∆gu Lemma

Z

M

(∆gu)²d vg= d d−1

Z

M

kLguk²d vg+ d d−1

Z

M

R(∇u,∇u)d vg

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

2∆|∇u|²=kHguk²+∇(∆gu)· ∇u+R(∇u,∇u)

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Elementary observations (2/2)

Lemma Z

M

∆gu|∇u|² u d vg

= d

d+ 2 Z

M

|∇u|⁴

u² d vg− 2d d+ 2

Z

M

[Lgu] :

∇u⊗ ∇u u

d vg

Lemma

Z

M

(∆gu)²d vg≥λ1

Z

M

|∇u|²d vg ∀u∈H²(M) andλ1is the optimal constant in the above inequality

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The key estimates

G[u] :=R

M

hθ(∆gu)²+ (κ+β−1) ∆gu^|∇u|_u ² +κ(β−1)^|∇u|_u2⁴

id vg

Lemma 1 2β²

d

dtF[u] =−(1−θ) Z

M

(∆gu)²d vg− G[u] +λ Z

M

|∇u|²d vg

Q^θ_gu:=Lgu−¹_θ ^d_d+2⁻¹(κ+β−1)h

∇u⊗∇u

u −^g_d ^|∇u|_u ²i Lemma

G[u] = θd d−1

Z

MkQ^θ_guk²d vg+ Z

M

R(∇u,∇u)d vg

−µ Z

M

|∇u|⁴ u² d vg

withµ:= 1 θ

d−1 d+ 2

2

(κ+β−1)²−κ(β−1)−(κ+β−1) d d+ 2

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The end of the proof

Assume thatd ≥2. Ifθ= 1, thenµis nonpositive if β₋(p)≤β≤β+(p) ∀p∈(1,2^∗) where β_±:= ^b^±^√_{2 a}^b²⁻^a witha = 2−p+h

(d−1) (p−1) d+2

i2

andb = ^d+3_d+2⁻^p Notice that β₋(p)< β+(p)ifp∈(1,2^∗)andβ₋(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^∗)(p6= 5or d 6= 2) 1

2β² d

dtF[u]≤(λ−Λ⋆) Z

M

|∇u|²d vg

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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...

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We shall also denote byRthe Ricci tensor, by Hgu the Hessian ofu and by

Lgu:=Hgu−g d ∆gu

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

d |∇u|² We define

λ⋆:= inf

u∈H²(M)\{0}

Z

M

h

kLgu−¹2Mguk²+R(∇u,∇u)i

e⁻^u/2d vg

Z

M|∇u|²e⁻^u/2d vg

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Theorem

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

−1

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

1

4k∇uk²L²(M)+λ Z

M

u d vg≥λlog Z

M

e^ud vg

∀u∈H¹(M) for some constantλ >0. Let us denote byλ1the first positive eigenvalue of−∆g

Corollary

If d = 2, then the MTO inequality holds withλ= Λ := min{4π, λ⋆}. Moreover, if Λis strictly smaller thanλ1/2, then the optimal constant in the MTO inequality is strictly larger thanΛ

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The flow

∂f

∂t = ∆g(e^−f^/2)−¹₂|∇f|²e^−f^/2 G^λ[f] :=

Z

MkLgf −¹2Mgf k²e⁻^f^/2d vg+ Z

M

R(∇f,∇f)e⁻^f^/2d vg

−λ Z

M

|∇f|²e⁻^f^/2d v_g Then for any λ≤λ⋆ we have

d

dtF^λ[f(t,·)] = Z

M −¹2∆gf +λ

∆g(e⁻^f^/2)−¹2|∇f|²e⁻^f^/2 d vg

=− G^λ[f(t,·)]

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

Z ∞

0 Gλ[f(t,·)]dt

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

On the Euclidean spaceR², given a general probability measureµ does the inequality

1 16π

Z

R2|∇u|²dx≥λ

log Z

Rd

e^udµ

− Z

Rd

u dµ

hold for someλ >0? Let

Λ⋆:= inf

x∈R2

−∆ logµ 8π µ Theorem

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

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Caffarelli-Kohn-Nirenberg inequalities

Work in progress with M.J. Esteban and M. Loss