Interpolation inequalities (Gagliardo-Nirenberg, Sobolev and Onofri inequalities): rigidity results,

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The line The Moser-Trudinger-Onofri inequality

Interpolation inequalities (Gagliardo-Nirenberg, Sobolev and Onofri inequalities): rigidity results,

nonlinear flows and applications

Jean Dolbeault

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

March 19, 2014

J. Dolbeault Interpolation inequalities: rigidity results, nonlinear flows and applications.

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Scope (1/3): rigidity results

Rigidity resultsfor semilinear elliptic PDEs on manifolds...

Let(M,g)be a smooth compact connected Riemannian manifold of dimensiond ≥2, no boundary,∆g is the Laplace-Beltrami operator the Ricci tensor Rhas good properties (which ones ?)

Letp∈(2,2^∗), with2^∗=_d^2d₋₂ ifd ≥3,2^∗=∞ifd= 2 For which values of λ>0 the equation

−∆gv+λv=v^p−¹

has a unique positive solutionv ∈C²(M): v≡λ^p−2¹ ? A typical rigidity resultis: there existsλ0>0 such that v ≡λ^p−2² ifλ∈(0, λ0]

Assumptions ? Optimal λ0 ?

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Scope (2/3): interpolation inequalities

Still on a smooth compact connected Riemannian manifold (M,g) we assume that volg(M) = 1

For any p∈(1,2)∪(2,2^∗)orp= 2^∗ ifd≥3, consider the interpolation inequality

k∇vk²L²(M)≥ λ p−2

hkvk²L^p(M)− kvk²L²(M)

i ∀v ∈H¹(M)

What is the largest possible value of λ? usingu= 1 +ε ϕ as a test function proves thatλ≤λ1

the minimum ofv 7→ k∇vk²L²(M)−_p−^λ2

i under the constraint kvkL^p(M) = 1is negative ifλis above the rigidity threshold

the threshold casep= 2is thelogarithmic Sobolev inequality k∇uk²L²(M) ≥λ

Z

M

u² log u² kuk²L²(M)

!

dvg ∀u∈H¹(M)

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Scope (3/3): flows

We shall consider a flow of porous media / fast diffusion type ut =u²⁻^2β

∆gu+κ|∇u|² u

, κ= 1 +β(p−2)

Ifv =u^β, then _dt^dkvk^L^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 as long asλis not too big. Hence, if the limit as t→ ∞ is0 (convergence to the constants), we know thatF[u]≥0

Structure ? Link with computations in the rigidity approach

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Some references (1/2)

Some references (incomplete) andgoals

1 rigidity results and elliptic PDEs: [Gidas-Spruck 1981], [Bidaut-Véron & Véron 1991],[Licois & Véron 1995]

−→ systematize and clarify the strategy

2 semi-group approach andΓ2 orcarr´e du champ method:

[Bakry-Emery 1985],[Bakry & Ledoux 1996],[Bentaleb et al., 1993-2010],[Fontenas 1997],[Brouttelande 2003],[Demange, 2005

& 2008]

−→ emphasize the role of the flow, get various improvements

−→ get rid of pointwise constraints on the curvature, discuss optimality

3 harmonic analysis, duality and spectral theory: [Lieb 1983], [Beckner 1993]

−→ apply results to get new spectral estimates

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Outline

1 The case of the sphere

Inequalities on the sphere Flows on the sphere Spectral consequences Improved inequalities

2 The case of Riemannian manifolds Flows

Spectral consequences

3 Inequalities on the line

Variational approaches Mass transportation Flows

4 The Moser-Trudinger-Onofri inequality Joint work with:

M.J. Esteban, G. Jankowiak, M. Kowalczyk, A. Laptev and M. Loss

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These slides can be found at

http://www.ceremade.dauphine.fr/∼dolbeaul/Conferences/

⊲Lectures

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The sphere

The case of the sphere as a simple example

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Inequalities on the sphere

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A family of interpolation inequalities on the sphere

The following interpolation inequality holds on the sphere:

p−2 d

Z

S^d

|∇u|²d vg+ Z

S^d

|u|²d vg ≥ Z

S^d

|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 functionuonS^d into a functionv onR^d using u(y) = _ρ^r^d⁻2²v(x) = ^r²₂⁺¹^d⁻2²v(x) = (1−z)⁻^d−2² v(x) p= 2^∗,Sd =¹₄d(d−2)|S^d|^2/d: Euclidean Sobolev inequality

Z

R^d

|∇v|²dx≥Sd

Z

R^d

|v|^d²⁻^d² dx ^d−_d²

∀v∈ D^1,2(R^d)

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Extended inequality

Z

S^d|∇u|²d vg ≥ d p−2

"Z

S^d|u|^pd vg

2/p

− Z

S^d|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 Logarithmic Sobolev inequality

Z

S^d|∇u|²d vg≥ d 2

Z

S^d|u|²log

|u|² R

S^d|u|²d vg

d vg ∀u∈H¹(S^d,dµ) casep= 2

Poincar´e inequality Z

S^d|∇u|²d vg ≥d Z

S^d|u−u¯|²d vg with ¯u:=

Z

S^d

u d vg ∀u∈H¹(S^d,dµ) casep= 1

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A spectral approach when p ∈ (1 , 2) – 1

^st

step

[Dolbeault-Esteban-Kowalczyk-Loss]adapted from[Beckner](case of Gaussian measures).

Nelson’s hypercontractivity result. Consider the heat equation

∂f

∂t = ∆gf

with initial datumf(t= 0,·) =u∈L^2/p(S^d), for somep∈(1,2], and letF(t) :=kf(t,·)kL^p(t)(S^d). The key computation goes as follows.

F^′ F = p^′

p²F^p Z

S^d

v²log

v² R

S^dv²d vg

d vg+ 4p−1 p^′

Z

S^d|∇v|²d vg

with v:=|f|^p(t)/2. With 4^p−_p′¹= _d² andt_∗>0e such thatp(t_∗) = 2, we have

kf(t_∗,·)k^L²(S^d)≤ kukL^2/p(S^d) if 1

p−1 =e^{2d t}^∗

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A spectral approach when p ∈ (1 , 2) – 2

^nd

step

Spectral decomposition. Letu=P

k∈Nuk be a spherical harmonics decomposition,λk =k(d+k−1),ak =kukk²L²(S^d) so that

kuk²L²(S^d)=P

k∈Nak andk∇uk²L²(S^d) =P

k∈Nλkak

kf(t_∗,·)k²L²(S^d)=X

k∈N

ake⁻^2λ^k^t^∗

kuk²L²(S^d)− kuk²L^p(S^d)

2−p ≤ kuk²L²(S^d)− kf(t_∗,·)k²L²(S^d)

2−p

= 1

2−p X

k∈N∗

λkak

1−e⁻^2λ^k^t^∗ λk

≤ 1−e⁻²^λ¹^t^∗ (2−p)λ1

X

k∈N∗

λkak = 1−e⁻²^λ¹^t^∗

(2−p)λ1 k∇uk²L²(S^d)

The conclusion easily follows if we notice that λ1=d, and e⁻^2λ¹^t^∗ =p−1 so that ¹⁻₍₂^e₋⁻²_p)^λ_λ¹₁^t^∗ =_d¹

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Optimality: a perturbation argument

The optimality of the constant can be checked by a Taylor expansion ofu= 1 +εv at order two in terms ofε >0, small

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

Q[u] := (p−2)k∇uk²L²(S^d)

kuk²L^p(S^d)− kuk²L²(S^d)

≥ inf

u∈H¹(S^d,dµ)Q[u] = 1 d

is achieved byQ[1 +εv]asε→0andv is an eigenfunction associated with the first nonzero eigenvalue of∆g

p>2no simple proof based on spectral analysis: [Beckner], an approach based on Lieb’s duality, the Funk-Hecke formula and some (non-trivial) computations

elliptic methods /Γ2formalism of Bakry-Emery / flow... they are the same (main contribution) and can be simplified (!) As a side result, you can go beyond these approaches and discuss optimality

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Schwarz symmetry 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

• 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

Withdνd=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

1 p

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

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Heat flow and the Bakry-Emery method

Withg =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 < 2d

d−2 = 2^∗

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... up to the critical exponent: a proof on 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 theporous 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(S^d)+_p₋^d₂

kfk²L²(S^d)− kfk²L^p(S^d)

≥0 ?

A:=− 1 2β²

d dt

Z 1

−1

|(u^β)^′|²ν+ d

p−2 u^2β−u^2β

dνd

= Z 1

−1

Lu+ (β−1)|u^′|²

u ν Lu+κ|u^′|² u ν

dνd

+ d

p−2 κ−1

β Z 1

−1|u^′|²ν dνd

= 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

= Z 1

−1

u^′′−p+ 2 6−p

|u^′|² u

2

ν²dνd ≥0 ifp= 2^∗andβ= 4 6−p Ais nonnegative for someβ if 8d²

(d+ 2)²(p−1) (2^∗−p)≥0

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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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Spectral consequences

A quantitative deviation with respect to the semi-classical regime

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Some references (2/2)

Consider the Schr¨odinger operatorH=−∆−V onR^d and denote by (λk)_k≥1its eigenvalues

Euclidean case[Keller, 1961]

|λ1|^γ ≤L¹_γ,d Z

R^d

V₊^γ+^d²

[Lieb-Thirring, 1976]

X

k≥1

|λk|^γ≤Lγ,d

Z

R^d

V₊^γ+^d²

γ≥1/2ifd= 1,γ >0 ifd = 2andγ≥0 ifd≥3[Weidl],[Cwikel], [Rosenbljum], [Aizenman],[Laptev-Weidl], [Helffer],[Robert], [Dolbeault-Felmer-Loss-Paturel]... [Dolbeault-Laptev-Loss 2008]

Compact manifolds: log Sobolev case: [Federbusch],[Rothaus];

caseγ= 0(Rozenbljum-Lieb-Cwikel inequality): [Levin-Solomyak];

[Lieb],[Levin],[Ouabaz-Poupaud]... [Ilyin]

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An interpolation inequality (I)

Lemma (Dolbeault-Esteban-Laptev)

Let q∈(2,2^∗). Then there exists a concave increasing function µ:R⁺→R⁺ with the following properties

µ(α) =α ∀α∈ 0,_q₋^d₂

and µ(α)< α ∀α∈ _q₋^d₂,+∞ µ(α) =µasymp(α) (1 +o(1)) as α→+∞, µasymp(α) :=Kq,d

κq,d

α¹⁻^ϑ such that

k∇uk²L²(S^d)+αkuk²L²(S^d) ≥µ(α)kuk²L^q(S^d) ∀u∈H¹(S^d) If d ≥3 and q= 2^∗, the inequality holds withµ(α) = min{α, α_∗}, α_∗:= ¹₄d(d−2)

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µasymp(α) :=^K_κ^q,d_q,d α¹⁻^ϑ,ϑ:=d ^q₂⁻_q² corresponds to the semi-classical regime andKq,d is the optimal constant in the Euclidean Gagliardo-Nirenberg-Sobolev inequality

Kq,dkvk²L^q(R^d)≤ k∇vk²L²(R^d)+kvk²L²(R^d) ∀v ∈H¹(R^d) Letϕbe a non-trivial eigenfunction of the Laplace-Beltrami operator corresponding the first nonzero eigenvalue

−∆ϕ=dϕ

Consideru= 1 +ε ϕasε→0Taylor expandQ^α aroundu= 1 µ(α)≤ Qα[1 +ε ϕ] =α+

d+α(2−q) ε²

Z

S^d

|ϕ|²d vg+o(ε²) By takingεsmall enough, we getµ(α)< αfor allα >d/(q−2) Optimizing on the value of ε >0(not necessarily small) provides an interesting test function...

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! " # $ %& %! %"

!

"

#

$

%&

%!

%"

α µ

µ=µ(α)

µ=α µ=µ±(α) µ=µasymp(α)

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Consider the Schr¨odinger operator−∆−V and the energy E[u] :=

Z

S^d

|∇u|²− Z

S^d

V|u|²

≥ Z

S^d

|∇u|²−µkuk²L^q(S^d)≥ −α(µ)kuk²L²(S^d) ifµ=kV+kL^p(S^d)

Theorem (Dolbeault-Esteban-Laptev) Let d ≥1, p∈ max{1,d/2},+∞

. Then there exists a convex increasing function αs.t. α(µ) =µif µ∈

0,^d₂(p−1)

andα(µ)> µif µ∈ ^d2(p−1),+∞

|λ1(−∆−V)| ≤α kVkL^p(S^d)

∀V ∈L^p(S^d) For large values ofµ, we haveα(µ)^p⁻^d² =L¹_p−d

2,d(κq,dµ)^p(1 +o(1)) and the above estimate is optimal

If p =d/2 and d≥3, the inequality holds withα(µ) =µiff µ∈[0, α_∗]

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A Keller-Lieb-Thirring inequality

Corollary (Dolbeault-Esteban-Laptev) Let d ≥1,γ=p−d/2

|λ1(−∆−V)|^γ .L¹_γ,d Z

S^d

V^γ+^d² as µ=kVk_Lγ+d

2(S^d)→ ∞ if either γ >max{0,1−d/2}orγ= 1/2and d = 1 However, if µ=kVk_Lγ+d

2(S^d) ≤¹₄d(2γ+d−2), then we have

|λ1(−∆−V)|^γ+^d² ≤ Z

S^d

V^γ+^d² for anyγ≥max{0,1−d/2}and this estimate is optimal

L¹_γ,d is the optimal constant in the Euclidean one bound state ineq.

|λ1(−∆−φ)|^γ ≤L¹γ,d

Z

R^d

φ^γ++ ^d² dx

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Another interpolation inequality (II)

Letd ≥1 andγ >d/2and assume that L¹₋_γ,d is the optimal constant in

λ1(−∆ +φ)⁻^γ ≤L¹₋_γ,d Z

R^d

φ^d²⁻^γ dx q= 2 2γ−d

2γ−d+ 2 and p= q

2−q =γ−d 2 Theorem (Dolbeault-Esteban-Laptev)

λ1(−∆ +W)−γ

.L¹₋_γ,d Z

S^d

W^d²⁻^γ as β=kW⁻¹k⁻¹

L^γ⁻^d2(Sd) → ∞ However, if γ≥ ^d2 + 1andβ=kW⁻¹k⁻¹

L^γ⁻^d2(S^d)≤ ¹4d(2γ−d+ 2) λ1(−∆ +W)^d₂−γ

≤ Z

S^d

W^d²⁻^γ and this estimate is optimal

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K^∗_q,d is the optimal constant in the Gagliardo-Nirenberg-Sobolev inequality

K^∗_q,dkvk²L²(R^d) ≤ k∇vk²L²(R^d)+kvk²L^q(R^d) ∀v ∈H¹(R^d) and L¹₋γ,d :=

K^∗_q,d−γ

withq= 2₂²_γ^γ₋⁻_d+2^d ,δ:= _2d₋²_q^q_(d₋₂₎ Lemma (Dolbeault-Esteban-Laptev)

Let q∈(0,2)and d ≥1. There exists a concave increasing functionν ν(β)≤β ∀β >0 and ν(β)< β ∀β ∈ 2−^dq,+∞

ν(β) =β ∀β∈ 0,₂₋^d_q

if q∈[1,2) ν(β) = K^∗_q,d (κq,dβ)^δ (1 +o(1)) as β→+∞ such that

k∇uk²L²(S^d)+βkuk²L^q(S^d)≥ν(β)kuk²L²(S^d) ∀u∈H¹(S^d)

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The threshold case: q = 2

Lemma (Dolbeault-Esteban-Laptev)

Let p>max{1,d/2}. There exists a concave nondecreasing functionξ ξ(α) =α ∀α∈(0, α0) and ξ(α)< α ∀α > α0

for someα0∈_d

2(p−1),^d₂p

, andξ(α)∼α¹⁻²^d^p as α→+∞ such that, for any u∈H¹(S^d)withkukL²(S^d)= 1

Z

S^d|u|²log|u|²d vg+plog ^ξ(α)_α

≤plog

1 + _α¹k∇uk²L²(S^d)

Corollary (Dolbeault-Esteban-Laptev) e⁻^λ¹⁽⁻^∆⁻^W^)/α≤ α

ξ(α) Z

S^d

e⁻^{p W}^/αd vg

1/p

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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

[Dolbeault-Esteban-Kowalczyk-Loss]

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What does “improvement” mean ?

An improvedinequality is dkuk²L²(S^d)Φ

e kuk²_L2(Sd)

≤i ∀u∈H¹(S^d) 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≥dkuk²L²(S^d)(Ψ◦Φ) e kuk²_L²_(Sd)

!

∀u∈H¹(S^d) Lemma (Generalized Csisz´ar-Kullback inequalities)

k∇uk²L²(S^d)− d p−2

hkuk²L^p(S^d)− kuk²L²(S^d)

i

≥dkuk²L²(S^d)(Ψ◦Φ)

C^k^u^k

2 (1−r) Ls(Sd)

kuk²_{L2 (}Sd) ku^r−u¯^rk²L^q(S^d)

∀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 With2^♯:= _(d²^d₋²⁺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 which after integration implies an inequality of the form

dΦ(e(0))≤i(0)

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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 = 1, 2

0 4 6 8 10

!2

!1 1 2

0 4 6 8 10

!3

!2

!1 1 2 3

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Admissible (p, β ) for d = 3, 4

0 1 3 4 5 6

!6

!4

!2 2 4 6

1 2 3 4

2 4 6 8 10 12 14

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Admissible (p, β ) for d = 5, 10

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1 2 3 4 5 6

1.0 1.5 2.0 2.5

0.5 1.0 1.5 2.0 2.5

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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 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∈S^d−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

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

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^dkvk^L^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

MkLguk²d 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] :=

Z

M

θ(∆gu)²+ (κ+β−1) ∆gu|∇u|²

u +κ(β−1)|∇u|⁴ u²

d 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−1

d+ 2(κ+β−1)

∇u⊗ ∇u

u −g

d

|∇u|² u

Lemma G[u] = θd

d−1 Z

M

kQ^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