Spectral estimates and entropy methods

(1)

Spectral estimates and entropy methods

Jean Dolbeault

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

April 21, 2015, Besan¸con Spectral theory and kinetic equations

April 20–24, 2015

(2)

Scope

We prove sharp estimates for Schr¨odinger operators using - a duality which reduces the problem to a nonlinear interpolation inequality

- rigidity methods based on a nonlinear flow

- generalized entropies and generalized Fisher informations

We start with compact manifolds for which rigidity statements are easy and extend the method to non-compact settings which are much more difficult

Spectral estimates on the sphere Interpolation inequalities on the sphere

A nonlinear flow and improvements of the inequalities The line

Compact manifolds The cylinder

Symmetry breaking issues in Caffarelli-Kohn-Nirenberg inequalities

J. Dolbeault Spectral estimates and entropy methods

(3)

Spectral estimates on the sphere

The Keller-Lieb-Tirring inequality is equivalent to an interpolation inequality of Gagliardo-Nirenberg-Sobolev type

We measure a quantitative deviation with respect to the semi-classical regime due to finite size effects

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

(4)

An introduction to Lieb-Thirring inequalities

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

Rd

V₊^γ+^d²

[Lieb-Thirring, 1976]

X

k≥1

|λk|^γ≤Lγ,d

Z

Rd

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]

⊲How does one take into account the finite size effects in the case of compact manifolds ?

(5)

A Keller-Lieb-Thirring inequality on the sphere

Letd ≥1,p∈

max{1,d/2},+∞ and µ_∗:= ^d₂(p−1) Theorem (Dolbeault-Esteban-Laptev)

There exists a convex increasing functionαs.t. α(µ) =µifµ∈ 0, µ_∗ andα(µ)> µifµ∈ µ_∗,+∞

and, for any p<d/2,

|λ1(−∆−V)| ≤α kVkL^p(Sd)

∀V ∈L^p(S^d)

This estimate is optimal For large values ofµ, we have

α(µ)^p⁻^d² =L¹_p−d

2,d(κq,dµ)^p(1 +o(1))

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

(6)

A Keller-Lieb-Thirring inequality: second formulation

Letd ≥1,γ=p−d/2

Corollary (Dolbeault-Esteban-Laptev)

|λ1(−∆−V)|^γ .L¹_γ,d Z

Sd

V^γ+^d² as µ=kVk_L^γ+^d₂₍_Sd)→ ∞ if either γ >max{0,1−d/2}orγ= 1/2and d = 1 However, if µ=kVk_L^γ+^d₂₍_Sd) ≤µ_∗, then we have

|λ1(−∆−V)|^γ+^d² ≤ Z

Sd

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

Rd

φ^γ++ ^d² dx

(7)

H¨older duality and link with interpolation inequalities

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

Z

Sd|∇u|²− Z

Sd

V|u|²

≥ Z

Sd|∇u|²−µkuk²L^q(Sd)

≥ −α(µ)kuk²L²(Sd) ifµ=kV+kL^p(Sd)

⊲Is it true that

k∇uk²L²(Sd)+αkuk²L²(Sd)≥µ(α)kuk²L^q(Sd) ? In other words, what are the properties of the minimum of

Qα[u] := k∇uk²L²(Sd)+αkuk²L²(Sd)

kuk²_Lq(Sd)

?

An important convention (for the numerical value of the constants):

we consider the uniform probability measureon the unit sphereS^d

(8)

! " # $ %& %! %"

!

"

#

$

%&

%!

%"

α µ

µ=µ(α)

µ=α µ=µ±(α) µ=µasymp(α)

(9)

µasymp(α) :=^K_κ^q,d

q,d α¹⁻^ϑ,ϑ:=d ^q−₂_q² corresponds to the semi-classical regime andKq,d is the optimal constant in the Euclidean Gagliardo-Nirenberg-Sobolev inequality

Kq,dkvk²L^q(Rd)≤ k∇vk²L²(Rd)+kvk²L²(Rd) ∀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)≤ Q^α[1 +ε ϕ] =α+

d+α(2−q) ε²

Z

Sd|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...

(10)

Another inequality

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

λ1(−∆ +φ)⁻^γ ≤L¹₋_γ,d Z

Rd

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

Sd

W^d²⁻^γ as β=kW⁻¹k⁻¹

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

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

≤ Z

Sd

W^d²⁻^γ and this estimate is optimal

(11)

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

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

K^∗_q,d−γ

withq= 2₂²_γ^γ_−d+2⁻^d ,δ:= _2d−q²^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−q^d ,+∞

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

(12)

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(Sd)= 1

Z

Sd|u|²log|u|²d vg+plog ^ξ(α)_α

≤plog

1 + _α¹k∇uk²L²(Sd)

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

ξ(α) Z

Sd

e⁻^{p W}^/αd vg

1/p

(13)

Interpolation inequalities on the sphere

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

(14)

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...

(15)

Stereographic projection

(16)

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)

(17)

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)

(18)

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)

kuk²L^p(Sd)− kuk²L²(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

(19)

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−1

d dθ,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

²p

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

(20)

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)

(21)

Flows on the sphere

Heat flow and the Bakry-Emery method

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

(22)

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,·)]

(23)

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^∗

(24)

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

d dz,L

u= (Lu)^′− Lu^′ =−2z u^′′−d u^′

Z 1

−1

(Lu)²dν_d = Z 1

−1|u^′′|²ν²dν_d+d Z 1

−1|u^′|²ν dν_d 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

(25)

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

(26)

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

(27)

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

(28)

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

(29)

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)

(30)

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

(31)

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

(32)

The line

A first example of a non-compact manifold

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

(33)

One-dimensional Gagliardo-Nirenberg-Sobolev inequalities

kfk^L^p(R)≤CGN(p)kf^′k^θL²(R)kfk¹L⁻²(^θR) if p∈(2,∞) kfk^L²(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)²⁻²^p,x∈(−π/2, π/2)

(34)

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)

(35)

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−¹² 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= _p²₋^p₂ ⇐⇒ p=_d^2d₋₂

(36)

Change of variables = stereographic projection + Emden-Fowler

(37)

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

(38)

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∈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

(39)

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 lim_p→1+θ(p) = 0 =⇒lim_p→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

(40)

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

(41)

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

(42)

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

(43)

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)

(44)

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

(45)

The key estimates

G[u] :=R

M

hθ(∆gu)²+ (κ+β−1) ∆gu^|∇_u^u^|² +κ(β−1)^|∇_u^u2^|⁴

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+2¹(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

(46)

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

(47)

Spectral consequences

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

The same kind of results as for the sphere. However, estimates are not, in general, sharp.

(48)

Manifolds: the first interpolation inequality

Let us define

κ:=volg(M)¹⁻^2/q Proposition

Assume that q∈(2,2^∗)if d ≥3, or q∈(2,∞)if d= 1or2. There exists a concave increasing functionµ:R⁺→R⁺such thatµ(α) =κ α for anyα≤q^Λ−2,µ(α)< κ αforα > _q₋^Λ₂ and

k∇uk²L²(M)+αkuk²L²(M)≥µ(α)kuk²L^q(M) ∀u∈H¹(M) The asymptotic behaviour ofµis given byµ(α)∼Kq,dα¹⁻^ϑ as α→+∞, withϑ=d^q_2q⁻² andKq,d defined by

Kq,d := inf

v∈H¹(Rd)\{0}

k∇vk²L²(Rd)+kvk²L²(Rd)

kvk²L^q(Rd)

(49)

Manifolds: the first Keller-Lieb-Thirring estimate

We consider kVkL^p(M)=µ7→α(µ) Z

M|∇u|²d vg− Z

M

V|u|²d vg+α(µ) Z

M|u|²d vg

≥ k∇uk²L²(M)−µkuk²L^q(M)+α(µ)kuk²L²(M)

p and ^q₂ are H¨older conjugate exponents Theorem

Let d ≥1, p∈(1,+∞)if d = 1and p∈(^d₂,+∞)if d≥2 and assume thatΛ⋆>0. With the above notations and definitions, for any

nonnegative V ∈L^p(M), we have

|λ1(−∆g−V)| ≤α kVk^L^p(M)

Moreover, we have α(µ)^p⁻^d² =L¹_γ,dµ^p(1 +o(1))asµ→+∞with L¹_γ,d := (Kq,d)^−p,γ=p−^d2

(50)

Manifolds: the second Keller-Lieb-Thirring estimate

Theorem

Let d ≥1, p∈(0,+∞). There exists an increasing concave function ν :R⁺→R⁺, satisfyingν(β) =β/κ, for anyβ∈(0,^p+1₂ κΛ)if p>1, such that for any positive potential W we have

λ1(−∆ +W)≥ν β

with β= R

MW^−pd vg1/p

Moreover, for large values of β, we have ν(β)⁻^(p+^d²⁾=L¹

−(p+^d₂),dβ^−p(1 +o(1))asβ →+∞

(51)

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...

(52)

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

(53)

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 v_g≥λlog Z

M

e^ud v_g

∀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

(54)

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 vg

Then for any λ≤λ⋆ we have d

dtFλ[f(t,·)] = Z

M

−¹₂∆gf +λ

∆g(e^−f^/2)−¹₂|∇f|²e^−f^/2 d vg

=− Gλ[f(t,·)]

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

Z ∞

0 G^λ[f(t,·)]dt

(55)

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

R²

e^udµ

− Z

R²

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

(56)

Spectral estimates on the cylinder

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

(57)

Spectral estimates and the symmetry breaking problem on the cylinder

Let(M,g)be a smooth compact connected Riemannian manifold of dimensiond−1(no boundary) with volg(M) = 1, and let

C:=R×M∋x = (s,z)

be the cylinder. λ^M₁ is the lowest positive eigenvalue of the Laplace-Beltrami operator,κ:= infMinfξ∈Sd−2Ric(ξ , ξ)

⊲Is

Λ(µ) := sup

λ^C₁[V] :V ∈L^q(C), kVkL^q(C)=µ equal to

Λ⋆(µ) := sup

λ^R₁[V] :V ∈L^q(R, kVkL^q(R)=µ ?

−λ^C₁[V] is the lowest eigenvalue of−∂_s²−∆g−V and−∂_s²−V onC

(58)

The Keller-Lieb-Thirring inequality on the line

Assume thatq∈(1,+∞),β =₂^2q_q₋₁,µ1:=q(q−1)√ πΓ(q) Γ(q+1/2)

^1/q . Λ⋆(µ) = (q−1)² µ/µ1β

∀µ >0,

IfV is a nonnegative real valued potential in L^q(R), then we have λ^R₁[V]≤Λ⋆(kVkL^q(R)) where Λ⋆(µ) = (q−1)² µ

µ1

β

∀µ >0 and equality holds if and only if, up to scalings, translations and multiplications by a positive constant,

V(s) = q(q−1)

(coshs)² =:V1(s) ∀s∈R

where kV1k^L^q(R) =µ1,λ^R₁[V1] = (q−1)² andϕ(s) = (coshs)¹⁻^q