Nonlinear flows, functional inequalities and applications
Jean Dolbeault
http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Universit´e Paris-Dauphine
February 3, 2015 – Nantes
Mathematical Physics conference (February 2-6, 2015 - Thematic semester on PDE and Large Time Asymptotics, LabEx Lebesgue)
GDR Dynqua Annual meeting
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
Scope (1/4): rigidity results
Rigidity resultsfor semilinear elliptic PDEs on manifolds...
Let(M,g)be a smooth compact 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∗=d2d−2 ifd ≥3,2∗=∞ifd= 2 For which values of λ>0 the equation
−∆gv+λv=vp−1
has a unique positive solutionv ∈C2(M): v≡λp−21 ? A typical rigidity resultis: there existsλ0>0 such that v ≡λp−22 ifλ∈(0, λ0]
Assumptions ? Optimal λ0 ?
J. Dolbeault Nonlinear flows, functional inequalities and applications
Scope (2/4): interpolation inequalities
Still on a smooth compact 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
hkvk2Lp(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 h
kvk2Lp(M)− kvk2L2(M)
i under the constraint kvkLp(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)
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
Scope (3/4): 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 =uβ, then dtdkvkLp(M) = 0and the functional F[u] :=
Z
M|∇(uβ)|2d vg+ λ p−2
"
Z
M
u2βd vg− Z
M
uβpd 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 A collaboration mostly with M. Esteban and M. Loss
J. Dolbeault Nonlinear flows, functional inequalities and applications
Scope (4/4): spectral estimates
1 Sharp interpolation inequalities are equivalent, by duality, to sharp estimates on the lowest eigenvalues of Schr¨odinger operators, the so-calledKeller-Lieb-Thirring inequalities
2 These spectral estimate differ from semi-classical inequalities because they take into accountfinite volumeeffects. The semi-classical regime is recovered only in the limit of large potentials
A collaboration mostly with M. Esteban, A. Laptev, and M. Loss
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
Some references (1/2)
Some references (incomplete) andgoals
1 rigidity results and elliptic PDEs: [Gidas-Spruck 1981], [Bidaut-V´eron & V´eron 1991],[Licois & V´eron 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
J. Dolbeault Nonlinear flows, functional inequalities and applications
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... + another flow Joint work with:
M.J. Esteban, G. Jankowiak, M. Kowalczyk, A. Laptev and M. Loss
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
The sphere
The case of the sphere as a simple example
J. Dolbeault Nonlinear flows, functional inequalities and applications
Inequalities on the sphere
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
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(Sd,dvg) for anyp∈(2,2∗]with2∗= d2d−2 ifd ≥3
for anyp∈(2,∞)ifd= 2
Here dvg is the uniform probability measure: vg(Sd) = 1 1 is the optimal constant, equality achieved by constants p= 2∗ corresponds to Sobolev’s inequality...
J. Dolbeault Nonlinear flows, functional inequalities and applications
Stereographic projection
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
Sobolev inequality
The stereographic projection of Sd ⊂Rd×R∋(ρ φ,z)ontoRd: to ρ2+z2= 1,z ∈[−1,1],ρ≥0,φ∈Sd−1 we associatex∈Rd such that r =|x|,φ= |x|x
z = r2−1
r2+ 1 = 1− 2
r2+ 1 , ρ= 2r r2+ 1
and transform any function uonSd into a functionv onRd using u(y) = ρrd−22
v(x) = r22+1d−22
v(x) = (1−z)−d−22v(x) p= 2∗,Sd =14d(d−2)|Sd|2/d: Euclidean Sobolev inequality
Z
Rd|∇v|2dx≥Sd
Z
Rd|v|d−2d2 dx d−2d
∀v∈ D1,2(Rd)
J. Dolbeault Nonlinear flows, functional inequalities and applications
Extended inequality
Z
Sd
|∇u|2d vg ≥ d p−2
"
Z
Sd
|u|pd vg
2/p
− Z
Sd
|u|2d vg
#
∀u∈H1(Sd,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(Sd,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(Sd,dµ)
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
A spectral approach when p ∈ (1 , 2) – 1
ststep
[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∈L2/p(Sd), for somep∈(1,2], and letF(t) :=kf(t,·)kLp(t)(Sd). The key computation goes as follows.
F′ F = p′
p2Fp Z
Sd
v2log
v2 R
Sdv2d vg
d vg+ 4p−1 p′
Z
Sd|∇v|2d vg
with v:=|f|p(t)/2. With 4p−p′1= d2 andt∗>0e such thatp(t∗) = 2, we have
kf(t∗,·)kL2(Sd)≤ kukL2/p(Sd) if 1
p−1 =e2d t∗
J. Dolbeault Nonlinear flows, functional inequalities and applications
A spectral approach when p ∈ (1 , 2) – 2
ndstep
Spectral decomposition. Letu=P
k∈Nuk be a spherical harmonics decomposition,λk =k(d+k−1),ak =kukk2L2(Sd) so that
kuk2L2(Sd)=P
k∈Nak andk∇uk2L2(Sd) =P
k∈Nλkak
kf(t∗,·)k2L2(Sd)=X
k∈N
ake−2λkt∗ kuk2L2(Sd)− kuk2Lp(Sd)
2−p ≤ kuk2L2(Sd)− kf(t∗,·)k2L2(Sd)
2−p
= 1
2−p X
k∈N∗
λkak
1−e−2λkt∗ λk
≤ 1−e−2λ1t∗ (2−p)λ1
X
k∈N∗
λkak = 1−e−2λ1t∗
(2−p)λ1 k∇uk2L2(Sd)
The conclusion easily follows if we notice that λ1=d, and e−2λ1t∗ =p−1 so that 1−e(2−p)−2λλ1t∗ = 1
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
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(Sd)
kuk2Lp(Sd)− kuk2L2(Sd)
≥ inf
u∈H1(Sd,dµ)Q[u] = 1 d
is achieved by Q[1 +εv]asε→0andv is an eigenfunction associated with the first nonzero eigenvalue of∆g
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 / flow... they are the same (main contribution) and can be simplified (!) As a side result, you can go beyond these approaches and discuss optimality
J. Dolbeault Nonlinear flows, functional inequalities and applications
Schwarz symmetry and the ultraspherical setting
(ξ0, ξ1, . . . ξd)∈Sd,ξd =z, Pd
i=0|ξi|2= 1[Smets-Willem]
Lemma
Up to a rotation, any minimizer ofQdepends only onξd =z
• Letdσ(θ) := (sinZθ)dd−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 2p
• 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) :=Zd−1νd2−1dz,ν(z) := 1−z2
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
The ultraspherical operator
With dνd=Zd−1νd2−1dz, ν(z) := 1−z2, consider the space L2((−1,1),dνd)with scalar product
hf1,f2i= Z 1
−1
f1f2dνd, kfkp= Z 1
−1
fp 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)
J. Dolbeault Nonlinear flows, functional inequalities and applications
Flows on the sphere
Heat flow and the Bakry-Emery method
Fast diffusion (porous media) flow and the choice of the exponents
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
Heat flow and the Bakry-Emery method
With g =fp,i.e. f =gαwithα= 1/p
(Ineq.) −hf,Lfi=−hgα,Lgαi=:I[g]≥d kgk21α− kg2αk1
p−2 =:F[g] Heat flow
∂g
∂t =Lg d
dt kgk1= 0, d
dt kg2αk1=−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
dtkg2αk1=−2dI[g(t,·)]
Ineq. ⇐⇒ d
dtF[g(t,·)]≤ −2dF[g(t,·)] ⇐= d
dtI[g(t,·)]≤ −2dI[g(t,·)]
J. Dolbeault Nonlinear flows, functional inequalities and applications
The equation forg =fp 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∗
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
... 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 the porous medium (fast diffusion)flow ut=u2−2β
Lu+κ|u′|2 u ν
Ifκ=β(p−2) + 1, the Lp norm is conserved d
dt Z 1
−1
uβpdνd =βp(κ−β(p−2)−1) Z 1
−1
uβ(p−2)|u′|2ν dνd= 0
J. Dolbeault Nonlinear flows, functional inequalities and applications
f =uβ,kf′k2L2(Sd)+p−d2
kfk2L2(Sd)− kfk2Lp(Sd)
≥0 ?
A:=− 1 2β2
d dt
Z 1
−1
|(uβ)′|2ν+ d
p−2 u2β−u2β
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
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
the rigidity point of view
Which computation have we done ? ut=u2−2β
Lu+κ|uu′|2ν
− Lu−(β−1)|u′|2
u ν+ λ
p−2u= λ p−2uκ Multiply by Luand integrate
...
Z 1
−1Lu uκdνd=−κ Z 1
−1
uκ|u′|2 u dνd
Multiply by κ|uu′|2 and integrate
...= +κ Z 1
−1
uκ|u′|2 u dνd
The two terms cancel and we are left only with the two-homogenous terms
J. Dolbeault Nonlinear flows, functional inequalities and applications
Spectral consequences
A quantitative deviation with respect to the semi-classical regime
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
Some references (2/2)
Consider the Schr¨odinger operatorH=−∆−V onRd and denote by (λk)k≥1its eigenvalues
Euclidean case[Keller, 1961]
|λ1|γ ≤L1γ,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]
J. Dolbeault Nonlinear flows, functional inequalities and applications
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(Sd)+αkuk2L2(Sd) ≥µ(α)kuk2Lq(Sd) ∀u∈H1(Sd) If d ≥3 and q= 2∗, the inequality holds withµ(α) = min{α, α∗}, α∗:= 14d(d−2)
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
µasymp(α) :=Kκq,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,dkvk2Lq(Rd)≤ k∇vk2L2(Rd)+kvk2L2(Rd) ∀v ∈H1(Rd) Letϕbe a non-trivial eigenfunction of the Laplace-Beltrami operator corresponding the first nonzero eigenvalue
−∆ϕ=dϕ
Consideru= 1 +ε ϕasε→0Taylor expandQα aroundu= 1 µ(α)≤ Qα[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...
J. Dolbeault Nonlinear flows, functional inequalities and applications
! " # $ %& %! %"
!
"
#
$
%&
%!
%"
α µ
µ=µ(α)
µ=α µ=µ±(α) µ=µasymp(α)
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
Consider the Schr¨odinger operator−∆−V and the energy E[u] :=
Z
Sd
|∇u|2− Z
Sd
V|u|2
≥ Z
Sd
|∇u|2−µkuk2Lq(Sd)≥ −α(µ)kuk2L2(Sd) ifµ=kV+kLp(Sd)
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 kVkLp(Sd)
∀V ∈Lp(Sd) For large values ofµ, we haveα(µ)p−d2 =L1p−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∗]
J. Dolbeault Nonlinear flows, functional inequalities and applications
A Keller-Lieb-Thirring inequality
Corollary (Dolbeault-Esteban-Laptev) Let d ≥1,γ=p−d/2
|λ1(−∆−V)|γ .L1γ,d Z
Sd
Vγ+d2 as µ=kVkLγ+d
2(Sd)→ ∞ if either γ >max{0,1−d/2}orγ= 1/2and d = 1 However, if µ=kVkLγ+d
2(Sd) ≤14d(2γ+d−2), then we have
|λ1(−∆−V)|γ+d2 ≤ Z
Sd
Vγ+d2 for anyγ≥max{0,1−d/2}and this estimate is optimal
L1γ,d is the optimal constant in the Euclidean one bound state ineq.
|λ1(−∆−φ)|γ ≤L1γ,d Z
Rd
φγ++ d2 dx
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
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
Rd
φd2−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
Wd2−γ as β=kW−1k−1
Lγ−d2(Sd) → ∞ However, if γ≥ d2 + 1andβ=kW−1k−1
Lγ−d2(Sd)≤ 14d(2γ−d+ 2) λ1(−∆ +W)d2−γ
≤ Z
Sd
Wd2−γ and this estimate is optimal
J. Dolbeault Nonlinear flows, functional inequalities and applications
K∗q,d is the optimal constant in the Gagliardo-Nirenberg-Sobolev inequality
K∗q,dkvk2L2(Rd) ≤ k∇vk2L2(Rd)+kvk2Lq(Rd) ∀v ∈H1(Rd) and L1−γ,d :=
K∗q,d−γ
withq= 222γγ−−dd+2,δ:= 2d−2qq(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−qd ,+∞
ν(β) =β ∀β∈ 0,2−qd
if q∈[1,2) ν(β) = K∗q,d (κq,dβ)δ (1 +o(1)) as β→+∞ such that
k∇uk2L2(Sd)+βkuk2Lq(Sd)≥ν(β)kuk2L2(Sd) ∀u∈H1(Sd)
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
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)∼α1−2dp as α→+∞ such that, for any u∈H1(Sd)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
J. Dolbeault Nonlinear flows, functional inequalities and applications
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]
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
What does “improvement” mean ?
An improvedinequality is dkuk2L2(Sd)Φ
e kuk2L2(Sd)
≤i ∀u∈H1(Sd) 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(Sd)(Ψ◦Φ) e kuk2L2(Sd)
!
∀u∈H1(Sd) Lemma (Generalized Csisz´ar-Kullback inequalities)
k∇uk2L2(Sd)− d p−2
hkuk2Lp(Sd)− kuk2L2(Sd)
i
≥dkuk2L2(Sd)(Ψ◦Φ)
Ckuk
2 (1−r) Ls(Sd)
kuk2L2 (Sd) kur−u¯rk2Lq(Sd)
∀u∈H1(Sd) 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
J. Dolbeault Nonlinear flows, functional inequalities and applications
Linear flow: improved Bakry-Emery method
Cf. [Arnold, JD]
wt =Lw+κ|w′|2 w ν With 2♯:= (d2d−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 After integration: dΦ(e(0))≤i(0)
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
Nonlinear flow: the H¨older estimate
wt =w2−2β
Lw+κ|w′|2 w
For allp∈[1,2∗],κ=β(p−2) + 1, dtd R1
−1wβpdνd = 0
−2β12 dtd R1
−1
|(wβ)′|2ν+p−d2 w2β−w2β
dνd ≥γR1
−1
|w′|4 w2 ν2dνd
Lemma
For all w ∈H1 (−1,1),dνd
, such thatR1
−1wβpdνd = 1 Z 1
−1
|w′|4
w2 ν2dνd ≥ 1 β2
R1
−1|(wβ)′|2νdνdR1
−1|w′|2ν dνd
R1
−1w2βdνd
δ
.... but there are conditions on β
J. Dolbeault Nonlinear flows, functional inequalities and applications
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
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
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
J. Dolbeault Nonlinear flows, functional inequalities and applications
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
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
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 Nonlinear flows, functional inequalities and applications
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−vp−1
= 0 has a unique positive solution v ∈C2( ): v ≡1
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
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=Sd sinceρ=d−1
(1−θ)λ1+θ dρ
d−1 ≤λ⋆≤λ1
J. Dolbeault Nonlinear flows, functional inequalities and applications
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
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
Optimal interpolation inequality
For any p∈(1,2)∪(2,2∗)orp= 2∗ ifd≥3 k∇vk2L2(M)≥ λ
p−2
hkvk2Lp(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
hkvk2Lp(M)− kvk2L2(M)
i under the constraintkvkLp(M)= 1is negative ifλ > λ1
J. Dolbeault Nonlinear flows, functional inequalities and applications
The flow
The key tools the flow ut =u2−2β
∆gu+κ|∇u|2 u
, κ= 1 +β(p−2) Ifv =uβ, then dtdkvkLp(M) = 0and the functional
F[u] :=
Z
M|∇(uβ)|2d vg+ λ p−2
"
Z
M
u2βd vg− Z
M
uβpd 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
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
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)
J. Dolbeault Nonlinear flows, functional inequalities and applications
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
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
The key estimates
G[u] :=R
M
hθ(∆gu)2+ (κ+β−1) ∆gu|∇uu|2 +κ(β−1)|∇uu2|4
id vg
Lemma 1 2β2
d
dtF[u] =−(1−θ) Z
M
(∆gu)2d vg− G[u] +λ Z
M
|∇u|2d vg
Qθgu:=Lgu−1θ dd+2−1(κ+β−1)h
∇u⊗∇u
u −gd |∇u|u 2i Lemma
G[u] = θd d−1
Z
M
kQθguk2d 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
J. Dolbeault Nonlinear flows, functional inequalities and applications
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 ab2−a witha = 2−p+h(d
−1) (p−1) d+2
i2
andb = d+3d+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
Sobolev and Hardy-Littlewood-Sobolev inequalities: duality, flows
The line
J. Dolbeault Nonlinear flows, functional inequalities and applications
One-dimensional Gagliardo-Nirenberg-Sobolev inequalities
kfkLp(R)≤CGN(p)kf′kθL2(R)kfk1L−2(θR) if p∈(2,∞) kfkL2(R) ≤CGN(p)kf′kηL2(R)kfk1L−p(R)η if p∈(1,2) withθ= p−2p2 andη=22+p−p
The threshold case corresponding to the limit as p→2 is the logarithmic Sobolev inequality
R
Ru2log
u2 kuk2L2 (R)
dx≤ 12kuk2L2(R) log
2 πe
ku′k2L2 (R)
kuk2L2 (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)