Sharp functional inequalities and nonlinear diffusions

Sharp functional inequalities and nonlinear diffusions

Jean Dolbeault

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

April 25, 2015, Shanghai

International Conference on Local and Nonlocal Nonlinear Partial Differential Equations

April 24 to 26, 2015, at NYU Shanghai

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Scope

Prove inequalities with sharp constants on: the sphere, the line, compact manifolds, cylinders

- 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

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 Spectral estimates on the sphere

Spectral consequences on Riemannian manifolds Spectral estimates on the cylinder

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

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...

Stereographic projection

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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^rd−₂²

v(x) = ^r2₂^+1d−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−2d2 dx ^d−2_d

∀v∈ D^1,2(R^d)

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)

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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 /Γ formalism of Bakry-Emery / nonlinear flows

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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)

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^αwithα= 1/p

(Ineq.) −hf,Lfi=−hg^α,Lg^αi=:I[g]≥d kgk²1^α− kg^2αk¹

p−2 =:F[g] Heat flow

∂g

∂t =Lg d

dt kgk¹= 0, d

dt kg^2αk¹=−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

dtkg^2αk1=−2dI[g(t,·)]

Ineq. ⇐⇒ d

dtF[g(t,·)]≤ −2dF[g(t,·)] ⇐= d

dtI[g(t,·)]≤ −2dI[g(t,·)]

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^∗

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

... 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=u^2−2β

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

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The rigidity point of view

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

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

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

Joint work with M.J. Esteban, M. Kowalczyk and M. LossJ. Dolbeault Sharp functional inequalities and nonlinear diffusions

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^kuk

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

Linear flow: improved Bakry-Emery method

Cf. [Arnold, JD]

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

wt =w^2−2β

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

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

One-dimensional Gagliardo-Nirenberg-Sobolev inequalities

kfk^Lp(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)^2−2p,x∈(−π/2, π/2)

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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= v^1−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)

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Change of variables = stereographic projection + Emden-Fowler

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 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

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

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^Lp(M)= 1is negative ifλ > λ1

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The flow

The key tools the flow ut =u^2−2β

∆gu+κ|∇u|² u

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

F[u] :=

Z

M|∇(u^β)|²d vg+ λ p−2

"

Z

M

u^2βd vg− Z

M

u^βpd 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

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 Sharp functional inequalities and nonlinear diffusions

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

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

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 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

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

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

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Caffarelli-Kohn-Nirenberg inequalities

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

Caffarelli-Kohn-Nirenberg inequalities and the symmetry breaking issue

LetDa,b:=n

v ∈L^p R^d,|x|^−bdx

: |x|^−a|∇v| ∈L² R^d,dxo Z

Rd

|v|^p

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

≤ Ca,b

Z

Rd

|∇v|²

|x|^2a dx ∀v∈ Da,b

hold under the conditions thata≤b≤a+ 1 ifd ≥3,a<b≤a+ 1if d = 2,a+ 1/2<b≤a+ 1ifd= 1, anda<ac := (d−2)/2

p= 2d

d−2 + 2 (b−a)

⊲With

v⋆(x) =

1 +|x|^{(p−2) (ac−a)}−p−2²

and C^⋆_a,b= k |x|^−bv⋆k²p

k |x|^−a∇v⋆k²2

do we have Ca,b= C^⋆_a,b (symmetry)

orCa,b>C^⋆_a,b (symmetry breaking) ?

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The Emden-Fowler transformation and the cylinder

v(r, ω) =r^a−acϕ(s, ω) with r =|x|, s=−logr and ω= x r With this transformation, the Caffarelli-Kohn-Nirenberg inequalities can be rewritten as

k∂sϕk²L²(C1)+k∇^ωϕk²L²(C1)+ Λkϕk²L²(C1)≥µ(Λ)kϕk²L^p(C1) ∀ϕ∈H¹(C)

where Λ := (ac−a)²,C=R×S^d−1 and the optimal constantµ(Λ)is µ(Λ) = 1

Ca,b

with a=ac±√

Λ and b= d p ±√

Λ

Numerical results

20 40 60 80 100

10 20 30 40 50

Λ(µ) symmetric non-symmetric asymptotic

bifurcation µ

Parametric plot of the branch of optimal functions for p= 2.8, d= 5, θ= 1. 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 Λas predicted by F. Catrina and Z.-Q. Wang

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

The symmetry result

bFS(a) := d(ac−a) 2p

(ac−a)²+d−1 +a−ac

Theorem

Let d ≥2and p≤4. If either a∈[0,ac)and b>0, or a<0 and b≥bFS(a), then the optimal functions for the Caffarelli-Kohn-Nirenberg inequalities are radially symmetric

a b

0

−1

1

b=a+ 1

b=a b=bFS(a)

b=b_direct(a) Symmetry region

Symmetry breaking region

ac=^d−2₂

The Felli-Schneider region, or symmetry breaking region, appears in dark grey and is defined bya<0,a≤b<bFS(a). We prove that symmetry holds in the light grey region defined by b≥bFS(a)whena<0and for any b∈[a,a+ 1]ifa∈[0,ac)

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Sketch of a proof

A change of variables

With (r =|x|, ω=x/r)∈R⁺×S^d−1, the Caffarelli-Kohn-Nirenberg inequality is

Z _∞

0

Z

Sd−1|v|^pr^{d−b p}dr r dω

²_p

≤Ca,b

Z ∞ 0

Z

Sd−1|∇v|² r^d−2adr r dω Change of variables r7→r^α,v(r, ω) =w(r^α, ω)

α^1−2p Z _∞

0

Z

Sd−1|w|^p r^{d−b pα} dr r dω

²_p

≤Ca,b

Z ∞ 0

Z

Sd−1

α^{2 ∂w}_∂r

2+_r¹2|∇ωw|²

r^{d−2aα−2+2}dr r dω Choice of α

n= d−b p

α = d−2a−2

α + 2

Then p= _n²₋ⁿ₂ is the critical Sobolev exponent associated withn

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

A Sobolev type inequality

The parametersαandnvary in the ranges0< α <∞andd<n<∞ and theFelli-Schneider curvein the(α,n)variables is given by

α=

rd−1

n−1 =:αFS

With

Dw =

α∂w

∂r ,1 r ∇ωw

, dµ:=rⁿ⁻¹dr dω the inequality becomes

α^1−2p Z

Rd|w|^pdµ _p²

≤Ca,b

Z

Rd|Dw|²dµ Proposition

Let d ≥4. Optimality is achieved by radial functions and Ca,b= C^⋆_a,bif α≤αFS

The case of Gagliardo-Nirenberg inequalities on general cylinders is

Notations

When there is no ambiguity, we will omit the indexω and from now on write that ∇=∇^ωdenotes the gradient with respect to the angular variable ω∈S^d−1and that ∆is the Laplace-Beltrami operator onS^d−1. We define the self-adjoint operatorL by

Lw :=−D^∗Dw =α²w^′′+α²n−1

r w^′+∆w r² The fundamental property of L is the fact that

Z

Rd

w1Lw2dµ=− Z

Rd

Dw1·Dw2dµ ∀w1,w2∈ D(R^d)

⊲Heuristics: we look for a monotonicity formula along a well chosen nonlinear flow, based on the analogy with the decay of the Fisher information along the fast diffusion flow in R^d

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Fisher information

Letu¹²⁻¹ⁿ =|w| ⇐⇒ u=|w|^p,p=_n²ⁿ₋₂ I[u] :=

Z

Rd

u|Dp|²dµ , p = m

1−mu^m−1 and m= 1−1 n Here I is theFisher informationand pis thepressure function Proposition

WithΛ = 4α²/(p−2)² and for some explicit numerical constantκ, we have

κ µ(Λ) = inf

I[u] : kuk^L1(Sd,dνn)

The fast diffusion equation

∂u

∂t =Lu^m, m= 1−1 n Barenblatt self-similar solutions

u⋆(t,r, ω) =t⁻ⁿ

c⋆+ r² 2 (n−1)α²t²

−n

Lemma

κ µ⋆(Λ) =I[u⋆(t,·)] ∀t>0

⊲Strategy:

1) prove that _dt^dI[u(t,·)]≤0,

2) prove that _dt^dI[u(t,·)] = 0means thatu=u⋆ up to a time shift

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Decay of the Fisher information along the flow ?

∂p

∂t = 1

npLp− |Dp|² Q[p] := 1

2L |Dp|²−Dp·DLp K[p] :=

Z

Rd

Q[p]−1 n(Lp)²

p¹⁻ⁿdµ

Lemma

d

dtI[u(t,·)] =−2 (n−1)ⁿ⁻¹K[p]

Ifu is a critical point, thenK[p] = 0 Boundary terms ! Regularity !

Proving decay (1/2)

k[p] :=Q(p)−1

n(Lp)²=1

2L |Dp|²−Dp·DLp−1 n(Lp)² kM[p] := 1

2∆|∇p|²− ∇p· ∇∆ p−_n−¹1(∆ p)²−(n−2)α²|∇p|² Lemma

Let n6= 1be any real number, d ∈N, d ≥2, and consider a function p∈C³((0,∞)×M), where(M,g)is a smooth, compact Riemannian manifold. Then we have

k[p] =α⁴

1−1

n p^′′−p^′

r − ∆ p

α²(n−1)r² 2

+ 2α² 1 r²

∇p^′−∇p r

2

+ 1 r⁴kM[p]

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

Proving decay (2/2)

Lemma

Assume that d ≥3, n>d andM=S^d−1. There is a positive constant ζ⋆ such that

Z

Sd−1

kM[p] p¹⁻ⁿdω≥ λ⋆−(n−2)α² Z

Sd−1|∇p|²p¹⁻ⁿdω +ζ⋆(n−d)

Z

Sd−1|∇p|⁴p¹⁻ⁿdω Proof based on the Bochner-Lichnerowicz-Weitzenb¨ock formula Corollary

Let d ≥2and assume thatα≤αFS. Then for any nonnegative function u∈L¹(R^d)withI[u]<+∞andR

Rdu dµ= 1, we have I[u]≥ I⋆

WhenM=S^d−1,λ⋆= (n−2)^d−1

A perturbation argument

Ifuis a critical point ofI under the mass constraintR

Rdu dµ= 1, then

o(ε) =I[u+εLu^m]− I[u] =−2 (n−1)ⁿ⁻¹εK[p] +o(ε) because εLu^m is an admissible perturbation. Indeed, we know that

Z

Rd

(u+εLu^m)dµ= Z

Rd

u dµ= 1

and, as we take the limit asε→0,u+εLu^m makes sense and, in particular, is positive

Ifα≤αFS, then K[p] = 0implies thatu=u⋆

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

A summary

the sphere: the flow tells us what to do, and provides a simple proof (choice of the exponents / of the nonlinearity) once the problem is reduced to the ultraspherical setting + improvements

[not presented here: Keller-Lieb-Thirring estimates] the spectral point of view on the inequality: how to measure the deviation with respect to the semi-classicalestimates, a nice example of bifurcation (andsymmetry breaking)

Riemannian manifolds: no sign is required on the Ricci tensor and an improved integral criterion is established. We extend the theory from pointwise criteria to a non-local Schr¨odinger type estimate (Rayleigh quotient). The method generically shows the

non-optimality of the improved criterion

the flow is a nice way of exploring an energy space: it explain how to produce a good test function at anycritical point. Arigidityresult tells you that a local result is actually global because otherwise the flow would relate (far away) extremal points while keeping the energy minimal

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

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

⊲Preprints(or arxiv, or HAL)

J.D., Maria J. Esteban, Ari Laptev, and Michael Loss. Spectral properties of Schr¨odinger operators on compact manifolds: rigidity, flows, interpolation and spectral estimates, C.R. Math., 351 (11-12): 437–440, 2013.

J.D., Maria J. Esteban, and Michael Loss. Nonlinear flows and rigidity results on compact manifolds. Journal of Functional Analysis, 267 (5):

1338-1363, 2014.

J.D., Maria J. Esteban and Ari Laptev. Spectral estimates on the sphere. Analysis & PDE, 7 (2): 435-460, 2014.

J.D., Maria J. Esteban, Michal Kowalczyk, and Michael Loss. Sharp interpolation inequalities on the sphere: New methods and consequences.

Chinese Annals of Mathematics, Series B, 34 (1): 99-112, 2013.

J.D., Maria J. Esteban, Ari Laptev, and Michael Loss. One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: Remarks on duality and flows.

Journal of the London Mathematical Society, 2014.

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

⊲Preprints(or arxiv, or HAL)

J.D., Maria J. Esteban, Michal Kowalczyk, and Michael Loss. Improved interpolation inequalities on the sphere, Preprint, 2013. Discrete and Continuous Dynamical Systems Series S (DCDS-S), 7 (4): 695724, 2014.

J.D., Maria J. Esteban, Gaspard Jankowiak.

The Moser-Trudinger-Onofri inequality, Preprint, 2014

J.D., Maria J. Esteban, Gaspard Jankowiak. Rigidity results for semilinear elliptic equation with exponential nonlinearities and Moser-Trudinger-Onofri inequalities on two-dimensional Riemannian manifolds, Preprint, 2014

J.D., Michal Kowalczyk. Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates, Preprint, 2014

J.D., and Maria J. Esteban. Branches of non-symmetric critical points and symmetry breaking in nonlinear elliptic partial differential equations.

Nonlinearity, 27 (3): 435, 2014.

J.D., Maria J. Esteban, and Michael Loss. Keller-Lieb-Thirring inequalities for Schr¨odinger operators on cylinders. Preprint, 2015

J. Dolbeault Sharp functional inequalities and nonlinear diffusions

These slides can be found at

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

⊲Lectures

Thank you for your attention !

J. Dolbeault Sharp functional inequalities and nonlinear diffusions