Some features of the method

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Jean Dolbeault

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

September 29, 2016

Asymptotic Patterns in Variational Problems:

PDE and Geometric Aspects (16w5065) Oaxaca, September 25 – 30, 2016

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Outline

Caffarelli-Kohn-Nirenberg inequalities Bthe symmetry issue

Bthe result

Bthe critical and the subcritical regime The proof of the symmetry result

BR´enyi entropy powersvs.the Bakry-Emery method BA rigidity result

BThe strategy of the proof

Inequalities and flows on compact manifolds: the sphere BFlows on the sphere, improvements in the subcritical range BCan one prove Sobolev’s inequalities with a heat flow ? BThebifurcationviewpoint; constraints, improved inequalities Fast diffusion equations with weights: large time asymptotics BRelative uniform convergence, asymptotic rates, asymptotic and global estimates

Entropy methods, gradient flows and rates of convergence

J. Dolbeault Symmetry in interpolation inequalities

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Collaborations

Collaboration with...

M.J. Esteban and M. Loss (symmetry, critical case) M.J. Esteban, M. Loss and M. Muratori (symmetry, subcritical case) M. Bonforte, M. Muratori and B. Nazaret (linearization and large time asymptotics for the evolution problem) and also: S. Filippas, A. Tertikas, G. Tarantello, M. Kowalczyk

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Some features of the method

BCharacterize bubble-like functions (in presence of weights) BA new way of testing elliptic equations, similar to Pohozaev’s method ?

BUniqueness (rigidity) and Bakry-Emery methods unified through a non-linear flow interpretation

BThe non-linear parabolic flow interpretation as a tool to extend local (bifurcation) results to global (uniqueness) results

BLinearization: why is the method optimal in case of Caffarelli-Kohn-Nirenberg inequalities ?

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Background references (entropy methods)

Rigidity methods, uniqueness in nonlinear elliptic PDE’s:

[B. Gidas, J. Spruck, 1981],[M.-F. Bidaut-V´eron, L. V´eron, 1991]

Probabilistic methods (Markov processes), semi-group theory and carr´e du champmethods (Γ2theory): [D. Bakry, M. Emery, 1984], [Bakry, Ledoux, 1996], [Demange, 2008],[JD, Esteban, Loss, 2014 & 2015] →D. Bakry, I. Gentil, and M. Ledoux.

Analysis and geometry of Markov diffusion operators (2014) Entropy methods in PDEs

BEntropy-entropy production inequalities: Arnold, Carrillo, Desvillettes, JD, Jüngel, Lederman, Markowich, Toscani, Unterreiter, Villani...,[del Pino, JD, 2001],[Blanchet, Bonforte, JD, Grillo, Vázquez]→A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations (2016)

BMass transportation: [Otto]→C. Villani, Optimal transport.

Old and new (2009)

BR´enyi entropy powers (information theory)[Savar´e, Toscani, 2014], [Dolbeault, Toscani]

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Caffarelli-Kohn-Nirenberg inequalities and symmetry issues

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Critical Caffarelli-Kohn-Nirenberg inequalities

LetD^a,b:=n

v ∈L^p R^d,|x|^−bdx

: |x|^−a|∇v| ∈L² R^d,dxo Z

R^d

|v|^p

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

≤ C_a,b Z

R^d

|∇v|²

|x|²^a dx ∀v∈ D^a,b

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

p= 2d

d−2 + 2 (b−a) (critical case) BAn optimal function among radial functions:

v_?(x) =

1 +|x|^{(p−2) (a}^c^−a)−_p−2²

and C^?_a,b= k |x|^−bv?k²p

k |x|^−a∇v?k²2

Question: C_a,b= C^?_a,b (symmetry) orC_a,b>C^?_a,b (symmetry breaking) ?

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Critical CKN: range of the parameters

Figure: d = 3 Z

R^d

|v|^p

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

≤ C_a,b Z

R^d

|∇v|²

|x|²^a dx

a b

0 1

−1

b=a

b=a+ 1

a=^d−2₂

a≤b≤a+ 1ifd≥3

a<b≤a+ 1ifd= 2,a+ 1/2<b≤a+ 1ifd= 1 anda<ac := (d−2)/2

p= 2d

d−2 + 2 (b−a)

[Glaser, Martin, Grosse, Thirring (1976)]

[F. Catrina, Z.-Q. Wang (2001)]

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Symmetry 1: moving planes and symmetrization

a b

0

[Chou, Chu],[Horiuchi]

[Betta, Brock, Mercaldo, Posteraro]

+ Perturbation results: [CS Lin, ZQ Wang],[Smets, Willem],[JD, Esteban, Tarantello 2007],[JD, Esteban, Loss, Tarantello, 2009]

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The threshold between symmetry and symmetry breaking

[F. Catrina, Z.-Q. Wang]

a b

0

[JD, Esteban, Loss, Tarantello, 2009]There is a curve which separates the symmetry region from the symmetry breaking region, which is parametrized by a functionp7→a+b

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Linear instability of radial minimizers:

the Felli-Schneider curve

a b

0

[Catrina, Wang],[Felli, Schneider]The functional

C^?_a,b Z

R^d

|∇v|²

|x|²^a dx− Z

R^d

|v|^p

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

is linearly instable atv =v_?

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Direct spectral estimates

a b

0

[JD, Esteban, Loss, 2011]: sharp interpolation on the sphere and a Keller-Lieb-Thirring spectral estimate on the line

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The Emden-Fowler transformation and the cylinder

B[F. Catrina, Z.-Q. Wang (2001)]With an Emden-Fowler

transformation, critical the Caffarelli-Kohn-Nirenberg inequality on the Euclidean space are equivalent to Gagliardo-Nirenberg inequalities on a cylinder

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

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

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

Ca,b with a=a_c±√

Λ and b= d p ±√

Λ

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The bifurcation point of view (p or b − a fixed)

Define

Q^Λ[ϕ] := k∂sϕk²L²(C)+k∇^ωϕk²L²(C)+ Λkϕk²L²(C)

kϕk²L^p(C)

and look for

µ(Λ) := inf

ϕ∈H¹(C)Q^Λ[ϕ]

compared to

µ?(Λ) := inf

ϕ∈H¹(R)Q^Λ[ϕ] =K_?Λ^α¹

AsΛ>0increases,symmetry breakingoccurs whenµ(Λ)< µ?(Λ)

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Numerical results

20 40 60 80 100

10 20 30 40 50

Λ(µ) symmetric non-symmetric asymptotic

bifurcation

J^θ(µ) J^θ(1)µ^α

Parametric plot of the branch of optimal functions forp= 2.8, d= 5.

Non-symmetric solutions bifurcate from symmetric ones at a bifurcation point computed by V. Felli and M. Schneider. The branch behaves for large values ofΛas predicted by F. Catrina and Z.-Q. Wang

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Other evidences

Further numerical results [JD, Esteban, 2012](coarse / refined / self-adaptive grids)

20 40 60 80 100 120

10 20 30 40 50 60

Formal commutation of the non-symmetric branch near the bifurcation point[JD, Esteban, 2013]

Asymptotic energy estimates [JD, Esteban, 2013]

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Symmetry versus symmetry breaking:

the sharp result in the critical case

A result based on entropies and nonlinear flows

a b

0

[JD, Esteban, Loss (Inventiones 2016)]

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The symmetry result in the critical case

The Felli & Schneider curve is defined by

b_FS(a) := d(a_c−a) 2p

(ac−a)²+d−1 +a−a_c

Theorem

Let d≥2and p<2^∗. If either a∈[0,a_c)and b>0, or a<0 and b≥b_FS(a), then the optimal functions for the critical

Caffarelli-Kohn-Nirenberg inequalities are radially symmetric

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Interpolation and subcritical Caffarelli-Kohn-Nirenberg

inequalities

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Caffarelli-Kohn-Nirenberg inequalities (with two weights)

Norms: kwk^L^q,γ⁽R^d) := R

R^d|w|^q|x|^−γdx^1/q

,kwk^L^q⁽R^d):=kwk^L^q,0⁽R^d)

(some)Caffarelli-Kohn-Nirenberg interpolation inequalities(1984) kwkL^2p,γ(R^d)≤Cβ,γ,pk∇wk^ϑL^2,β(R^d)kwk^1−ϑ_L^p+1,γ(R^d) (CKN) HereC_β,γ,p denotes the optimal constant, the parameters satisfy

d≥2, γ−2< β < ^d−2_d γ , γ∈(−∞,d), p∈(1,p_?] withp_?:= _d−β−2^d−γ and the exponentϑis determined by the scaling invariance, i.e.,

ϑ= (d−γ) (p−1)

p d+β+2−2γ−p(d−β−2)

[JD, Muratori, Nazaret] β= 0,γ >0small...

General case: is the equality case achieved by the Barenblatt / Aubin-Talenti type function

w_?(x) = 1 +|x|^2+β−γ^−1/(p−1)

∀x ∈R^d ?

Do we know (symmetry) that the equality case is achieved among radial functions?

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Range of the parameters

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With two weights: a symmetry breaking result

Let us define

βFS(γ) :=d−2−p

(d−γ)²−4 (d−1) Theorem

Symmetry breaking holds in (CKN) if

γ <0 and βFS(γ)< β < d−2

d γ

In the rangeβFS(γ)< β <^d−2_d γ,w_?(x) = 1 +|x|^2+β−γ−1/(p−1)

is notoptimal.

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Symmetry and symmetry breaking

[JD, Esteban, Loss, Muratori, 2016]

Let us defineβFS(γ) :=d−2−p

(d−γ)²−4 (d−1) Theorem

Symmetry breaking holds in (CKN) ifand only if

γ <0 and βFS(γ)< β < d−2

d γ

In the rangeβFS(γ)< β <^d−2_d γ,w_?(x) = 1 +|x|^2+β−γ−1/(p−1)

is notoptimal.

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The grey area corresponds to the admissible cone. The light grey area is the region of symmetry, while the dark grey area is the region of symmetry breaking. The threshold is determined by the hyperbola

(d−γ)²−(β−d+ 2)²−4 (d−1) = 0

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A useful change of variables

With

α= 1 + β−γ

2 and n= 2 d−γ β+ 2−γ, (CKN) can be rewritten for a functionv(|x|^α−1x) =w(x)as

kvk^L^2p,d−n⁽R^d)≤Kα,n,pkDαvk^ϑL^2,d−n(R^d)kvk^1−ϑL^p+1,d−n(R^d)

with the notationss=|x|,D_αv = α^∂v_∂s,¹_s ∇^ωv

. Parameters are in the range

d≥2, α >0, n>d and p∈(1,p_?], p_?:= n n−2 By our change of variables,w_? is changed into

v_?(x) := 1 +|x|²−1/(p−1)

∀x∈R^d

The symmetry breaking condition (Felli-Schneider) now reads

α < αFS with αFS:=

rd−1 n−1

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The second variation

J[v] :=ϑlog kD_αvk^L^2,d−n⁽R^d)

+ (1−ϑ) log kvk^L^p+1,d−n⁽R^d)

+ log K_α,n,p−log kvkL^2p,d−n(R^d)

Let us definedµδ:=µδ(x)dx, where µδ(x) := (1 +|x|²)^−δ. Sincev_? is a critical point ofJ, a Taylor expansion at orderε²shows that

kDαv?k²L^2,d−n(R^d)J

v?+ε µδ/2f

=¹₂ε²ϑQ[f]+o(ε²) withδ=_p−1²^p and

Q[f] =R

R^d|D_αf|²|x|^n−ddµδ− ^4pp−1^α²

R

R^d|f|²|x|^n−ddµδ+1

We assume thatR

R^df|x|^n−ddµδ+1= 0(mass conservation)

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Symmetry breaking: the proof

Proposition (Hardy-Poincar´e inequality)

Let d≥2,α∈(0,+∞), n>d andδ≥n. If f has0 average, then Z

R^d

|Dαf|²|x|^n−ddµδ ≥Λ Z

R^d

|f|²|x|^n−ddµδ+1

with optimal constantΛ = min{2α²(2δ−n),2α²δ η} whereη is the unique positive solution toη(η+n−2) = (d−1)/α². The corresponding eigenfunction is not radially symmetric ifα²>_n₍₂^(d−1)_{δ−n) (δ−1)}^δ² .

Q ≥0iff ^4p_p−1^α² ≤Λand symmetry breaking occurs in (CKN) if 2α²δ η < 4pα²

p−1 ⇐⇒ η <1

⇐⇒ d−1

α² =η(η+n−2)<n−1 ⇐⇒ α > αFS

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Relative uniform convergence, asymptotic rates, global estimates

Fast diffusion equations with weights: a symmetry result

R´enyi entropy powers The symmetry result The strategy of the proof

Joint work with M.J. Esteban, M. Loss in the critical case β=d−2 + ^γ−d_p Joint work with M.J. Esteban, M. Loss and M. Muratori in the

subcritical case d−2 + ^γ−d_p < β <^d−2_d γ

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R´ enyi entropy powers (no weights)

We consider the flow ^∂u_∂t = ∆u^m and the Gagliardo-Nirenberg inequalities (GN)

kwk^L^2p⁽R^d) ≤ Cp,d^GNk∇wk^θL²(R^d)kwk^1−θL^p+1(R^d)

whereu=w^2p, that is,w =u^m−1/2withp=₂_m−1¹ . Straightforward computations show that (GN) can be brought into the form

Z

R^d

u dx

(σ+1)m−1

≤CI E^σ−1 where σ= 2

d(1−m)−1 whereE:=R

R^du^mdx andI:=R

R^du|∇P|²dx,P = _1−m^m u^m−1is the pressure variable. IfF=E^σ is theR´enyi entropy powerand σ=_d² _1−m¹ −1, thenF⁰⁰ is proportional to

−2 (1−m)D Tr

HessP−¹d∆PId²E

+ (1−m)²(1−σ)D

(∆P− h∆Pi)²E where we have used the notationhAi:=R

R^du^mA dx/R

R^du^mdx

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A rigidity result

We actually prove a rigidity (uniqueness) result Bcritical case: [JD, Esteban, Loss; Inventiones]

Bsubcritical case: [JD, Esteban, Loss, Muratori]

Theorem

Assume thatβ≤βFS(γ). Then all positive solutions inH^p_β,γ(R^d)of

−div |x|^−β∇w

=|x|^−γ w^2p−1−w^p

in R^d\ {0} are radially symmetric and, up to a scaling and a multiplication by a constant, equal to w?(x) = 1 +|x|^2+β−γ−1/(p−1)

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The strategy of the proof (1/3: changing the dimension)

We rephrase our problem in a space of higher,artificial dimension n>d (herenis a dimension at least from the point of view of the scaling properties), or to be precise we consider a weight|x|^n−d which is the same in all norms. With

v(|x|^α−1x) =w(x), α= 1 +β−γ

2 and n= 2 d−γ β+ 2−γ, we claim that Inequality (CKN) can be rewritten for a function v(|x|^α−1x) =w(x)as

kvkL^2p,d−n(R^d) ≤K_α,n,pkD_αvk^ϑL^2,d−n(R^d)kvk^1−ϑL^p+1,d−n(R^d) ∀v ∈H^p_d−n,d−n(R^d) with the notationss=|x|,D_αv = α^∂v_∂s,¹_s ∇^ωv

and d≥2, α >0, n>d and p∈(1,p_?] . By our change of variables,w_? is changed into

v_?(x) := 1 +|x|²−1/(p−1)

∀x∈R^d

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The strategy of the proof (2/3: R´ enyi entropy)

The derivative of the generalizedR´enyi entropy powerfunctional is

G[u] :=

Z

R^d

u^mdµ σ−1Z

R^d

u|D_αP|²dµ whereσ= ²_d _1−m¹ −1. Heredµ=|x|^n−ddx and the pressure is

P := m 1−mu^m−1

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WithL^α=−D^∗_αDα=α² u⁰⁰+ⁿ⁻¹_s u⁰

+_s¹2∆ωu, we consider the fast diffusion equation

∂u

∂t =L^αu^m

in the subcritical range1−1/n<m<1. The key computation is the proof that

−dt^d G[u(t,·)] R

R^du^mdµ1−σ

≥(1−m) (σ−1)R

R^du^m L^αP−

R

Rdu|DαP|²dµ R

Rdu^mdµ

2

dµ + 2R

R^d

α⁴ 1−¹n

P⁰⁰−^Ps⁰ −α²(n−1)^∆^ω^Ps²

2

+^2α_s₂²

∇^ωP⁰−^∇^ωs^P

2 u^mdµ + 2R

R^d

(n−2) α²_FS−α²

|∇^ωP|²+c(n,m,d)^|∇_P^ω2^P|⁴

u^mdµ=:H[u]

for some numerical constantc(n,m,d)>0. Hence ifα≤αFS, the r.h.s.H[u] vanishes if and only ifPis an affine function of|x|², which proves the symmetry result.

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(3/3: elliptic regularity, boundary terms)

This method has a hidden difficulty: integrations by parts ! Hints:

use elliptic regularity: Moser iteration scheme, Sobolev regularity, local H¨older regularity, Harnack inequality, and get global regularity using scalings

use the Emden-Fowler transformation, work on a cylinder, truncate, evaluate boundary terms of high order derivatives using Poincar´e inequalities on the sphere

Summary: ifusolves the Euler-Lagrange equation, we test byL^αu^m 0 =

Z

R^d

dG[u]· L^αu^mdµ≥H[u]≥0

H[u]is the integral of a sum of squares (with nonnegative constants in front of each term)... or test by|x|^γdiv |x|^−β∇w^1+p

the equation

(p−1)²

p(p+1)w^1−3pdiv |x|^−βw^2p∇w^1−p

+|∇w^1−p|²+|x|^−γ c1w^1−p−c2

= 0

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Comments and open questions

BFlows on the sphere, improvements in the subcritical range

BCan one prove Sobolev’s inequalities with a heat flow ? R´enyi entropy powersvs.the Bakry-Emery method

BThebifurcationpoint of view

BSome open problems: constraints, improved inequalities, nodal solutions

BA parabolic theory in the Euclidean case (so far: formal)

[Bakry, Emery, 1984]

[Bidault-V´eron, V´eron, 1991],[Bakry, Ledoux, 1996]

[Demange, 2008],[JD, Esteban, Loss, 2014 & 2015]

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The bifurcation point of view (on the sphere)

µ(λ)is the optimal constant in the functional inequality

k∇uk²L²(S^d)+λkuk²L²(S^d)≥µ(λ)kuk²L^p(S^d) ∀u∈H¹(S^d,dµ)

2 4 6 8 10

2 4 6 8

Here d= 3 andp= 4

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Antipodal symmetry (on the sphere)

With the additional restriction ofantipodal symmetry, that is u(−x) =u(x) ∀x∈S^d

Theorem

If p∈(1,2)∪(2,2^∗), we have Z

S^d

|∇u|²dµ≥ d p−2

1 + (d²−4) (2^∗−p) d(d+ 2) +p−1

kuk²L^p(S^d)− kuk²L²(S^d)

for any u∈H¹(S^d,dµ)with antipodal symmetry. The limit case p= 2 corresponds to the improved logarithmic Sobolev inequality

Z

S^d

|∇u|²dµ≥ d 2

(d+ 3)² (d+ 1)² Z

S^d

|u|²log |u|² kuk²_L²(S^d)

! dµ

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Branches of antipodal solutions

8 9 10 11 12 13 14

Case d= 5,p= 3: values of the shooting parametera as a function ofλ

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Fast diffusion equations with weights: large time asymptotics

Relative uniform convergence Asymptotic rates of convergence From asymptotic to global estimates

Herev solves the Fokker-Planck type equation

∂tv+|x|^γ∇ ·h

|x|^−βv∇ v^m−1− |x|^2+β−γi

= 0 (WFDE-FP)

Joint work with M. Bonforte, M. Muratori and B. Nazaret

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Relative uniform convergence

ζ:= 1− 1−(1−m)^2−mq

1−^2−m1−mθ θ:= (1−m) (2+β−γ)

(1−m) (2+β)+2+β−γ is in the range0< θ < ^1−m_2−m <1 Theorem

For “good” initial data, there exist positive constantsKand t₀such that, for all q∈_2−m

1−m,∞

, the function w=v/Bsatisfies

kw(t)−1kL^q,γ(R^d) ≤ Ke⁻²^(1−m)2^2−m ^Λ^ζ^(t−t⁰⁾ ∀t ≥t0

in the caseγ∈(0,d), and

kw(t)−1kL^q,γ(R^d)≤ Ke⁻²^(1−m)2^2−m ^{Λ (t−t}⁰⁾ ∀t ≥t0

in the caseγ≤0

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0

Λ0,1

Λ1,0

Λess

Essential spectrum

δ δ4

δ1 δ₂ δ₅

Λ0,1

Λ1,0 Λess Essential spectrum

δ4 δ5:=n 2−η

The spectrum ofLas a function ofδ= _1−m¹ , withn= 5. The essential spectrum corresponds to the grey area, and its bottom is determined by the parabolaδ7→Λ_ess(δ). The two eigenvaluesΛ_0,1andΛ_1,0are given by the plain, half-lines, away from the essential spectrum.

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Main steps of the proof:

Existence of weak solutions, L^1,γ contraction, Comparison Principle, conservation of relative mass

Self-similar variables and the Ornstein-Uhlenbeck equation in relative variables: the ratiow(t,x) :=v(t,x)/B(x)solves







|x|^−γ∂tw =−B¹ ∇ ·

|x|^−βBw∇ (w^m−1−1)B^m−1

inR⁺×R^d

w(0,·) =w0:=v0/B inR^d

Regularity, relative uniform convergence (without rates) and asymptotic rates (linearization)

The relative free energy and the relative Fisher information:

linearized free energy and linearized Fisher information A Duhamel formula and a bootstrap

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(1/2) Harnack inequality and H¨ older regularity

We change variables: x 7→ |x|^α−1x and adapt the ideas of F. Chiarenza and R. Serapioni to

∂tu+ D^∗_αh

a (Du+ Bu)i

= 0 in R⁺×R^d

Proposition (A parabolic Harnack inequality)

Let d≥2,α >0and n>d . If u is a bounded positive solution, then for all(t0,x0)∈R⁺×R^d and r >0such that Qr(t0,x0)⊂R⁺×B1, we have

sup

Q_r⁻(t0,x0)

u≤H inf

Q⁺_r(t₀,x₀)

u

The constant H>1depends only on the local bounds on the coefficients a,Band on d ,α, and n

By adapting the classical method`a la De Giorgito our weighted framework: H¨older regularity at the origin

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Regularity (2/2): from local to global estimates

Lemma

If w is a solution of the the Ornstein-Uhlenbeck equation with initial datum w₀bounded from above and from below by a Barenblatt profile (+ relative mass condition) = “good solutions”, then there exist ν∈(0,1)and a positive constant K>0, depending on d , m,β,γ, C , C₁, C₂such that:

k∇v(t)kL^∞(B_2λ\B_λ) ≤ Q₁

λ^2+β−γ^1−m ⁺¹ ∀t≥1, ∀λ >1 sup

t≥1kwk^C^k^((t,t+1)×Bε^c)<∞ ∀k ∈N, ∀ε >0 sup

t≥1kw(t)kC^ν(R^d)<∞ sup

τ≥t|w(τ)−1|C^ν(R^d)≤ Ksup

τ≥tkw(τ)−1kL^∞(R^d) ∀t ≥1

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Asymptotic rates of convergence

Corollary

Assume that m∈(0,1), with m6=m_∗ with m_∗:=. Under the relative mass condition, for any “good solution” v there exists a positive constantCsuch that

F[v(t)]≤ Ce⁻^{2 (1−m) Λ}^t ∀t≥0.

With Csisz´ar-Kullback-Pinsker inequalities, these estimates provide a rate of convergence in L^1,γ(R^d)

Improved estimates cane be obtained using “best matching techniques”

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From asymptotic to global estimates

When symmetry holds (CKN) can be written as anentropy – entropy productioninequality

(2 +β−γ)²F[v]≤ m 1−mI[v] so that

F[v(t)]≤ F[v(0)]e⁻^{2 (1−m) Λ}^?^t ∀t ≥0 with Λ_?:= ^(2+β−γ)_{2 (1−m)}² Let us consider again theentropy – entropy production inequality

K(M)F[v]≤ I[v] ∀v ∈ L^1,γ(R^d) such that kvkL^1,γ(R^d)=M, whereK(M)is the best constant: withΛ(M) := ^m₂(1−m)⁻²K(M)

F[v(t)]≤ F[v(0)]e⁻2 (1−m) Λ(M)t

∀t≥0

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The symmetry breaking region (again)

... consequences for the entropy – entropy production inequalities (a measurement in relative entropy of the global rates of convergence)

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Symmetry breaking and global entropy – entropy production inequalities

Proposition

•In the symmetry breaking range of (CKN), for any M >0, we have 0<K(M)≤m² (1−m)²Λ_0,1

•If symmetry holds in (CKN) andβ 6=βFS(γ),then K(M)> ^1−m_m (2 +β−γ)² Corollary

Assume that m∈[m₁,1)

(i) For any M>0, if Λ(M) = Λ_? thenβ=βFS(γ)

(ii) Ifβ > βFS(γ)thenΛ_0,1<Λ_? andΛ(M)∈(0,Λ_0,1]for any M>0 (iii) For any M>0, if β < βFS(γ)and if symmetry holds in (CKN), then

Λ(M)>Λ_?

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Summary and comments

BLinearization / linear instability determines the symmetry breaking range

BAd hocflows can be build to obtain strict monotonicity results (parabolic point of view) and prove the uniqueness of critical points (elliptic point of view). These flows allow us to extend local results (bifurcations) to global results (uniqueness)

BStudying large time asymptotics amounts to linearize around asymptotic profiles

BLarge time asymptoticsdo notdetermine the global rates as measured byentropy methods

B... but large time asymptoticsdodetermine thesymmetry range: why ?

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Entropy methods, gradient flows and rates of convergence

BThe Bakry-Emery method (Fokker-Planck and Ornstein-Uhlenbeck) BGradient flow interpretation

BLinearization: why is the method optimal ?

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The Fokker-Planck equation

The linear Fokker-Planck (FP) equation in the Euclidean case,in presence of a potential / drift term

∂u

∂t = ∆u+∇ ·(u∇φ)

on a domainΩ⊂R^d, with no-flux boundary conditions (∇u+u∇φ)·ν = 0 on ∂Ω

Unique stationary solution (mass normalized to1):γ= e^−φ R

Ωe^−φdx Withv =u/γ, (FP) is equivalent to the Ornstein-Uhlenbeck (OU) equation

∂v

∂t = ∆v− ∇φ· ∇v =:Lv

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The Bakry-Emery method

Withv such thatR

Ωv dγ= 1,q∈(1,2], theq-entropyis defined by E[v] := 1

q−1 Z

Ω

(v^q−1−q(v−1))dγ Under the action of (OU), withw =v^q/2,I[v] := _q⁴R

Ω|∇w|²dγ, d

dtE[v(t,·)] =− I[v(t,·)] and d

dt (I[v]− 2λE[v])≤0 with λ:= inf

w∈H¹(Ω,dγ)\{0}

R

Ω

2^q−1_q kHesswk²+Hessφ: ∇w⊗ ∇w dγ R

Ω|w|²dγ Proposition

[Bakry, Emery, 1984] [JD, Nazaret, Savar´e, 2008]LetΩbe convex.

Ifλ >0and v is a solution of (OU), thenI[v(t,·)]≤ I[v(0,·)]e⁻²^λ^t andE[v(t,·)]≤ E[v(0,·)]e⁻²^λ^t for any t≥0 and, as a consequence,

I[v]≥ 2λE[v] ∀v∈H¹(Ω,dγ)

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Some remarks

Grisvard’s lemma: by convexity ofΩ, boundary terms have the right sign

On a manifold, the curvature tensor replaces Hessφ What is still to be covered:

- The gradient flow interpretation (Euclidean space, linear flow) - The explanation of why the method for proving symmetry is optimal

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Gradient flow interpretation (Ornstein-Uhlenbeck)

R^d case: a question by F. Poupaud. Letφs.t. Hessφ≥λI,µ:=e^−φL^d

•Entropy : E(ρ) :=R

R^dψ(ρ)dµ

•Action density : φ(ρ,w) := ^|w^|²

h(ρ),1/h=ψ⁰⁰

•Action functional : Φ(ρ,w) :=R

R^dφ(ρ,w)dγ

•Γ(µ0, µ1): (µs,ν_s)_s∈[0,1]is anadmissible pathconnectingµ0toµ1 if there is a solution(µs,ν_s)_s∈[0,1]to thecontinuity equation

∂sµs+∇ ·ν_s = 0, s∈[0,1]

•h-Wasserstein distance W_h²(µ0, µ1) := infnZ 1

0

Φ(µs,νs)ds : (µ,ν)∈Γ(µ0, µ1)o [JD, Nazaret, Savar´e]: (OU) is the gradient flow ofE w.r.t.W_h

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Optimality of the method in GN inequalities: linearization

Without weights, but with a nonlinear flow: let ube a solution to

∂u

∂t +∇ · u(∇u^m−1−2x)

= 0

Gagliardo-Nirenberg inequalities Barenblatt functionsB(x) = (1 +|x|²)^1/(m−1)are stationary solutions Scalar products

hf1,f2i^B = Z

R^d

f1f2B^2−mdx and hhf1,f2ii^B = Z

R^d

∇f1· ∇f2B dx Linearization

u_ε=B

1 +ε_m−1^m B^1−mf Linearized evolution equation

∂f

∂t =L^uf where L^uf :=m u^m−2∇ ·(u∇f)

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Optimality of the method in GN inequalities: spectral gap

∂f

∂t =L^uf where L^uf :=m u^m−2∇ ·(u∇f) At formal level, it is straightforward to check that

1 2

d

dthf,fi^B = Z

R^d

fL^uf B^2−mdx=− Z

R^d

|∇f|²B dx =− hhf,fii^B With the spectral gap

hhf,fii^B ≥Λ_?hf,fi^B

ast→+∞one gets theasymptotic rates of convergence F[u] := 1

m−1 Z

R^d

u^m−B^m−m B^m−1(u−B) dx

∼ ε²

2 hf,fi^B ≤ ε²

2 Λ?hhf,fii^B ∼ 1 2 Λ? I[u]

where I[u] := m² (m−1)²

Z

R^d

u

∇u^m−1− ∇B^m−1

2dx=−d dtF[u]

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Optimality of the method in GN inequalities: global rates

Global rates of convergence: from _dt^dF[u] =− I[u]and theentropy – entropy productioninequality

I[u]≥2 Λ_?F[u]

we get that

F[u(t,·)]≤F[u0]e⁻^{2 Λ}^?^t Optimality in the variational problem

infu

I[u]

F[u] = 1 2 Λ_? is achieved in the limit corresponding touε=B

1 +ε_m−1^m B^1−mf withε→0because the optimality in the variational problem

inf

u

K[u]

I[u] where K[u] :=− d

dtI[u(t,·)]

is also achieved in the limit asε→0

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Optimality of the method in GN inequalities: conclusion

Optimality is achieved in the asymptotic regime 1

2ε² d

dtI[u(t,·)]

∼ 1 2

d

dthhf,fii^B = Z

R^d

∇f · ∇L^uf u dx

=− hhf,L^ufii^B ∼ 1

2ε²K[u(t,·)]

Consider the eigenfunction ofL associated withλ1

− L^uf1=λ1f1

In the asymptotic regime, the optimal contant in the entropy-entropy production inequality has to be given by the same eigenvalue problem

λ1=hf₁,Lf₁i^B

hf1,f1i^B =hhf₁,f₁ii^B hf1,f1i^B ≥inf

u

I[u]

F[u] ≥inf

u

K[u]

I[u] = hhf₁,Lf₁ii^B hhf1,f1ii^B =λ1

... λ1= Λ_? is the best constant

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Optimality of the method in CKN inequalities: weights !

With weights BThe infimuminf

u

K[u]

I[u] is achieved in the asymptotic regime, and given by ₂¹_λ

1, that is, the linearized problem (spectral gap) BThis gives anon-optimal estimate of

infu

I[u]

F[u] =: 1 2 Λ_? ≤ 1

2λ1

but equality holds if and only ifβ =βFS(γ). Otherwise, as far as rates of convergence are concerned, this estimateis notoptimal

BThe inequalityK[u]≥2λ1I[u]proves the monotonicity that provides the symmetry result as long asλ1≥0. The conditionλ1<0 determines the symmetry breaking region. As far as symmetry issues are concerned, this estimateisoptimal

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Computing the second derivative of the entropy is known as the Γ2method, orcarr´e du champmethod, orBakry-Emerymethod, but

the so-called CD(ρ,N)condition encodes integrations by parts which are not easy to justify if the potential has singularities or if the manifold is not compact

Figure: c N. Ghoussoub, dec. 2014

Non-linear flows (as shown by the R´enyi entropy powers) are essential to cover ranges of exponents up to the critical exponent, and have not so much to do with the CD(ρ,N)condition (a priori!)

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These slides can be found at

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

BLectures The papers can be found at

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

BPreprints / papers

For final versions, useDolbeaultas login andJeanas password

Some features of the method

Outline

Collaborations

Some features of the method

Background references (entropy methods)

Caffarelli-Kohn-Nirenberg inequalities and symmetry issues

Critical Caffarelli-Kohn-Nirenberg inequalities

Critical CKN: range of the parameters

Symmetry 1: moving planes and symmetrization

The threshold between symmetry and symmetry breaking

Linear instability of radial minimizers:

the Felli-Schneider curve

Direct spectral estimates

The Emden-Fowler transformation and the cylinder

The bifurcation point of view (p or b − a fixed)

Numerical results

Other evidences

Symmetry versus symmetry breaking:

the sharp result in the critical case

The symmetry result in the critical case

Interpolation and subcritical Caffarelli-Kohn-Nirenberg

inequalities

Caffarelli-Kohn-Nirenberg inequalities (with two weights)

Range of the parameters

With two weights: a symmetry breaking result

Symmetry and symmetry breaking

A useful change of variables

The second variation

Symmetry breaking: the proof

Fast diffusion equations with weights: a symmetry result

R´ enyi entropy powers (no weights)

A rigidity result

The strategy of the proof (1/3: changing the dimension)

The strategy of the proof (2/3: R´ enyi entropy)

(3/3: elliptic regularity, boundary terms)

Comments and open questions

The bifurcation point of view (on the sphere)

Antipodal symmetry (on the sphere)

Branches of antipodal solutions

Fast diffusion equations with weights: large time asymptotics

Relative uniform convergence

(1/2) Harnack inequality and H¨ older regularity

Regularity (2/2): from local to global estimates

Asymptotic rates of convergence

From asymptotic to global estimates

The symmetry breaking region (again)

Symmetry breaking and global entropy – entropy production inequalities

Summary and comments

Entropy methods, gradient flows and rates of convergence

The Fokker-Planck equation

The Bakry-Emery method

Some remarks

Gradient flow interpretation (Ornstein-Uhlenbeck)

Optimality of the method in GN inequalities: linearization

Optimality of the method in GN inequalities: spectral gap

Optimality of the method in GN inequalities: global rates

Optimality of the method in GN inequalities: conclusion

Optimality of the method in CKN inequalities: weights !

Thank you for your attention !