Nonlinear ﬂows and entropy - entropy production methods on unbounded domains

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Nonlinear flows and entropy - entropy production methods on unbounded domains

Jean Dolbeault

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

November 2, 2016 Kaust

Conference on Novel Developments in Evolutionary Partial Differential Equations, November 1–4, 2016

Peter Markowich’s 60^th birthday

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Entropy methods: the compact case Entropy methods: the non-compact case Weighted nonlinear flows and CKN inequalities

Outline

BEntropy methods in case of compact domains Interpolation inequalities on the sphere The bifurcation point of view

Neumann boundary conditions

BUnbounded domains: inequalities without weights and fast diffusion equations without weights

R´enyi entropy powers

Self-similar variables and relative entropies Equivalence of the methods ?

BUnbounded domains and weighted nonlinear flows:

Caffarelli-Kohn-Nirenberg inequalities Elliptic and parabolic proofs

Large time asymptotics and spectral gaps Optimality cases

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Interpolation inequalities on the sphere The bifurcation point of view Neumann boundary conditions

Entropy methods in case of compact domains

BInterpolation inequalities on the sphere Carr´e du champ

Can one prove Sobolev’s inequalities with a heat flow ? Some open problems: constraints and improved inequalities [Beckner, 1993],[J.D., Zhang, 2016]

[Bakry, Emery, 1984]

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

[Demange, 2008][J.D., Esteban, Loss, 2014 & 2015]

BNeumann boundary conditions [J.D., Kowalczyk]

J. Dolbeault Nonlinear flows and entropy methods

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Interpolation inequalities on the sphere

On thed-dimensional sphere, let us consider the interpolation inequality

k∇uk²L²(S^d)+ d

p−2kuk²L²(S^d) ≥ d

p−2kuk²L^p(S^d) ∀u∈H¹(S^d,dµ) where the measuredµis the uniform probability measure on S^d⊂R^d+1 corresponding to the measure induced by the Lebesgue measure onR^d+1, and the exposantp≥1, p6= 2, is such that

p≤2^∗:= 2d d−2

ifd ≥3. We adopt the convention that2^∗=∞ifd = 1ord= 2.

The casep= 2 corresponds to the logarithmic Sobolev inequality k∇uk²L²(S^d) ≥d

2 Z

S^d

|u|²log |u|² kuk²L²(S^d)

!

dµ ∀u∈H¹(S^d,dµ)\ {0}

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The Bakry-Emery method

Entropy functional E^p[ρ] := _p−2¹ h

R

S^dρ²^p dµ− R

S^dρdµ²_pi

if p6= 2 E²[ρ] :=R

S^dρlog

ρ kρk_{L1 (S}d)

dµ

Fisher informationfunctional I^p[ρ] :=R

S^d|∇ρ¹^p|²dµ

Bakry-Emery (carr´e du champ) method: use the heat flow

∂ρ

∂t = ∆ρ

and compute _dt^dE^p[ρ] =− I^p[ρ]and _dt^dI^p[ρ]≤ −dI^p[ρ]to get d

dt(I^p[ρ]−dE^p[ρ])≤0 =⇒ I^p[ρ]≥dE^p[ρ]

withρ=|u|^p, ifp≤2^#:= _(d−1)^2d²⁺¹2 J. Dolbeault Nonlinear flows and entropy methods

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The evolution under the fast diffusion flow

To overcome the limitationp≤2^#, one can consider a nonlinear diffusion of fast diffusion / porous medium type

∂ρ

∂t = ∆ρ^m. (1)

[Demange],[J.D., Esteban, Kowalczyk, Loss]: for anyp∈[1,2^∗] K^p[ρ] := d

dt

I^p[ρ]−dE^p[ρ]

≤0

1.0 1.5 2.5 3.0

0.0 0.5 1.5 2.0

(p,m)admissible region,d = 5

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Cylindrical coordinates, Schwarz symmetrization, stereographic projection...

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... and the ultra-spherical operator

Change of variablesz = cosθ,v(θ) =f(z),dνd :=ν^d²⁻¹dz/Zd, ν(z) := 1−z²

The self-adjointultrasphericaloperator is

Lf := (1−z²)f⁰⁰−d z f⁰=νf⁰⁰+d 2 ν⁰f⁰ which satisfieshf₁,Lf₂i=−R1

−1f₁⁰f₂⁰νdνd

Proposition

Let p∈[1,2)∪(2,2^∗], d ≥1. For any f ∈H¹([−1,1],dνd),

− hf,Lfi= Z 1

−1|f⁰|²ν dνd ≥dkfk²L^p(S^d)− kfk²L²(S^d)

p−2

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The heat equation ^∂g_∂t =Lg forg =f^p can be rewritten in terms off

as ∂f

∂t =Lf + (p−1)|f⁰|² f ν

−1 2

d dt

Z 1

−1|f⁰|²ν dνd =1 2

d

dt hf,Lfi=hLf,Lfi+(p−1) |f⁰|²

f ν,Lf

d

dtI[g(t,·)] + 2dI[g(t,·)]= d dt

Z 1

−1|f⁰|²νdνd+ 2d Z 1

−1|f⁰|²νdνd

=−2 Z 1

−1

|f⁰⁰|²+ (p−1) d d+ 2

|f⁰|⁴

f² −2 (p−1)d−1 d+ 2

|f⁰|²f⁰⁰ f

ν²dνd

is nonpositive if

|f⁰⁰|²+ (p−1) d d+ 2

|f⁰|⁴

f² − 2 (p−1)d−1 d+ 2

|f⁰|²f⁰⁰ 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^∗

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Improved functional inequalities

The range 2^#<p≤2^∗ is covered using the adapted fast diffusion eq.

∂ρ

∂t = ∆ρ^m ρ=|u|^βp m= 1 +²_p

1 β −1

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1 2 3 4 5 6

(p, β)representation of the admissible range of parameters whend= 5 [J.D., Esteban, Kowalczyk, Loss]

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The bifurcation point of view

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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A critical point ofu7→Q^λ[u] := k∇uk²L²(S^d)+λkuk²L²(S^d)

kuk²L^p(S^d)

solves

−∆u+λu=|u|^p−2u (EL) up to a multiplication by a constant (and a conformal transformation ifp= 2^∗)

The best constantµ(λ) = inf

u∈H¹(S^d,dµ)\{0}Q^λ[u]is such that µ(λ)< λifλ > _p−2^d , and µ(λ) =λifλ≤ p−2^d so that

d

p−2 = min{λ >0 : µ(λ)< λ}

Rigidity: the unique positive solution of (EL) isu=λ^1/(p−2)if λ≤ p−2^d

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Constraints and improvements

Taylor expansion:

d = inf

u∈H¹(S^d,dµ)\{0}

(p−2)k∇uk²L²(S^d)

kuk²L^p(S^d)− kuk²L²(S^d)

is achieved in the limit asε→0 withu= 1 +ε ϕ1such that

−∆ϕ1=dϕ1

BThis suggest that improved inequalities can be obtained under appropriate orthogonality constraints...

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Integral constraints

With the heat flow...

Proposition

For any p∈(2,2^#), the inequality Z 1

−1|f⁰|²ν dνd+ λ

p−2kfk²2≥ λ p−2kfk²p

∀f ∈H¹((−1,1),dνd)s.t.

Z 1

−1

z|f|^pdνd = 0 holds with

λ≥d+ (d−1)²

d(d+ 2)(2^#−p) (λ^?−d)

... and with a nonlinear diffusion flow ?

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Antipodal symmetry

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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The larger picture: 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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The optimal constant in the antipodal framework

æ

æ æ

1.5 2.0 2.5 3.0

6 7 8 9 10 11 12

Numerical computation of the optimal constant whend= 5 and 1≤p≤10/3≈3.33. The limiting value of the constant is numerically found to be equal to λ?= 2^1−2/pd≈6.59754with d= 5 andp= 10/3

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Neumann boundary conditions: the Lin-Ni problem

Ωis a smooth bounded domain inR^d, |Ω|= 1. With

1<p<2^∗−1 = (d+ 2)/(d−2), let us consider the 3 problems (P1) For which values ofλ >0does the equation

−∆u+λu=u^p in Ω, ∂nu= 0 on ∂Ω has a unique positive solution (rigidity) ?

(P2) For which valuesλ >0 do we haveµ(λ) =λif

µ(λ) := inf

u∈H¹(Ω)\{0}

k∇uk²L²(R^d)+λkuk²L²(R^d)

kuk²L^p+1(R^d)

(P3) Letq= ^p+1_p−1 and denote byλ1(Ω,−φ)the lowest eigenvalue of the Schr¨odinger operator−∆−φ. For which values ofµ >0 do we know thatν(µ) =µif (Keller-Lieb-Thirring)

λ1(Ω,−φ)≥ −ν kφk^L^q⁽R^d)

∀φ∈ L^q⁺(Ω)

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An estimate based on the spectral gap inequality

λ1(Ω,0) = 0, λ2:=λ2(Ω,0)>0 θ?(p,d) = (d−1)²p

d(d+ 2) +p Theorem

Ωis convex and d ≥2. Ifλ < µ1, then

−∆u+λu=u^p in Ω, ∂nu= 0 on ∂Ω has no non-constant positive solution, whereµ1is such that

1−θ?(p,d)

|p−1| λ2≤µ1≤ λ2

|p−1| for any p∈(0,1)∪(1,2^∗−1)

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Unbounded domains: inequalities without weights and fast diffusion

equations

BR´enyi entropy powers

BSelf-similar variables and relative entropies BEquivalence of the methods ?

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R´ enyi entropy powers and fast diffusion

BR´enyi entropy powers, the entropy approach without rescaling:

[Savar´e, Toscani]: scalings, nonlinearity and a concavity property inspired by information theory

Bfaster rates of convergence: [Carrillo, Toscani],[JD, Toscani]

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The fast diffusion equation in original variables

Consider the nonlinear diffusion equation inR^d,d ≥1

∂v

∂t = ∆v^m

with initial datumv(x,t = 0) =v₀(x)≥0such thatR

R^dv₀dx= 1 and R

R^d|x|²v₀dx<+∞. The large time behavior of the solutions is governed by the source-type Barenblatt solutions

U^?(t,x) := 1

κt^1/µ^dB^? x κt^1/µ

where

µ:= 2 +d(m−1), κ:=

2µm m−1

1/µ

andB^? is the Barenblatt profile

B^?(x) :=







C_?− |x|²1/(m−1)

+ ifm>1 C?+|x|²1/(m−1)

ifm<1

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The R´ enyi entropy power F

Theentropy is defined by

E :=

Z

R^d

v^mdx

and theFisher informationby

I :=

Z

R^d

v|∇p|²dx with p = m m−1v^m−1 Ifv solves the fast diffusion equation, then

E⁰ = (1−m) I To computeI⁰, we will use the fact that

∂p

∂t = (m−1) p ∆p +|∇p|² F := E^σ with σ= µ

d(1−m) = 1+ 2 1−m

1

d +m−1

= 2 d

1 1−m−1 has a linear growth asymptotically ast →+∞

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The concavity property

Theorem

[Toscani-Savar´e]Assume that m≥1−¹d if d>1 and m>0 if d= 1.

Then F(t)is increasing,(1−m) F⁰⁰(t)≤0 and

t→+∞lim 1

t F(t) = (1−m)σ lim

t→+∞E^σ−1I = (1−m)σE^σ−1_? I?

[Dolbeault-Toscani]The inequality

E^σ−1I≥E^σ−1_? I_?

is equivalent to the Gagliardo-Nirenberg inequality

k∇wk^θL²(R^d)kwk^1−θL^q+1(R^d)≥C_GNkwk^L^2q⁽R^d)

if1−¹d ≤m<1. Hint: v^m−1/2= _kwk^w

L2q(Rd),q=₂_m−1¹

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The proof

Lemma

If v solves ^∂v_∂t = ∆v^m with ¹_d ≤m<1, then I⁰= d

dt Z

R^d

v|∇p|²dx =−2 Z

R^d

v^m

kD²pk²+ (m−1) (∆p)² dx

Explicit arithmetic geometric inequality

kD²pk²− 1

d (∆p)²=

D²p− 1 d∆pId

2

There are no boundary terms in the integrations by parts

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Remainder terms

F⁰⁰=−σ(1−m) R[v]. Thepressure variableisP = _1−m^m v^m−1

R[v] := (σ−1) (1−m) E^σ−1 Z

R^d

v^m

∆P− R

R^dv|∇P|²dx R

R^dv^mdx

2

dx

+ 2 E^σ−1 Z

R^d

v^m

D²P−d¹∆PId

2dx

Let

G[v] := F[v] σ(1−m) =

Z

R^d

v^mdx σ−1Z

R^d

v|∇P|²dx The Gagliardo-Nirenberg inequality is equivalent toG[v₀]≥G[v_?] Proposition

G[v₀]≥G[v_?] + Z ∞

0

R[v(t,·)]dt

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Self-similar variables and relative entropies

The large time behavior of the solution of ^∂v_∂t = ∆v^m is governed by the source-typeBarenblatt solutions

v_?(t,x) := 1 κ^d(µt)^d/µB^?

x κ(µt)^1/µ

where µ:= 2 +d(m−1) whereB^?is the Barenblatt profile (with appropriate mass)

B^?(x) := 1 +|x|²^1/(m−1) A time-dependent rescaling: self-similar variables

v(t,x) = 1 κ^dR^d u

τ, x κR

where dR

dt =R^1−µ, τ(t) := ¹₂ log R(t)

R0

Then the functionusolvesa Fokker-Planck type equation

∂u

∂τ +∇ ·h

u ∇u^m−1− 2xi

= 0

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A computation on a large ball, with boundary terms

∂u

∂τ +∇ ·h

u ∇u^m−1− 2xi

= 0 τ >0, x ∈BR

whereB_R is a centered ball in R^d with radiusR>0, and assume that usatisfies zero-flux boundary conditions

∇u^m−1−2x

· x

|x| = 0 τ >0, x ∈∂BR. Withz(τ,x) :=∇Q(τ,x) :=∇u^m−1− 2x, therelative Fisher informationis such that

d dτ

Z

B_R

u|z|²dx+ 4 Z

B_R

u|z|²dx

+ 2^1−m_m Z

BR

u^m D²Q

2− (1−m) (∆Q)² dx

= Z

∂B_R

u^m ω· ∇|z|²

dσ≤0 by Grisvard’s lemma)

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Another improvement of the GN inequalities

Let us define therelative entropy

E[u] := 1 m

Z

R^d

u^m− B^m? − mB?^m−1(u− B^?) dx

therelative Fisher information

I[u] :=

Z

R^d

u|z|²dx= Z

R^d

u

∇u^m−1−2x

2dx

and R[u] := 21−m m

Z

R^d

u^m D²Q

2− (1−m) (∆Q)² dx

Proposition

If1−1/d≤m<1and d ≥2, then I[u0]− 4E[u0]≥

Z ∞

0 R[u(τ,·)]dτ

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Still another improvement of the GN inequalities

We can use the computation of R´enyi entropies to estimate the decay of the relative Fisher information using the time-dependent change of variables

R^?[u] = 2^1−m_m Z

R^d

u^m

D²u^m−1− d¹∆u^m−1Id

2dx

+ 2 (m−m1)^1−m_m Z

R^d

u^m

∆u^m−1−2d

2dx

Proposition

If1−1/d<m<1and d ≥2, then I[u₀]−4E[u₀] =

Z ∞

0 R^?[u(τ,·)]dτ

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Symmetry, symmetry breaking and branches of solutions A parabolic proof ?

Large time asymptotics and spectral gaps Linearization and optimality

Unbounded domains and weighted nonlinear flows:

Caffarelli-Kohn-Nirenberg inequalities

BCaffarelli-Kohn-Nirenberg inequalities: symmetry results, elliptic and parabolic proofs

BLarge time asymptotics and spectral gaps BOptimality cases

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Caffarelli-Kohn-Nirenberg inequalities and the symmetry

breaking issue

BSymmetry, symmetry breaking and branches of solutions BThe sharp result on symmetry

BBifurcation and branches

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

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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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Proving symmetry breaking

[F. Catrina, Z.-Q. Wang],[V. Felli, M. Schneider (2003)]

a b

0

[J.D., 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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Moving planes and symmetrization techniques

a b

0

[Chou, Chu],[Horiuchi]

[Betta, Brock, Mercaldo, Posteraro]

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

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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

[J.D., 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

BWith an Emden-Fowler transformation, Caffarelli-Kohn-Nirenberg inequalities 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

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

C_a,b with a=ac±√

Λ and b= d p ±√

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:

µ(Λ) = Λ^1/αis the solution is symmetric. Bifurcation point: V. Felli and M. Schneider. Large values ofΛ: F. Catrina and Z.-Q. Wang

Further numerical results [J.D., Esteban, 2012]

Formal computation near the bifurcation point & asymptotic energy estimates[J.D., Esteban, 2013]

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Symmetry versus symmetry breaking:

the sharp result

A result based on entropies and nonlinear flows

a b

0

[J.D., Esteban, Loss, 2015]: http://arxiv.org/abs/1506.03664

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The symmetry result

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 Caffarelli-Kohn-Nirenberg inequalities are radially symmetric

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Subcritical Caffarelli-Kohn-Nirenberg inequalities

Norms: kwk^L^q,γ⁽R^d) := R

R^d|w|^q|x|^−γ dx^1/q

,kwk^L^q⁽R^d) :=kwk^L^q,0⁽R^d)

(some)Caffarelli-Kohn-Nirenberg interpolation inequalities(1984) kwk^L^2p,γ⁽R^d)≤C_β,γ,pk∇wk^ϑL^2,β(R^d)kwk^1−ϑL^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)

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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CKN and entropy – entropy production inequalities

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

1−m

m (2 +β−γ)²F[v]≤ I[v]

and equality is achieved byB_β,γ. Here thefree energyand the relative Fisher informationare defined by

F[v] := 1 m−1

Z

R^d

v^m−B^m_β,γ−mB^m−1_β,γ (v−B_β,γ) dx

|x|^γ I[v] :=

Z

R^d

v

∇v^m−1− ∇B^m−1_β,γ

2 dx

|x|^β. Ifv solves theFokker-Planck type equation

vt+|x|^γ∇ ·h

|x|^−βv∇ v^m−1− |x|^2+β−γi

= 0 (WFDE-FP)

then d

dtF[v(t,·)] =− m

1−mI[v(t,·)]

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Proposition

Let m=^p+1_2p and consider a solution to (WFDE-FP) with nonnegative initial datum u₀∈L^1,γ(R^d)such that ku₀^mkL^1,γ(R^d) and

R

R^du0|x|^2+β−2γdx are finite. Then

F[v(t,·)]≤ F[u₀]e^{−(2+β−γ)}²^t ∀t≥0 if one of the following two conditions is satisfied:

(i) either u₀ is a.e. radially symmetric (ii) or symmetry holds in (CKN)

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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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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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The symmetry result

Bcritical case: [J.D., Esteban, Loss; Inventiones]

Bsubcritical case: [J.D., Esteban, Loss, Muratori; CR Math.]

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)

The first step is based on achange of variableswhich amounts to rephrase our problem in a space of higher,artificial dimensionn>d (herenis a dimension at least from the point of view of the scaling properties), or to be precise to 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

kvk^L^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): concavity of the R´ enyi entropy power

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

Proving the symmetry in the inequality amounts to proving the monotonicity of G[u]

along a well chosen fast diffusion flow