When do spectral problems determine optimal constants in nonlinear interpolation inequalities ?

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When do spectral problems determine optimal constants in nonlinear interpolation inequalities ?

Jean Dolbeault

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

October 20, 2016 Institut Mittag-Leffler

Fall program Interactions between Partial Differential Equations

& Functional Inequalities

Stockholm, September 1^st – December 16, 2016

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Interpolation inequalities on the sphere Caffarelli-Kohn-Nirenberg inequalities Fast diffusion equations, large time asymptotics, spectrum

Outline

BInterpolation inequalities on the sphere

BCaffarelli-Kohn-Nirenberg inequalities: the bifurcation point of view

BGagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg inequalities: an approach based on nonlinear flows

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Spectral methods Nonlinear flows on the sphere The bifurcation point of view

Interpolation inequalities on the sphere

BA spectral point of view on fractional and non-fractional interpolation inequalities

BThebifurcationpoint of view BFlows 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]

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Non-fractional interpolation inequalities

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 induced by the Lebesgue measure onR^d+1

1≤p<2 or 2<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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Optimal interpolation inequalities for fractional operators

The sharpHardy-Littlewood-Sobolev inequality onSⁿ[Lieb, 1983]

ZZ

Sⁿ×Sⁿ

F(ζ)|ζ−η|^−λF(η)dµ(ζ)dµ(η)≤ Γ(n) Γ ^n−λ₂ 2^λΓ ⁿ₂

Γ ⁿ_pkFk²L^p(Sⁿ)

λ∈(0,n),p=₂_n−λ²ⁿ ∈(1,2) λ= ²_qⁿ

? where _p¹+_q¹

? = 1

Asubcritical interpolation inequality dµis the uniform probability measure onSⁿ

L^s is the fractional Laplace operator of orders∈(0,n) q∈[1,2)∪(2,q_?],q_?= _n−s²ⁿ

kFk²L^q(Sⁿ)− kFk²L²(Sⁿ)

q−2 ≤C_q,s Z

Sⁿ

FL^sF dµ ∀F ∈H^s/2(Sⁿ)

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The sharp constants

Theorem

[J.D., Zhang]Let n≥1. If either s∈(0,n], q∈[1,2)∪(2,q?], or s=n and q ∈[1,2)∪(2,∞), then

C_q,s = ^n−s_2s ^Γ

n−s 2

Γ ^n+s₂

C⁻¹_q,s =λ₁(L^s)= inf

F∈H^s/2(Sⁿ)\RQ[F], Q[F] := (q−2)R

SⁿFL^sF dµ kFk²L^q(Sⁿ)− kFk²L²(Sⁿ)

Sharp subcritical fractional logarithmic Sobolev inequalities Corollary

[J.D., Zhang]Let s∈(0,n]

Z

|F|² log

|F| k k

dµ≤C_2,s Z

FL^sF dµ ∀F ∈H^s/2(Sⁿ)

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From HLS to Sobolev

Lieb’s approach... Decomposition on spherical harmonics:

F=P∞ k=0F_(k₎ Funk-Hecke formula

ZZ

Sⁿ×Sⁿ

F(ζ)|ζ−η|^−λF(η)dµ(ζ)dµ(η)

= Γ(n) Γ ^n−λ₂ 2^λΓ ⁿ₂

Γ ⁿ_p X∞

k=0

Γ(ⁿ_p) Γ(_pⁿ0 +k) Γ(_pⁿ0) Γ(ⁿ_p+k) Z

Sⁿ

|F_(k₎|²dµ

Thefractional Sobolev inequality

kFk²L^q?(Sⁿ)≤ Z

Sⁿ

FK^sF dµ:=

X∞ k=0

γk n q_?

Z

Sⁿ

|F_(k)|²dµ is dual of HLS, whereq_?= 2n

n−s is the critical exponent and γk(x) := Γ(x) Γ(n−x+k)

Γ(n−x) Γ(x+k) =(n+k−1−x) (n+k−2−x). . .(n−x) (k−1 +x) (k−2 +x). . .x

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The subcritical inequalities

L^s := 1 κn,s

(K^s−Id) with κn,s := ^Γ

n q?

Γ n−_q?ⁿ = ^Γ

n−s 2

Γ ^n+s₂ Subcritical interpolation inequalities: q∈[1,2)∪(2,q_?]

kFk²L^q(Sⁿ)− kFk²L²(Sⁿ)

q−2 ≤C_q,s Z

Sⁿ

FL^sF dµ ∀F ∈H^s/2(Sⁿ) Lemma

[J.D., Zhang]For any n≥1, the function q 7→γk n q

−1

q−2 is monotone increasing on(1,∞)for any k≥2

[Beckner, 1993]: ifq∈(2,q_?(2)],q_?=q_?(2) = 2n/(n−2), then δk(x) = 1

κn,s

γk n q

−1

=:δk n q

≤δk n q?

=k(k+n−1)

results inkFk² − kFk² ≤^q−2k∇Fk²

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Figure: The optimal constantCq,s is independent of q and determined for any given s by the critical case q=q?(s)which corresponds to the

Hardy-Littlewood-Sobolev inequality if s∈(−n,0)and to the Sobolev inequality if s∈(0,n)

The case s= 0corresponds to the critical fractional logarithmic Sobolev inequality if s= 0[Beckner, 1993]and the subcritical fractional logarithmic Sobolev inequality if s∈(0,n].

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Sketch of the proof

q7→γk(n/q)is strictly convexwith respect to qiff xγ_k⁰⁰+ 2γ_k⁰ >0 ∀x∈(0,n)

⇐⇒αk(x) :=−^γγ⁰^kk^(x)(x) =Pk−1

j=0 βj(x)withβj(x) =_n₊¹_j−x +_j₊¹_x solves α²_k−α⁰_k−²xαk >0

BA proof by induction: from

α²k ≥2β0

Xk−1

j=1

βj+

k−1X

j=0

βj²

β₀²−β₀⁰−²x β₀= 0 and

2β0βj+βj²−βj⁰−x²βj =(n−x) (n+j−x) (j+x)^{2 (}ⁿ⁺^j^{) (}ⁿ⁺²^j⁾ ²

we deduce that

α²−α⁰ −²αk ≥

Xk−1 2 (n+j) (n+2j)

∀k ≥2

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The non-fractional interpolation inequalities (again)

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

!

dµ ∀u∈H¹(S^d,dµ)\ {0}

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The Bakry-Emery method

Entropy functional E^p[ρ] := _p−2¹ hR

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^#:= ₍²_d−1)^d²⁺¹₂

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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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Can one prove Sobolev’s inequalities with a heat flow ?

1.0 1.5 2.0 2.5 3.0 3.5

1 2 3 4 5 6

(p, β)representation when d= 5. In the dark grey area, the functional is not monotone under the action of the heat flow[J.D., Esteban,

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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)

kuk²_L^p_(S^d₎− kuk²_L2(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

! 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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Symmetry, symmetry breaking and branches of solutions The sharp result on symmetry

Generalizations and comments

Symmetries, symmetry breaking and bifurcations

in Caffarelli-Kohn-Nirenberg inequalities

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

BBifurcation and branches

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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,dx o Z

R^d

|v|^p

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

≤ Ca,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<a_c := (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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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 [J.D., 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[J.D., Esteban, 2013]

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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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 := (a_c−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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Generalized Caffarelli-Kohn-Nirenberg inequalities (CKN)

Let2^∗=∞ifd= 1 ord= 2,2^∗= 2d/(d−2) ifd≥3 and define ϑ(p,d) := d(p−2)

2p

[Caffarelli-Kohn-Nirenberg-84]Letd≥1. For anyθ∈[ϑ(p,d),1], withp= d−2+2 (b−a)²^d , there exists a positive constantC_CKN(θ,p,a) such that

Z

R^d

|u|^p

|x|^{b p} dx ²_p

≤CCKN(θ,p,a) Z

R^d

|∇u|²

|x|²^a dx ^θZ

R^d

|u|²

|x|^{2 (a+1)} dx 1−θ

In the radial case, withΛ = (a−a_c)², the best constant when the inequality is restricted to radial functions isC^∗_CKN(θ,p,a)and

C_CKN(θ,p,a)≥C^∗_CKN(θ,p,a) = C^∗_CKN(θ,p) Λ^p−2^2p ^−θ C^∗_CKN(θ,p) =h

2π^d/2 Γ(d/2)

i2^p−1_p h _(p−2)2

2+(2θ−1)p

i^p−2₂_p h₂₊₍₂_θ−1)p

2pθ

iθh

4 p+2

i^6−p₂_p

Γ(_p−2² +¹₂)

√πΓ( ² ) ^p−2_p

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The method of Catrina-Wang / Felli-Schneider

Among functionsw ∈H¹(C)which depend only on s, the minimum of J[w] :=R

C |∇w|²+¹₄(d−2−2a)²|w|²

dx−[C^∗(θ,p,a)]⁻^θ¹ (^RC|w|^pdx)^p²^θ (^RC|w|²dx)^1−θ^θ is achieved byw(y) :=

cosh(λs)−_p−2²

,y = (s, ω)∈R×S=C with λ:= ¹₄(d−2−2a) (p−2)q _p+2

2pθ−(p−2) as a solution of λ²(p−2)²w⁰⁰−4w+ 2p|w|^p−2w = 0 Spectrum ofL:=−∆ +κw^p−2+µis given forp

1 + 4κ/λ²≥2j+ 1 byλi,j =µ+i(d+i−2)−^λ₄²p

1 + 4κ/λ²−(1 + 2j)2

∀i, j ∈N The eigenspace ofLcorresponding to λ0,0is generated byw The eigenfunctionφ_(1,0)associated toλ1,0is not radially symmetric and such thatR

Cwφ_(1,0)dx= 0andR

Cw^p−1φ_(1,0)dx= 0 Ifλ_1,0<0, optimal functions for (CKN) cannot be radially symmetricandC(θ,p,a)>C^∗(θ,p,a)

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A parametrization of the solutions

All optimal functions can be computed Bby solving (numerically)

−∆u+µu=u^p−1 Bby computingΛ = Λ^θ(µ)(reparametrization) and the corresponding optimal constant is given by

J^θ(µ) :=Q^θΛ^θ(µ)[u_µ] where

Q^θΛ[u] :=

k∇uk²L²(C)+ Λkuk²L²(C)

θ

kuk^{2 (1−θ)}L²(C)

kuk²L^p(C)

Ifuµ is symmetric, then

J^θ(µ) =J^θ(1)µ^α

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Parametric plot of µ 7→ (Λ

^θ

( µ ) , J

^θ

( µ )) for p = 2 . 8, d = 5, θ = 1

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

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Parametric plot of µ 7→ (Λ

^θ

( µ ) , J

^θ

( µ )) for p = 2 . 8, d = 5, θ = 0 . 8

Λ^θ(µ) J^θ(µ)

symmetric non-symmetric asymptotic

bifurcation

J^θ(1)µ^α

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Parametric plot of µ 7→ (Λ

^θ

( µ ) , J

^θ

( µ )) for p = 2 . 8, d = 5, θ = 0 . 72

Λ^θ(µ) J^θ(µ)

symmetric non-symmetric asymptotic

bifurcation

J^θ(1)µ^α

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Enlargement for p = 2 . 8, d = 5, θ = 0 . 95

non-symmetric symmetric

bifurcation J^θ(µ)

Λ^θ(µ)

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Enlargement for p = 2 . 8, d = 5, θ = 0 . 72

Λ^θ(µ) J^θ(µ)

symmetric

non-symmetric bifurcation

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Critical case θ = ϑ (p , d )

J^θ(µ) bifurcation

symmetric

non-symmetric

symmetric

Λ^θ(µ)

KGN

KCKN(ϑ(p,d), ΛFS(p,θ),p) 1/

1/

∗

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Parametric plot of µ 7→ (Λ

^θ

( µ ) , J

^θ

( µ )) for p = 3 . 15, d = 5, θ = 1

bifurcation

J^θ(µ)

Λ^θ(µ) J^θ(1)µ^α

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Parametric plot of µ 7→ (Λ

^θ

( µ ) , J

^θ

( µ )) for p = 3 . 15, d = 5, θ = 0 . 95

Λ^θ(µ) J^θ(µ)

bifurcation J^θ(1)µ^α

J. Dolbeault Optimal constants and spectral gaps

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Case p = 3 . 15, d = 5, θ = ϑ (3 . 15 , 5) ≈ 0 . 9127

bifurcation J^θ(µ)

Λ^θ(µ) J^θ(1)µ^α

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Local and asymptotic criteria for θ = ϑ (p , d) – numerical

2.9 3.0 3.1 3.2 3.3

!0.10

!0.05 0.05 0.10 0.15

Local criterion: based on an expansion of the solutions near the bifurcation point, it decides whether the branch goes to the right to to the left.

Asymptotic criterion: based on the energy of the branch asΛ→+∞

and an analysis in a semi-classical regime

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Weighted fast diffusion flows Large time asymptotics Linearization and optimality

Fast diffusion equations, large time asymptotics, spectrum

BWeighted fast diffusion equations

The equation and the self-similar solutions Without weights

A perturbation result Symmetry breaking More on symmetry

BLarge time asymptotics BLinearization and optimality

Most recent results: joint work with M. Bonforte, M. Muratori and B. Nazaret, and with M.J. Esteban and M. Loss

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Fast diffusion equations with weights: self-similar solutions

Let us consider thefast diffusion equation with weights u_t+|x|^γ∇ · |x|^−βu∇u^m−1

= 0 (t,x)∈R⁺×R^d Hereβ andγ are two real parameters, andm∈[m₁,1)with

m₁:= ²^{d−2−β−γ}_{2 (}_d−γ) GeneralizedBarenblatt self-similar solutions

u?(ρt,x) =t⁻^ρ^(d−γ)Bβ,γ t^−ρx

, Bβ,γ(x) = 1 +|x|^2+β−γ_m−1¹ where1/ρ= (d−γ) (m−m_c)withm_c:= ^d−2−β_d−γ <m₁<1

Self-similar solutions are known to govern the asymptotic behavior of the solutions when(β, γ) = (0,0)

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Mass conservation d dt

Z

R^d

u dx

|x|^γ = 0 and self-similar solutions suggest to introduce the

Time-dependent rescaling u(t,x) =R^γ−dv

(2 +β−γ)⁻¹logR, x R

withR=R(t)defined by dR

dt = (2 +β−γ)R⁽^{m−1)(γ−d})−(2+β−γ)+1, R(0) = 1 R(t) =

1 + ^2+β−γ_ρ tρ

with1/ρ= (1−m) (γ−d) + 2 +β−γ= (d−γ) (m−m_c) AFokker-Planck type equation

vt+|x|^γ∇ ·h

|x|^−βv∇ v^m−1− |x|^2+β−γi

= 0 with initial conditionv(t= 0,·) =u₀

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Without weights: time-dependent rescaling, free energy

Time-dependent rescaling: Takeu(τ,y) =R^−d(τ)v(t,y/R(τ)) where

dR

dτ =R^d(1−m)−1, t =1 2logR The functionv solves a Fokker-Planck type equation

∂v

∂t = ^m−1_m ∆v^m+∇ ·(x v) [Ralston, Newman, 1984] Lyapunov functional:

Generalized entropyor Free energy

F[v] :=

Z

R^d

−v^m m +|x|²v

dx− F⁰

Entropy production is measured by theGeneralized Fisher information

d

dtF[v] =−I[v], I[v] :=

Z

R^d

v∇v^m−1+ 2x² dx

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Without weights: relative entropy, entropy production

Stationary solution: chooseC such thatkv∞kL¹=kukL¹=M >0 v∞(x) := C+|x|²−1/(1−m)

+

Relative entropy: FixF0 so thatF[v_∞] = 0 Entropy – entropy production inequality

Theorem

d≥3, m∈[^d−1_d ,+∞), m>¹₂, m6= 1 I[v]≥4F[v]

Corollary

A solution v with initial data u₀∈L¹₊(R^d)such that |x|²u₀∈L¹(R^d), u₀^m∈L¹(R^d)satisfies

F[v(t,·)]≤ F[u₀]e⁻^4t

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More simple facts...

BThe entropy – entropy production inequality is equivalent to the Gagliardo-Nirenberg inequality

[del Pino, J.D.]With1<p≤ _d−2^d (fast diffusion case) andd ≥3 kwkL^2p(R^d)≤ Cp,d^GNk∇wk^θL²(R^d)kwk^1−θL^p+1(R^d)

Proofs: variational methods[del Pino, J.D.], orcarr´e du champ method(Bakry-Emery): [Carrillo, Toscani],[Carrillo, V´azquez], [CJMTU]

BSharp asymptotic rates are determined by thespectral gap in the linearized entropy – entropy production (Hardy–Poincar´e) inequality [Blanchet, Bonforte, J.D., Grillo, V´azquez]

BHigher order matching asymptotics can be achieved bybest matchingmethods: [Bonforte, J.D., Grillo, V´azquez],[J.D., Toscani]

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...

BImproved entropy – entropy production inequalitiesϕ(F[v])≤ I[v] can be proved[J.D., Toscani],[Carrillo, Toscani]

BR´enyi entropy powers: concavity, asymptotic regime (self-similar solutions) and Gagliardo-Nirenberg inequalities in scale invariant form [Savar´e, Toscani],[J.D., Toscani]

BConcavity of second moment estimates anddelays [J.D., Toscani]

BStabilityof entropy – entropy production inequalities (scaling methods), and improved rates of convergence[Carrillo, Toscani], [J.D., Toscani]