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non-lin´ eaires

Jean Dolbeault

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

8 f´evrier 2019 Universit´e de Monastir

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A result of uniqueness on a classical example

On the sphereSd, let us consider the positive solutions of

−∆u+λu=up−1 p∈[1,2)∪(2,2]ifd≥3,2=d−22d

p∈[1,2)∪(2,+∞)ifd = 1,2 Theorem

Ifλ≤d , u≡λ1/(p−2)is the unique solution

[Gidas & Spruck, 1981],[Bidaut-V´eron & V´eron, 1991]

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Bifurcation point of view

2 4 6 8 10

2 4 6 8

Figure:(p−2)λ7→(p−2)µ(λ)with d= 3 k∇uk2L2(Sd)+λkuk2L2(Sd)≥µ(λ)kuk2Lp(Sd)

Taylor expansion ofu= 1 +ε ϕ1as ε→0with−∆ϕ1=dϕ1

µ(λ)< λ if and only if λ > d p−2

BThe inequality holds withµ(λ) =λ= p−2d [Bakry & Emery, 1985]

[Beckner, 1993],[Bidaut-V´eron & V´eron, 1991, Corollary 6.1]

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Inequalities without weights and fast diffusion equations: optimality and

uniqueness of the critical points

The Bakry-Emery method (compact manifolds) BThe Fokker-Planck equation

BThe Bakry-Emery method on the sphere: a parabolic method BThe Moser-Trudiger-Onofri inequality (on a compact manifold)

Fast diffusion equations on the Euclidean space (without weights) BEuclidean space: R´enyi entropy powers

BEuclidean space: self-similar variables and relative entropies BThe role of the spectral gap

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The Fokker-Planck equation

The linear Fokker-Planck (FP) equation

∂u

∂t = ∆u+∇ ·(u∇φ)

on a domainΩ⊂Rd, with no-flux boundary conditions (∇u+u∇φ)·ν = 0 on ∂Ω is equivalent to the Ornstein-Uhlenbeck (OU) equation

∂v

∂t = ∆v− ∇φ· ∇v =:Lv

(Bakry, Emery, 1985), (Arnold, Markowich, Toscani, Unterreiter, 2001)

With mass normalized to1, the unique stationary solution of (FP) is us = e−φ

R

e−φdx ⇐⇒ vs = 1

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The Bakry-Emery method

Withdγ=usdxandv such thatR

v dγ= 1,q∈(1,2], theq-entropy is defined by

Eq[v] := 1 q−1

Z

(vq−1−q(v−1))dγ Under the action of (OU), withw =vq/2,Iq[v] := 4qR

|∇w|2dγ, d

dtEq[v(t,·)] =− Iq[v(t,·)] and d dt

Iq[v]−2λEq[v]

≤0 with λ:= inf

w∈H1(Ω,dγ)\{0}

R

(2q−1q kHesswk2+Hessφ:∇w⊗∇w)

R

|∇w|2

Proposition

(Bakry, Emery, 1984) (JD, Nazaret, Savar´e, 2008)LetΩbe convex.

Ifλ >0and v is a solution of (OU), thenIq[v(t,·)]≤ Iq[v(0,·)]e−2λt andEq[v(t,·)]≤ Eq[v(0,·)]e−2λt for any t ≥0 and, as a consequence,

Iq[v]≥ 2λEq[v] ∀v ∈H1(Ω,dγ)

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A proof of the interpolation inequality by the carr´ e du champ

method

k∇uk2L2(Sd)≥ d p−2

kuk2Lp(Sd)− kuk2L2(Sd)

∀u∈H1(Sd)

p∈[1,2)∪(2,2]ifd≥3,2=d−22d p∈[1,2)∪(2,+∞)ifd = 1,2

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The Bakry-Emery method on the sphere

Entropy functional Ep[ρ] := p−21 hR

Sdρ2p dµ− R

Sdρdµ2pi

if p6= 2 E2[ρ] :=R

Sdρlog

ρ kρkL1(Sd)

Fisher informationfunctional Ip[ρ] :=R

Sd|∇ρ1p|2

[Bakry & Emery, 1985]carr´e du champmethod: use the heat flow

∂ρ

∂t = ∆ρ and observe that dtdEp[ρ] =− Ip[ρ],

d dt

Ip[ρ]−dEp[ρ]

≤0 =⇒ Ip[ρ]≥dEp[ρ]

withρ=|u|p, ifp≤2#:= (2d−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 = ∆ρm

(Demange),(JD, Esteban, Kowalczyk, Loss): for anyp∈[1,2] Kp[ρ] := d

dt

Ip[ρ]−dEp[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 :=νd2−1dz/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 ≥dkfk2Lp(Sd)− kfk2L2(Sd)

p−2

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The heat equation ∂g∂t =Lg forg =fp 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

ν2d

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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The elliptic point of view (nonlinear flow)

∂u

∂t =u2−2β

Lu+κ|uu0|2ν

,κ=β(p−2) + 1

− L u−(β−1)|u0|2

u ν+ λ

p−2u= λ p−2uκ Multiply byLuand integrate

...

Z 1

−1Lu uκd=−κ Z 1

−1

uκ|u0|2 u dνd

Multiply byκ|uu0|2 and integrate ...= +κ

Z 1

−1

uκ|u0|2 u dνd

The two terms cancel and we are left only with Z 1

−1

u00−p+ 2 6−p

|u0|2 u

2

ν2d= 0 ifp= 2andβ = 4 6−p

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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...

=⇒

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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

hkLgu−12Mguk2+R(∇u,∇u)i

e−u/2d vg Z

M|∇u|2e−u/2d vg

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Theorem

Assume that d= 2andλ?>0. If u is a smooth solution to

−1

2∆gu+λ=eu then u is a constant function ifλ∈(0, λ?) The Moser-Trudinger-Onofri inequality onM

1

4k∇uk2L2(M)+λ Z

M

u d vg ≥λlog Z

M

eud vg

∀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, ifΛis strictly smaller thanλ1/2, then the optimal constant in the MTO inequality is strictly larger thanΛ

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The flow

∂f

∂t = ∆g(e−f/2)−12|∇f|2e−f/2 Gλ[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 anyλ≤λ? we have

d

dtFλ[f(t,·)] = Z

M12gf +λ ∆g(e−f/2)−12|∇f|2e−f/2 d vg

=− Gλ[f(t,·)]

SinceFλ is nonnegative andlimt→∞Fλ[f(t,·)] = 0, we obtain that Fλ[u]≥

Z

0 Gλ[f(t,·)]dt

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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

Rd

eu

− Z

Rd

u dµ

hold for someλ >0? Let

Λ?:= inf

x∈R2

−∆ logµ 8π µ

Theorem

Assume thatµis a radially symmetric function. Then any radially symmetric solution to the EL equation is a constant ifλ <Λ? and the inequality holds withλ= Λ? if equality is achieved among radial functions

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Euclidean space: R´enyi entropy powers and fast diffusion

The Euclidean space without weights

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

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

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The fast diffusion equation in original variables

Consider the nonlinear diffusion equation inRd,d ≥1

∂v

∂t = ∆vm

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

Rdv0dx= 1 and R

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

U?(t,x) := 1

κt1/µdB? x κt1/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

Rd

vmdx and theFisher informationby

I :=

Z

Rd

v|∇p|2dx with p = m m−1vm−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 := Eσ 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 variation of the Fisher information

Lemma

If v solves ∂v∂t = ∆vm with1−1d ≤m<1, then I0= d

dt Z

Rd

v|∇p|2dx =−2 Z

Rd

vm

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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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→+∞Eσ−1I = (1−m)σEσ−1? I?

[Dolbeault-Toscani]The inequality

Eσ−1I≥Eσ−1? I?

is equivalent to the Gagliardo-Nirenberg inequality k∇wkθL2(Rd)kwk1−θLq+1(Rd)≥CGNkwkL2q(Rd)

if1−1d ≤m<1. Hint: vm−1/2= kwkw

L2q(Rd),q=2m−11

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Euclidean space: self-similar variables and relative entropies

In the Euclidean space, it is possible to characterize the optimal constants using a spectral gap property

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Self-similar variables and relative entropies

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

v?(t,x) := 1 κd(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|21/(m−1)

A time-dependent rescaling: self-similar variables v(t,x) = 1

κdRd u τ, x

κR

where dR

dt =R1−µ, τ(t) := 12 log R(t)

R0

Then the functionusolvesa Fokker-Planck type equation

∂u

∂τ +∇ ·h

u ∇um−1− 2x i

= 0

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Free energy and Fisher information

The function usolves a Fokker-Planck type equation

∂u

∂τ +∇ ·h

u ∇um−1− 2x i

= 0 (Ralston, Newman, 1984)Lyapunov functional:

Generalized entropyor Free energy E[u] :=

Z

Rd

−um

m +|x|2u

dx− E0

Entropy production is measured by theGeneralized Fisher information

d

dtE[u] =− I[u], I[u] :=

Z

Rd

u∇um−1+ 2x2 dx

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Without weights: relative entropy, entropy production

Stationary solution: chooseC such thatkukL1 =kukL1 =M>0 u(x) := C+ |x|2−1/(1−m)

+

Entropy – entropy production inequality (del Pino, JD)

Theorem

d≥3, m∈[d−1d ,+∞), m>12, m6= 1 I[u]≥4E[u]

p= 2m−11 ,u=w2p: (GN) k∇wkθL2(Rd)kwk1−θLq+1(Rd)≥CGNkwkL2q(Rd)

Corollary

(del Pino, JD)A solution u with initial data u0∈L1+(Rd)such that

|x|2u0∈L1(Rd), u0m∈L1(Rd)satisfiesE[u(t,·)]≤ E[u0]e4t

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A computation on a large ball, with boundary terms

∂u

∂τ +∇ ·h

u ∇um−1− 2x i

= 0 τ >0, x ∈BR

whereBR is a centered ball in Rd with radiusR>0, and assume that usatisfies zero-flux boundary conditions

∇um−1−2x

· x

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

d dτ

Z

BR

u|z|2dx+ 4 Z

BR

u|z|2dx + 21−mm

Z

BR

um

D2Q2− (1−m) (∆Q)2 dx

= Z

∂BR

um ω· ∇|z|2

dσ≤0 (by Grisvard’s lemma)

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Spectral gap: sharp asymptotic rates of convergence

Assumptions on the initial datumv0 (H1)VD0 ≤v0≤VD1 for someD0>D1>0

(H2)if d≥3 andm≤m,(v0−VD)is integrable for a suitable D∈[D1,D0]

Theorem

(Blanchet, Bonforte, JD, Grillo, V´azquez)Under Assumptions (H1)-(H2), if m<1 and m6=m:= d−4d−2, the entropy decays according to

E[v(t,·)]≤C e−2 (1−m) Λα,dt ∀t ≥0

whereΛα,d >0is the best constant in the Hardy–Poincar´e inequality Λα,d

Z

Rd

|f|2α−1≤ Z

Rd

|∇f|2α ∀f ∈H1(dµα), Z

Rd

fdµα−1= 0 withα:= 1/(m−1)<0, dµα:=hαdx , hα(x) := (1 +|x|2)α

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Spectral gap and best constants

mc=d−2d 0

m1=d−1d

m2=d+1d+2

! m2:=d+4d+6

m 1

2 4

Case 1 Case 2 Case 3

γ(m)

(d= 5)

! m1:=d+2d

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Caffarelli-Kohn-Nirenberg, symmetry and symmetry breaking results, and weighted nonlinear flows

BThe critical Caffarelli-Kohn-Nirenberg inequality [JD, Esteban, Loss]

[BA family of sub-critical Caffarelli-Kohn-Nirenberg inequalities]

[JD. Esteban, Loss, Muratori]

BLarge time asymptotics and spectral gaps BOptimality cases

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Critical Caffarelli-Kohn-Nirenberg inequality

LetDa,b:=n

v ∈Lp Rd,|x|−bdx

: |x|−a|∇v| ∈L2 Rd,dx o Z

Rd

|v|p

|x|b p dx 2/p

≤ Ca,b

Z

Rd

|∇v|2

|x|2a dx ∀v ∈ Da,b holds under conditions ona andb

p= 2d

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

v?(x) =

1 +|x|(p−2) (ac−a)p−22

and C?a,b= k |x|−bv?k2p

k |x|−a∇v?k22

Question: Ca,b= C?a,b (symmetry) orCa,b>C?a,b (symmetry breaking) ?

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Critical CKN: range of the parameters

Figure: d = 3 Z

Rd

|v|p

|x|b p dx 2/p

≤ Ca,b

Z

Rd

|∇v|2

|x|2a dx

a b

0 1

−1

b=a

b=a+ 1

a=d−22 p

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)) (Caffarelli, Kohn, Nirenberg (1984)) [F. Catrina, Z.-Q. Wang (2001)]

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Linear instability of radial minimizers:

the Felli-Schneider curve

The Felli & Schneider curve bFS(a) := d(ac−a)

2p

(ac−a)2+d−1 +a−ac

a b

0

[Smets],[Smets, Willem],[Catrina, Wang],[Felli, Schneider]

The functional C?a,b

Z

Rd

|∇v|2

|x|2a dx− Z

Rd

|v|p

|x|b p dx 2/p

is linearly instable atv =v?

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Symmetry versus symmetry breaking:

the sharp result in the critical case

[JD, Esteban, Loss (2016)]

a b

0

Theorem

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

Caffarelli-Kohn-Nirenberg inequalities are radially symmetric

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The Emden-Fowler transformation and the cylinder

BWith 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) =ra−acϕ(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)kϕk2Lp(C) ∀ϕ∈H1(C)

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

Ca,b

with a=ac±√

Λ and b= d p ±√

Λ

(37)

Linearization around symmetric critical points

Up to a normalization and a scaling

ϕ?(s, ω) = (coshs)p−21

is a critical point of

H1(C)3ϕ7→ k∂sϕk2L2(C)+k∇ωϕk2L2(C)+ Λkϕk2L2(C)

under a constraint onkϕk2Lp(C)

ϕ? is notoptimal for (CKN) if the P¨oschl-Teller operator

−∂s2−∆ω+ Λ−ϕp−2? =−∂s2−∆ω+ Λ− 1 (coshs)2 hasa negative eigenvalue, i.e., forΛ>Λ1(explicit)

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The variational problem on the cylinder

Λ7→µ(Λ) := min

ϕ∈H1(C)

k∂sϕk2L2(C)+k∇ωϕk2L2(C)+ Λkϕk2L2(C)

kϕk2Lp(C)

is a concave increasing function Restricted to symmetric functions, the variational problem becomes

µ?(Λ) := min

ϕ∈H1(R)

k∂sϕk2L2(Rd)+ Λkϕk2L2(Rd)

kϕk2Lp(Rd)

?(1) Λα

Symmetry meansµ(Λ) =µ?(Λ)

Symmetry breaking meansµ(Λ)< µ?(Λ)

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Numerical results

20 40 60 80 100

10 20 30 40 50

symmetric non-symmetric asymptotic

bifurcation µ

Λα µ(Λ)

(Λ) =µ

(1) Λα

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

Non-symmetric solutions bifurcate from symmetric ones at a bifurcation pointΛ1computed by V. Felli and M. Schneider. The branch behaves for large values ofΛ as predicted by F. Catrina and Z.-Q. Wang

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what we have to to prove / discard...

non-symmetric symmetric

bifurcation Jθ(µ)

Λθ(µ) Λθ(µ)

Jθ(µ)

symmetric non-symmetric bifurcation

Jθ(µ) bifurcation

symmetric

non-symmetric

symmetric

Λθ(µ) KGN

KCKN(ϑ(p,d), ΛFS(p,θ),p) 1/

1/

When the local criterion (linear stability) differs from global results in a larger family of inequalities (center, right)...

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The uniqueness result and the

strategy of the proof

(42)

The elliptic problem: rigidity

The symmetry issue can be reformulated as a uniqueness (rigidity) issue. An optimal function for the inequality

Z

Rd

|v|p

|x|b p dx 2/p

≤ Ca,b

Z

Rd

|∇v|2

|x|2a dx solves the (elliptic) Euler-Lagrange equation

−∇ · |x|−2a∇v

=|x|−bpvp−1

(up to a scaling and a multiplication by a constant). Is any nonnegative solution of such an equation equal to

v?(x) = 1 +|x|(p−2) (ac−a)p−22 (up to invariances) ? On the cylinder

−∂2sϕ−∂ωϕ+ Λϕ=ϕp−1 Up to a normalization and a scaling

ϕ?(s, ω) = (coshs)p−21

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Symmetry in one slide: 3 steps

Achange of variables: v(|x|α−1x) =w(x),Dαv= α∂v∂s,1sωv kvkL2p,d−n(Rd) ≤Kα,n,pkDαvkϑL2,d−n(Rd)kvk1−ϑLp+1,d−n(Rd) ∀v ∈Hpd−n,d−n(Rd)

Concavity of the R´enyi entropy power: with Lα=−DαDα2 u00+n−1s u0

+s12ωuand ∂u∂t =Lαum

dtd G[u(t,·)] R

Rdum1−σ

≥(1−m) (σ−1)R

Rdum LαP−

R

Rdu|DαP|2 R

Rdum

2

dµ + 2R

Rd

α4 1−1n P00Ps0α2(n−1)ωPs2

2+s22

ωP0ωsP

2 umdµ + 2R

Rd

(n−2) α2FS−α2

|∇ωP|2+c(n,m,d)|∇Pω2P|4

umdµ Elliptic regularity and the Emden-Fowler transformation: justifying theintegrations by parts

=⇒

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Proof of symmetry (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. Withβ = 2a andγ=b p,

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

kvkL2p,d−n(Rd) ≤Kα,n,pkDαvkϑL2,d−n(Rd)kvk1−ϑLp+1,d−n(Rd) ∀v ∈Hpd−n,d−n(Rd) with the notationss=|x|,Dαv= α∂v∂s,1sωv

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∈Rd

(45)

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

The derivative of the generalizedR´enyi entropy powerfunctional is G[u] :=

Z

Rd

umσ−1Z

Rd

u|DαP|2

whereσ= 2d 1−m1 −1. Heredµ=|x|n−ddx and the pressure is P := m

1−mum−1

Looking for an optimal function in (CKN) is equivalent to minimizeG under a mass constraint

(46)

WithLα=−DαDα2 u00+n−1s u0

+s12ωu, we consider the fast diffusion equation

∂u

∂t = Lαum

critical casem= 1−1/n; subcritical range1−1/n<m<1 The key computation is the proof that

dtd G[u(t,·)] R

Rdum1−σ

≥(1−m) (σ−1)R

Rdum LαP−

R

Rdu|DαP|2 R

Rdum

2dµ + 2R

Rd

α4 1−1n P00Ps0α2(n−1)ωPs2

2+s22

ωP0ωsP

2 umdµ + 2R

Rd

(n−2) α2FS−α2

|∇ωP|2+c(n,m,d)|∇Pω2P|4

umdµ=: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. A quantifier elimination problem (Tarski, 1951)?

(47)

(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 byLαum 0 =

Z

Rd

dG[u]·Lαumdµ≥ 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∇w1+p

the equation (p−1)2

p(p+ 1)w1−3pdiv |x|−βw2p∇w1−p

+|∇w1−p|2+|x|−γ c1w1−p−c2

= 0

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Fast diffusion equations with weights: large time asymptotics

The entropy formulation of the problem [Relative uniform convergence]

Asymptotic rates of convergence From asymptotic to global estimates

Herev solves the Fokker-Planck type equation vt+|x|γ∇ ·h

|x|−βv∇ vm−1− |x|2+β−γi

= 0 (WFDE-FP)

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

(49)

CKN and entropy – entropy production inequalities

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

1−m

m (2 +β−γ)2E[v]≤ I[v]

and equality is achieved byBβ,γ(x) := 1 +|x|2+β−γm−11

Here thefree energyand therelative Fisher informationare defined by E[v] := 1

m−1 Z

Rd

vm−Bmβ,γ−mBm−1β,γ (v−Bβ,γ) dx

|x|γ I[v] :=

Z

Rd

v∇vm−1− ∇Bm−1β,γ 2 dx

|x|β Ifv solves theFokker-Planck type equation

vt+|x|γ∇ ·h

|x|−βv∇ vm−1− |x|2+β−γi

= 0 (WFDE-FP) then d

dtE[v(t,·)] =− m

1−mI[v(t,·)]

(50)

0

Λ0,1

Λ1,0

Λess

Essential spectrum

δ4 δ

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→Λess(δ). The two eigenvaluesΛ0,1and Λ1,0are given by the plain, half-lines, away from the essential

spectrum. The spectral gap determines the asymptotic rate of convergence to the Barenblatt functions

=⇒

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Global vs. asymptotic estimates

Estimates on the global rates. When symmetry holds (CKN) can be written as anentropy – entropy production inequality

(2 +β−γ)2E[v]≤ m 1−mI[v]

so that

E[v(t)]≤ E[v(0)]e2 (1−m) Λ?t ∀t ≥0 with Λ?:= (2 +β−γ)2 2 (1−m)

Optimal global rates. Let us consider again theentropy – entropy productioninequality

K(M)E[v]≤ I[v] ∀v ∈L1,γ(Rd) such that kvkL1,γ(Rd)=M, whereK(M)is the best constant: withΛ(M) := m2(1−m)−2K(M)

E[v(t)]≤ E[v(0)]e2 (1−m) Λ(M)t

∀t≥0

(52)

Linearization and optimality

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

(53)

Linearization and scalar products

Withuεsuch that

uε=B? 1 +εfB?1−m

and Z

Rd

uεdx=M? at first order inε→0 we obtain thatf solves

∂f

∂t =Lf where Lf := (1−m)Bm−2? |x|γDα |x|−βB?Dαf Using the scalar products

hf1,f2i= Z

Rd

f1f2B2−m? |x|−γ dx and hhf1,f2ii= Z

Rd

Dαf1·Dαf2B?|x|−βdx we compute

1 2

d

dthf,fi=hf,Lfi= Z

Rd

f (Lf)B?2−m|x|−γ dx=− Z

Rd

|Dαf|2B?|x|−βdx for anyf smooth enough: with hf,Lfi=− hhf,fii

1 2

d

dthhf,fii= Z

Rd

Dαf ·Dα(Lf)B?|x|−βdx=− hhf,Lfii

(54)

Linearization of the flow, eigenvalues and spectral gap

Now let us consider an eigenfunction associated with the smallest positive eigenvalueλ1ofL

− Lf11f1

so thatf1realizes the equality case in theHardy-Poincar´e inequality hhg,gii:=− hg,Lgi ≥λ1kg−g¯k2, g¯:=hg,1i/h1,1i (P1)

− hhg,Lgii ≥λ1hhg,gii (P2) Proof by expansion of the square

− hh(g−g¯),L(g −g¯)ii=hL(g−g¯),L(g −g¯)i=kL(g −g¯)k2 (P1) is associated with the symmetry breaking issue (P2) is associated with thecarr´e du champmethod The optimal constants / eigenvalues are the same

Key observation: λ1≥4 ⇐⇒ α≤αFS:=q

d−1 n−1

(55)

Three references

Lecture notes onSymmetry and nonlinear diffusion flows...

a course on entropy methods (see webpage)

[JD, Maria J. Esteban, and Michael Loss]Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs

... the elliptic point of view: arXiv: 1711.11291, Proc. Int. Cong. of Math., Rio de Janeiro, 3: 2279-2304, 2018.

[JD, Maria J. Esteban, and Michael Loss]Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization... the parabolic point of view

Journal of elliptic and parabolic equations, 2: 267-295, 2016.

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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

Thank you for your attention !

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