### non-lin´ eaires

Jean Dolbeault

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

8 f´evrier 2019 Universit´e de Monastir

### A result of uniqueness on a classical example

On the sphereS^{d}, let us consider the positive solutions of

−∆u+λu=u^{p−1}
p∈[1,2)∪(2,2^{∗}]ifd≥3,2^{∗}=_{d−2}^{2}^{d}

p∈[1,2)∪(2,+∞)ifd = 1,2 Theorem

Ifλ≤d , u≡λ^{1/(p−2)}is the unique solution

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

### 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∇uk^{2}L^{2}(S^{d})+λkuk^{2}L^{2}(S^{d})≥µ(λ)kuk^{2}L^{p}(S^{d})

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−2}^{d} [Bakry & Emery, 1985]

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

## 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

### The Fokker-Planck equation

The linear Fokker-Planck (FP) equation

∂u

∂t = ∆u+∇ ·(u∇φ)

on a domainΩ⊂R^{d}, 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

Ωe^{−φ}dx ⇐⇒ vs = 1

### The Bakry-Emery method

Withdγ=usdxandv such thatR

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

E^{q}[v] := 1
q−1

Z

Ω

(v^{q}−1−q(v−1))dγ
Under the action of (OU), withw =v^{q/2},I^{q}[v] := ^{4}_{q}R

Ω|∇w|^{2}dγ,
d

dtE^{q}[v(t,·)] =− I^{q}[v(t,·)] and d
dt

I^{q}[v]−2λE^{q}[v]

≤0 with λ:= inf

w∈H^{1}(Ω,dγ)\{0}

R

Ω(^{2}^{q−1}q kHesswk^{2}+Hessφ:∇w⊗∇w)^{dγ}

R

Ω|∇w|^{2}dγ

Proposition

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

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

I^{q}[v]≥ 2λE^{q}[v] ∀v ∈H^{1}(Ω,dγ)

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

## method

k∇uk^{2}L^{2}(S^{d})≥ d
p−2

kuk^{2}L^{p}(S^{d})− kuk^{2}L^{2}(S^{d})

∀u∈H^{1}(S^{d})

p∈[1,2)∪(2,2^{∗}]ifd≥3,2^{∗}=_{d−2}^{2}^{d}
p∈[1,2)∪(2,+∞)ifd = 1,2

### The Bakry-Emery method on the sphere

Entropy functional
E^{p}[ρ] := _{p−2}^{1} hR

S^{d}ρ^{2}^{p} dµ− R

S^{d}ρdµ^{2}_{p}i

if p6= 2
E^{2}[ρ] :=R

S^{d}ρlog

ρ
kρk_{L1(S}_{d}_{)}

dµ

Fisher informationfunctional
I^{p}[ρ] :=R

S^{d}|∇ρ^{1}^{p}|^{2}dµ

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

∂ρ

∂t = ∆ρ
and observe that _{dt}^{d}E^{p}[ρ] =− I^{p}[ρ],

d dt

I^{p}[ρ]−dE^{p}[ρ]

≤0 =⇒ I^{p}[ρ]≥dE^{p}[ρ]

withρ=|u|^{p}, ifp≤2^{#}:= _{(}^{2}_{d−1)}^{d}^{2}^{+1}2

### 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}

(Demange),(JD, 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

### Cylindrical coordinates, Schwarz symmetrization,

### stereographic projection...

### ... and the ultra-spherical operator

Change of variablesz = cosθ,v(θ) =f(z),dνd :=ν^{d}^{2}^{−1}dz/Zd,
ν(z) := 1−z^{2}

The self-adjointultrasphericaloperator is

Lf := (1−z^{2})f^{00}−d z f^{0}=νf^{00}+d
2 ν^{0}f^{0}
which satisfieshf_{1},Lf_{2}i=−R1

−1f_{1}^{0}f_{2}^{0}νdνd

Proposition

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

− hf,Lfi= Z 1

−1|f^{0}|^{2}ν dνd ≥dkfk^{2}L^{p}(S^{d})− kfk^{2}L^{2}(S^{d})

p−2

The heat equation ^{∂g}_{∂t} =Lg forg =f^{p} can be rewritten in terms off

as ∂f

∂t =Lf + (p−1)|f^{0}|^{2}
f ν

−1 2

d dt

Z 1

−1|f^{0}|^{2}ν dνd =1
2

d

dt hf,Lfi=hLf,Lfi+(p−1)
|f^{0}|^{2}

f ν,Lf

d

dtI[g(t,·)] + 2dI[g(t,·)]= d dt

Z 1

−1|f^{0}|^{2}νdνd+ 2d
Z 1

−1|f^{0}|^{2}νdνd

=−2 Z 1

−1

|f^{00}|^{2}+ (p−1) d
d+ 2

|f^{0}|^{4}

f^{2} −2 (p−1)d−1
d+ 2

|f^{0}|^{2}f^{00}
f

ν^{2}dνd

is nonpositive if

|f^{00}|^{2}+ (p−1) d
d+ 2

|f^{0}|^{4}

f^{2} − 2 (p−1)d−1
d+ 2

|f^{0}|^{2}f^{00}
f
is pointwise nonnegative, which is granted if

(p−1)d−1 d+ 2

2

≤(p−1) d

d+ 2 ⇐⇒ p≤ 2d^{2}+ 1

(d−1)^{2} = 2^{#}< 2d
d−2 = 2^{∗}

### The elliptic point of view (nonlinear flow)

∂u

∂t =u^{2−2β}

Lu+κ^{|u}_{u}^{0}^{|}^{2}ν

,κ=β(p−2) + 1

− L u−(β−1)|u^{0}|^{2}

u ν+ λ

p−2u= λ
p−2u^{κ}
Multiply byLuand integrate

...

Z 1

−1Lu u^{κ}dνd=−κ
Z 1

−1

u^{κ}|u^{0}|^{2}
u dνd

Multiply byκ^{|u}_{u}^{0}^{|}^{2} and integrate
...= +κ

Z 1

−1

u^{κ}|u^{0}|^{2}
u dνd

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

−1

u^{00}−p+ 2
6−p

|u^{0}|^{2}
u

2

ν^{2}dνd= 0 ifp= 2^{∗}andβ = 4
6−p

## 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...

=⇒

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∈H^{2}(M)\{0}

Z

M

hkLgu−^{1}_{2}Mguk^{2}+R(∇u,∇u)i

e^{−u/2}d v_{g}
Z

M|∇u|^{2}e^{−u/2}d vg

Theorem

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

−1

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

1

4k∇uk^{2}L^{2}(M)+λ
Z

M

u d v_{g} ≥λlog
Z

M

e^{u}d v_{g}

∀u∈H^{1}(M)
for some constantλ >0. Let us denote byλ_{1}the first positive
eigenvalue of−∆g

Corollary

If d = 2, then the MTO inequality holds withλ= Λ := min{4π, λ?}.
Moreover, ifΛis strictly smaller thanλ_{1}/2, then the optimal constant
in the MTO inequality is strictly larger thanΛ

### The flow

∂f

∂t = ∆g(e^{−f}^{/2})−^{1}2|∇f|^{2}e^{−f}^{/2}
G^{λ}[f] :=

Z

MkLgf −^{1}_{2}Mgf k^{2}e^{−f}^{/2}d v_{g}+
Z

M

R(∇f,∇f)e^{−f}^{/2}d v_{g}

−λ Z

M|∇f|^{2}e^{−f}^{/2}d v_{g}
Then for anyλ≤λ? we have

d

dtF^{λ}[f(t,·)] =
Z

M −^{1}2∆gf +λ ∆g(e^{−f}^{/2})−^{1}2|∇f|^{2}e^{−f}^{/2}
d vg

=− G^{λ}[f(t,·)]

SinceF^{λ} is nonnegative andlimt→∞F^{λ}[f(t,·)] = 0, we obtain that
F^{λ}[u]≥

Z ∞

0 G^{λ}[f(t,·)]dt

### Weighted Moser-Trudinger-Onofri inequalities on the two-dimensional Euclidean space

On the Euclidean spaceR^{2}, given a general probability measureµ
does the inequality

1 16π

Z

R^{2}

|∇u|^{2}dx≥λ

log Z

R^{d}

e^{u}dµ

− Z

R^{d}

u dµ

hold for someλ >0? Let

Λ?:= inf

x∈R^{2}

−∆ 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

## 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

### 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) =v_{0}(x)≥0such thatR

R^{d}v_{0}dx= 1 and
R

R^{d}|x|^{2}v_{0}dx<+∞. 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|^{2}1/(m−1)

+ ifm>1
C_{?}+|x|^{2}1/(m−1)

ifm<1

### The R´ enyi entropy power F

Theentropy is defined by E :=

Z

R^{d}

v^{m}dx
and theFisher informationby

I :=

Z

R^{d}

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

E^{0} = (1−m) I
To computeI^{0}, 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 →+∞

### The variation of the Fisher information

Lemma

If v solves ^{∂v}_{∂t} = ∆v^{m} with1−^{1}d ≤m<1, then
I^{0}= d

dt Z

R^{d}

v|∇p|^{2}dx =−2
Z

R^{d}

v^{m}

kD^{2}pk^{2}+ (m−1) (∆p)^{2}
dx
Explicit arithmetic geometric inequality

kD^{2}pk^{2}− 1

d (∆p)^{2}=

D^{2}p− 1
d∆pId

2

.... there are no boundary terms in the integrations by parts ?

=⇒

### The concavity property

Theorem

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

Then F(t)is increasing,(1−m) F^{00}(t)≤0 and

t→+∞lim 1

t F(t) = (1−m)σ lim

t→+∞E^{σ−1}I = (1−m)σE^{σ−1}_{?} I?

[Dolbeault-Toscani]The inequality

E^{σ−1}I≥E^{σ−1}_{?} I?

is equivalent to the Gagliardo-Nirenberg inequality
k∇wk^{θ}L^{2}(R^{d})kwk^{1−θ}L^{q+1}(R^{d})≥CGNkwkL^{2q}(R^{d})

if1−^{1}d ≤m<1. Hint: v^{m−1/2}= _{kwk}^{w}

L2q(Rd),q=_{2}_{m−1}^{1}

## 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

### 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/µ}B^{?}

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^{d}R^{d} u
τ, x

κR

where dR

dt =R^{1−µ}, τ(t) := ^{1}_{2} log
R(t)

R_{0}

Then the functionusolvesa Fokker-Planck type equation

∂u

∂τ +∇ ·h

u ∇u^{m−1}− 2x i

= 0

### Free energy and Fisher information

The function usolves a Fokker-Planck type equation

∂u

∂τ +∇ ·h

u ∇u^{m−1}− 2x i

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

Generalized entropyor Free energy E[u] :=

Z

R^{d}

−u^{m}

m +|x|^{2}u

dx− E0

Entropy production is measured by theGeneralized Fisher information

d

dtE[u] =− I[u], I[u] :=

Z

R^{d}

u∇u^{m−1}+ 2x^{2} dx

### Without weights: relative entropy, entropy production

Stationary solution: chooseC such thatku_{∞}k^{L}^{1} =kuk^{L}^{1} =M>0
u_{∞}(x) := C+ |x|^{2}−1/(1−m)

+

Entropy – entropy production inequality (del Pino, JD)

Theorem

d≥3, m∈[^{d−1}_{d} ,+∞), m>^{1}_{2}, m6= 1
I[u]≥4E[u]

p= _{2m−1}^{1} ,u=w^{2p}: (GN) k∇wk^{θ}L^{2}(R^{d})kwk^{1−θ}L^{q+1}(R^{d})≥CGNkwkL^{2q}(R^{d})

Corollary

(del Pino, JD)A solution u with initial data u_{0}∈L^{1}_{+}(R^{d})such that

|x|^{2}u_{0}∈L^{1}(R^{d}), u_{0}^{m}∈L^{1}(R^{d})satisfiesE[u(t,·)]≤ E[u0]e^{−}^{4}^{t}

### A computation on a large ball, with boundary terms

∂u

∂τ +∇ ·h

u ∇u^{m−1}− 2x i

= 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 ∈∂B_{R}.
Withz(τ,x) :=∇Q(τ,x) :=∇u^{m−1}− 2x, therelative Fisher
informationis such that

d dτ

Z

B_{R}

u|z|^{2}dx+ 4
Z

B_{R}

u|z|^{2}dx
+ 2^{1−m}_{m}

Z

BR

u^{m}

D^{2}Q^{2}− (1−m) (∆Q)^{2}
dx

= Z

∂BR

u^{m} ω· ∇|z|^{2}

dσ≤0 (by Grisvard’s lemma)

### Spectral gap: sharp asymptotic rates of convergence

Assumptions on the initial datumv_{0}
(H1)VD_{0} ≤v_{0}≤VD_{1} for someD_{0}>D_{1}>0

(H2)if d≥3 andm≤m_{∗},(v_{0}−V_{D})is integrable for a suitable
D∈[D_{1},D_{0}]

Theorem

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

E[v(t,·)]≤C e^{−2 (1−m}^{) Λ}^{α,d}^{t} ∀t ≥0

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

Z

R^{d}

|f|^{2}dµ_{α−1}≤
Z

R^{d}

|∇f|^{2}dµα ∀f ∈H^{1}(dµα),
Z

R^{d}

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

### Spectral gap and best constants

m_{c}=^{d−2}_{d} 0

m1=^{d−1}_{d}

m_{2}=^{d+1}_{d+2}

!
m2:=^{d+4}_{d+6}

m 1

2 4

Case 1 Case 2 Case 3

γ(m)

(d= 5)

!
m1:=_{d+2}^{d}

## 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

### Critical Caffarelli-Kohn-Nirenberg inequality

LetD^{a,b}:=n

v ∈L^{p} R^{d},|x|^{−b}dx

: |x|^{−a}|∇v| ∈L^{2} 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 conditions ona andb

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−2}^{2}

and C^{?}_{a,b}= k |x|^{−b}v_{?}k^{2}p

k |x|^{−a}∇v_{?}k^{2}2

Question: Ca,b= C^{?}_{a,b} (symmetry) orCa,b>C^{?}_{a,b} (symmetry breaking) ?

### Critical CKN: range of the parameters

Figure: d = 3 Z

R^{d}

|v|^{p}

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

≤ Ca,b

Z

R^{d}

|∇v|^{2}

|x|^{2}^{a} dx

a b

0 1

−1

b=a

b=a+ 1

a=^{d−2}_{2}
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)]

### Linear instability of radial minimizers:

### the Felli-Schneider curve

The Felli & Schneider curve
b_{FS}(a) := d(ac−a)

2p

(ac−a)^{2}+d−1 +a−a_{c}

a b

0

[Smets],[Smets, Willem],[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?

### 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≥b_{FS}(a), then the optimal functions for the critical

Caffarelli-Kohn-Nirenberg inequalities are radially symmetric

### 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) =r^{a−a}^{c}ϕ(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∂sϕk^{2}L^{2}(C)+k∇^{ω}ϕk^{2}L^{2}(C)+ Λkϕk^{2}L^{2}(C)≥µ(Λ)kϕk^{2}L^{p}(C) ∀ϕ∈H^{1}(C)

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

Ca,b

with a=ac±√

Λ and b= d p ±√

Λ

### Linearization around symmetric critical points

Up to a normalization and a scaling

ϕ?(s, ω) = (coshs)^{−}^{p−2}^{1}

is a critical point of

H^{1}(C)3ϕ7→ k∂sϕk^{2}L^{2}(C)+k∇^{ω}ϕk^{2}L^{2}(C)+ Λkϕk^{2}L^{2}(C)

under a constraint onkϕk^{2}L^{p}(C)

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

−∂s^{2}−∆ω+ Λ−ϕ^{p−2}_{?} =−∂s^{2}−∆ω+ Λ− 1
(coshs)^{2}
hasa negative eigenvalue, i.e., forΛ>Λ1(explicit)

### The variational problem on the cylinder

Λ7→µ(Λ) := min

ϕ∈H^{1}(C)

k∂sϕk^{2}L^{2}(C)+k∇^{ω}ϕk^{2}L^{2}(C)+ Λkϕk^{2}L^{2}(C)

kϕk^{2}L^{p}(C)

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

µ?(Λ) := min

ϕ∈H^{1}(R)

k∂sϕk^{2}L^{2}(R^{d})+ Λkϕk^{2}L^{2}(R^{d})

kϕk^{2}L^{p}(R^{d})

=µ?(1) Λ^{α}

Symmetry meansµ(Λ) =µ?(Λ)

Symmetry breaking meansµ(Λ)< µ?(Λ)

### Numerical results

20 40 60 80 100

10 20 30 40 50

symmetric non-symmetric asymptotic

bifurcation µ

Λ^{α}
µ(Λ)

(Λ) =µ

(1) Λ^{α}

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

### what we have to to prove / discard...

non-symmetric symmetric

bifurcation J^{θ}(µ)

Λ^{θ}(µ) Λ^{θ}(µ)

J^{θ}(µ)

symmetric non-symmetric bifurcation

J^{θ}(µ)
bifurcation

symmetric

non-symmetric

symmetric

Λ^{θ}(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)...

## The uniqueness result and the

## strategy of the proof

### The elliptic problem: rigidity

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

Z

R^{d}

|v|^{p}

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

≤ Ca,b

Z

R^{d}

|∇v|^{2}

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

−∇ · |x|^{−2a}∇v

=|x|^{−bp}v^{p−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) (a}^{c}^{−a)}^{−}_{p−2}^{2}
(up to invariances) ? On the cylinder

−∂^{2}_{s}ϕ−∂ωϕ+ Λϕ=ϕ^{p−1}
Up to a normalization and a scaling

ϕ?(s, ω) = (coshs)^{−}^{p−2}^{1}

### Symmetry in one slide: 3 steps

Achange of variables: v(|x|^{α−1}x) =w(x),Dαv= α^{∂v}_{∂s},^{1}_{s}∇^{ω}v
kvkL^{2p,d−n}(R^{d}) ≤Kα,n,pkDαvk^{ϑ}L^{2,d−n}(R^{d})kvk^{1−ϑ}L^{p+1,d−n}(R^{d}) ∀v ∈H^{p}_{d−n,d−n}(R^{d})

Concavity of the R´enyi entropy power: with
L^{α}=−D^{∗}_{α}Dα=α^{2} u^{00}+^{n−1}_{s} u^{0}

+_{s}^{1}2∆ωuand ^{∂u}_{∂t} =L^{α}u^{m}

−dt^{d} G[u(t,·)] R

R^{d}u^{m}dµ1−σ

≥(1−m) (σ−1)R

R^{d}u^{m} L^{α}P−

R

Rdu|DαP|^{2}dµ
R

Rdu^{m}dµ

2

dµ + 2R

R^{d}

α^{4} 1−^{1}n P^{00}−^{P}s^{0} −α^{2}(n−1)^{∆}^{ω}^{P}s^{2}

^{2}+^{2α}_{s}2^{2}

∇^{ω}P^{0}−^{∇}^{ω}s^{P}

^{2}
u^{m}dµ
+ 2R

R^{d}

(n−2) α^{2}_{FS}−α^{2}

|∇^{ω}P|^{2}+c(n,m,d)^{|∇}_{P}^{ω}2^{P|}^{4}

u^{m}dµ
Elliptic regularity and the Emden-Fowler transformation: justifying
theintegrations by parts

=⇒

### 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−1}x) =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−1}x) =w(x)as

kvk^{L}^{2p,d−n}(R^{d}) ≤Kα,n,pkDαvk^{ϑ}L^{2,d−n}(R^{d})kvk^{1−ϑ}L^{p+1,d−n}(R^{d}) ∀v ∈H^{p}_{d−n,d−n}(R^{d})
with the notationss=|x|,Dαv= α^{∂v}_{∂s},^{1}_{s}∇^{ω}v

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}

### 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^{m}dµ
σ−1Z

R^{d}

u|DαP|^{2}dµ

whereσ= ^{2}_{d} _{1−m}^{1} −1. Heredµ=|x|^{n−d}dx and the pressure is
P := m

1−mu^{m−1}

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

WithLα=−D^{∗}_{α}Dα=α^{2} u^{00}+^{n−1}_{s} u^{0}

+_{s}^{1}2∆ωu, we consider the fast
diffusion equation

∂u

∂t = Lαu^{m}

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

−dt^{d} G[u(t,·)] R

R^{d}u^{m}dµ1−σ

≥(1−m) (σ−1)R

R^{d}u^{m}
LαP−

R

Rdu|DαP|^{2}dµ
R

Rdu^{m}dµ

^{2}dµ
+ 2R

R^{d}

α^{4} 1−^{1}n P^{00}−^{P}s^{0} −α^{2}(^{∆}n−1)^{ω}^{P}s^{2}

^{2}+^{2α}_{s}2^{2}

∇^{ω}P^{0}−^{∇}^{ω}s^{P}

^{2}
u^{m}dµ
+ 2R

R^{d}

(n−2) α^{2}_{FS}−α^{2}

|∇^{ω}P|^{2}+c(n,m,d)^{|∇}_{P}^{ω}2^{P|}^{4}

u^{m}dµ=: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)?

### (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 byLαu^{m}
0 =

Z

R^{d}

dG[u]·Lαu^{m}dµ≥ 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|^{γ}div |x|^{−β}∇w^{1+p}

the equation
(p−1)^{2}

p(p+ 1)w^{1−3p}div |x|^{−β}w^{2p}∇w^{1−p}

+|∇w^{1−p}|^{2}+|x|^{−γ} c_{1}w^{1−p}−c_{2}

= 0

## 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|^{−β}v∇ v^{m−1}− |x|^{2+β−γ}i

= 0 (WFDE-FP)

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

### CKN and entropy – entropy production inequalities

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

1−m

m (2 +β−γ)^{2}E[v]≤ I[v]

and equality is achieved byBβ,γ(x) := 1 +|x|^{2+β−γ}_{m−1}^{1}

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

m−1 Z

R^{d}

v^{m}−B^{m}_{β,γ}−mB^{m−1}_{β,γ} (v−Bβ,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

v_{t}+|x|^{γ}∇ ·h

|x|^{−β}v∇ v^{m−1}− |x|^{2+β−γ}i

= 0 (WFDE-FP) then d

dtE[v(t,·)] =− m

1−mI[v(t,·)]

0

Λ0,1

Λ1,0

Λess

Essential spectrum

δ4 δ

δ_{1} δ_{2} δ5

Λ0,1

Λ1,0 Λess Essential spectrum

δ4 δ5:=n 2−η

The spectrum ofLas a function ofδ=_{1−m}^{1} , withn= 5. The
essential spectrum corresponds to the grey area, and its bottom is
determined by the parabolaδ7→Λess(δ). The two eigenvaluesΛ_{0,1}and
Λ_{1,0}are given by the plain, half-lines, away from the essential

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

=⇒

### Global vs. asymptotic estimates

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

(2 +β−γ)^{2}E[v]≤ m
1−mI[v]

so that

E[v(t)]≤ E[v(0)]e^{−}^{2 (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 ∈L^{1,γ}(R^{d}) such that kvkL^{1,γ}(R^{d})=M,
whereK(M)is the best constant: withΛ(M) := ^{m}_{2}(1−m)^{−2}K(M)

E[v(t)]≤ E[v(0)]e^{−}2 (1−m) Λ(M)t

∀t≥0

## Linearization and optimality

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

### Linearization and scalar products

Withu_{ε}such that

u_{ε}=B^{?} 1 +εfB?^{1−m}

and Z

R^{d}

u_{ε}dx=M_{?}
at first order inε→0 we obtain thatf solves

∂f

∂t =Lf where Lf := (1−m)B^{m−2}? |x|^{γ}D^{∗}_{α} |x|^{−β}B^{?}Dαf
Using the scalar products

hf_{1},f_{2}i=
Z

R^{d}

f_{1}f_{2}B^{2−m}? |x|^{−γ} dx and hhf_{1},f_{2}ii=
Z

R^{d}

Dαf_{1}·Dαf_{2}B^{?}|x|^{−β}dx
we compute

1 2

d

dthf,fi=hf,Lfi= Z

R^{d}

f (Lf)B?^{2−m}|x|^{−γ} dx=−
Z

R^{d}

|Dαf|^{2}B^{?}|x|^{−β}dx
for anyf smooth enough: with hf,Lfi=− hhf,fii

1 2

d

dthhf,fii= Z

R^{d}

Dαf ·Dα(Lf)B^{?}|x|^{−β}dx=− hhf,Lfii

### Linearization of the flow, eigenvalues and spectral gap

Now let us consider an eigenfunction associated with the smallest
positive eigenvalueλ_{1}ofL

− Lf_{1}=λ_{1}f_{1}

so thatf_{1}realizes the equality case in theHardy-Poincar´e inequality
hhg,gii:=− hg,Lgi ≥λ_{1}kg−g¯k^{2}, 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¯)k^{2}
(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

### 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.

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

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