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 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]
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]
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⊂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
Ωe−φdx ⇐⇒ vs = 1
The Bakry-Emery method
Withdγ=usdxandv such thatR
Ωv dγ= 1,q∈(1,2], theq-entropy is defined by
Eq[v] := 1 q−1
Z
Ω
(vq−1−q(v−1))dγ Under the action of (OU), withw =vq/2,Iq[v] := 4qR
Ω|∇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
Ω(2q−1q kHesswk2+Hessφ:∇w⊗∇w)dγ
R
Ω|∇w|2dγ
Proposition
(Bakry, Emery, 1984) (JD, Nazaret, Savar´e, 2008)LetΩbe convex.
Ifλ >0and v is a solution of (OU), 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)
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
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)
dµ
Fisher informationfunctional Ip[ρ] :=R
Sd|∇ρ1p|2dµ
[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
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
Cylindrical coordinates, Schwarz symmetrization,
stereographic projection...
... 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
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
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∗
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 uκdνd=−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
ν2dνd= 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∈H2(M)\{0}
Z
M
hkLgu−12Mguk2+R(∇u,∇u)i
e−u/2d vg Z
M|∇u|2e−u/2d vg
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
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
M −12∆gf +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
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
eudµ
− 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
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 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
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 →+∞
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 ?
=⇒
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
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 = ∆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
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
Without weights: relative entropy, entropy production
Stationary solution: chooseC such thatku∞kL1 =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]e−4t
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)
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|2dµα−1≤ Z
Rd
|∇f|2dµα ∀f ∈H1(dµα), Z
Rd
fdµα−1= 0 withα:= 1/(m−1)<0, dµα:=hαdx , hα(x) := (1 +|x|2)α
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
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
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) ?
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)]
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?
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
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
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)
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)
=µ?(1) Λα
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
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
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)...
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
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
Symmetry in one slide: 3 steps
Achange of variables: v(|x|α−1x) =w(x),Dαv= α∂v∂s,1s∇ωv 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=−D∗αDα=α2 u00+n−1s u0
+s12∆ωuand ∂u∂t =Lαum
−dtd G[u(t,·)] R
Rdumdµ1−σ
≥(1−m) (σ−1)R
Rdum LαP−
R
Rdu|DαP|2dµ R
Rdumdµ
2
dµ + 2R
Rd
α4 1−1n P00−Ps0 −α2(n−1)∆ωPs2
2+2αs22
∇ωP0−∇ωsP
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
=⇒
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,1s∇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∈Rd
The strategy of the proof (2/3: R´ enyi entropy)
The derivative of the generalizedR´enyi entropy powerfunctional is G[u] :=
Z
Rd
umdµ σ−1Z
Rd
u|DαP|2dµ
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
WithLα=−D∗αDα=α2 u00+n−1s u0
+s12∆ωu, 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
Rdumdµ1−σ
≥(1−m) (σ−1)R
Rdum LαP−
R
Rdu|DαP|2dµ R
Rdumdµ
2dµ + 2R
Rd
α4 1−1n P00−Ps0 −α2(∆n−1)ωPs2
2+2αs22
∇ωP0−∇ωsP
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)?
(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
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
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,·)]
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
=⇒
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)]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 ∈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)]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
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|γD∗α |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
Linearization of the flow, eigenvalues and spectral gap
Now let us consider an eigenfunction associated with the smallest positive eigenvalueλ1ofL
− Lf1=λ1f1
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
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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