Entropy methods and sharp functional inequalities:
new results
Jean Dolbeault
http://www.ceremade.dauphine.fr/ ∼ dolbeaul Ceremade, Universit´e Paris-Dauphine
September 8, 2015
Nonlinear PDEs
Brussels, September 7-11, 2015
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Outline
Entropy and the fast diffusion equation: from functional inequalities and characterization of optimal rates to best matching Barenblatt functions and improved inequalities Fast diffusion equations on manifolds and sharp functional inequalities: rigidity results, the carr´e du champ or Bakry-Emery method, and the use of nonlinear diffusion equations
An equivalent point of view: optimal Keller-Lieb-Thirring estimates on manifolds
Symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities; an introduction to the lecture of Michael Loss
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Entropy and the fast diffusion equation
A summary —
⊲ Relative entropy, linearization, functional inequalities,
improvements, improved rates of convergence, delays
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
The fast diffusion equation
The fast diffusion equation corresponds to m < 1 u t = ∆u m x ∈ R d , t > 0
Self-similar (Barenblatt) functions attract all solutions as t → + ∞ [Friedmann, Kamin]
⊲ Entropy methods allow to measure the speed of convergence of any solution to U in norms which are adapted to the equation
⊲ Entropy methods provide explicit constants
The Bakry-Emery method [Carrillo, Toscani], [Juengel, Markowich, Toscani], [Carrillo, Juengel, Markowich, Toscani, Unterreiter], [Carrillo, V´ azquez]
The variational approach and Gagliardo-Nirenberg inequalities:
[del Pino, JD]
Mass transportation and gradient flow issues: [Otto et al.]
Large time asymptotics and the spectral approach: [Blanchet, Bonforte, JD, Grillo, V´ azquez], [Denzler, Koch, McCann], [Seis]
Refined relative entropy methods
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Time-dependent rescaling, free energy
Time-dependent rescaling: Take u(τ, y ) = R −d (τ) v (t , y /R(τ)) where
dR
dτ = R d(1 −m) − 1 , R(0) = 1 , t = log R The function v solves a Fokker-Planck type equation
∂v
∂t = ∆v m + ∇ · (x v) , v | τ=0 = u 0
[Ralston, Newman, 1984] Lyapunov functional:
Generalized entropy or Free energy F [v ] :=
Z
R
dv m m − 1 + 1
2 | x | 2 v
dx − F 0
Entropy production is measured by the Generalized Fisher information
d F [v ] = −I [v ] , I [v ] :=
Z v
∇ v m
+ x
2
dx
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Relative entropy and entropy production
Stationary solution: choose C such that k v ∞ k L
1= k u k L
1= M > 0 v ∞ (x ) :=
C + 1 − m 2 m | x | 2
− 1/(1 −m) +
Relative entropy: Fix F 0 so that F [v ∞ ] = 0 Entropy – entropy production inequality
Theorem
d ≥ 3, m ∈ [ d − d 1 , + ∞ ), m > 1 2 , m 6 = 1 I [v] ≥ 2 F [v ] Corollary
A solution v 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 ) satisfies F [v(t , · )] ≤ F [u 0 ] e − 2 t
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
An equivalent formulation: Gagliardo-Nirenberg inequalities
F [v] = Z
R
dv m m − 1 + 1
2 | x | 2 v
dx −F 0 ≤ 1 2 Z
R
dv
∇ v m v + x
2
dx = 1 2 I [v]
Rewrite it with p = 2m− 1 1 , v = w 2p , v m = w p+1 as 1
2 2m
2m − 1 2 Z
R
d|∇ w | 2 dx + 1
1 − m − d Z
R
d| w | 1+p dx − K ≥ 0 Theorem
[Del Pino, J.D.] With 1 < p ≤ d − d 2 (fast diffusion case) and d ≥ 3
k w k L
2p( R
d) ≤ C p,d GN k∇ w k θ L
2( R
d) k w k 1 L −
p+1θ ( R
d)
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Improved asymptotic rates
[Denzler, McCann], [Denzler, Koch, McCann], [Seis]
[Blanchet, Bonforte, J.D., Grillo, V´ azquez], [Bonforte, J.D., Grillo, V´ azquez], [J.D., Toscani]
mc=d−2d 0
m1=d−1d m2=d+1d+2
e m2:=d+4d+6
m 1
2 4
Case 1 Case 2 Case 3
γ(m)
(d= 5)
e m1:=d+2d
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Fast diffusion equations:
some recent results
1
improved inequalities and scalings
2
scalings and a concavity property
3
improved rates and best matching
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Improved inequalities and scalings
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Gagliardo-Nirenberg inequalities and the FDE
k∇ w k ϑ L
2( R
d) k w k 1 L −
q+1ϑ ( R
d) ≥ C GN k w k L
2q( R
d)
With the right choice of the constants, the functional J[w ] := 1 4 (q 2 − 1) R
R
d|∇ w | 2 dx+β R
R
d| w | q+1 dx −K C α GN R
R
d| w | 2q dx
2qαis nonnegative and J[w ] ≥ J[w ∗ ] = 0
Theorem
[Dolbeault-Toscani] For some nonnegative, convex, increasing ϕ J[w ] ≥ ϕ
β R
R
d| w ∗ | q+1 dx − R
R
d| w | q+1 dx for any w ∈ L q+1 ( R d ) such that R
R
d|∇ w | 2 dx < ∞ and R
R
d| w | 2q | x | 2 dx = R
R
dw ∗ 2q | x | 2 dx
Consequence for decay rates of relative R´enyi entropies: faster rates of
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Scalings and a concavity property
⊲ R´enyi entropies, the entropy approach without rescaling: [Savar´e, Toscani]
⊲ faster rates of convergence: [Carrillo, Toscani], [JD, Toscani]
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
The fast diffusion equation in original variables
Consider the nonlinear diffusion equation in R d , d ≥ 1
∂u
∂t = ∆u m
with initial datum u(x, t = 0) = u 0 (x) ≥ 0 such that R
R
du 0 dx = 1 and R
R
d| x | 2 u 0 dx < + ∞ . The large time behavior of the solutions is governed by the source-type Barenblatt solutions
U ⋆ (t, x) := 1 κ t 1/µ d B ⋆
x κ t 1/µ
where
µ := 2 + d (m − 1) , κ :=
2 µ m m − 1
1/µ
and B ⋆ is the Barenblatt profile B ⋆ (x) :=
C ⋆ − | x | 2 1/(m− 1)
+ if m > 1
2 1/(m− 1)
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
The entropy
The entropy is defined by E :=
Z
R
du m dx and the Fisher information by
I :=
Z
R
du |∇ p | 2 dx with p = m m − 1 u m − 1
p is the pressure variable. If u solves the fast diffusion equation, then E ′ = (1 − m) I
To compute I ′ , 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 as t → + ∞
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
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 ′′ (t) ≤ 0 and
t→ lim + ∞
1
t F(t ) = (1 − m) σ lim
t→ + ∞ E σ − 1 I = (1 − m) σ E σ ⋆ − 1 I ⋆
[Dolbeault-Toscani] The inequality
σ − 1 F ′ = E σ − 1 I ≥ E σ ⋆ − 1 I ⋆
is equivalent to the Gagliardo-Nirenberg inequality k∇ w k θ L
2( R
d) k w k 1 L −
q+1θ ( R
d) ≥ C GN k w k L
2q( R
d)
if 1 − 1 ≤ m < 1. Hint: u m − 1/2 = w , q = 1
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
The proof
Lemma
If u solves ∂u ∂t = ∆u m with 1 d ≤ m < 1, then I ′ = d
dt Z
R
du |∇ p | 2 dx = − 2 Z
R
du m
k D 2 p k 2 + (m − 1) (∆p) 2 dx
k D 2 p k 2 = 1
d (∆p) 2 +
D 2 p − 1 d ∆p Id
2
1
σ (1 − m) E 2 − σ (E σ ) ′′ = (1 − m) (σ − 1) Z
R
du |∇ p | 2 dx 2
− 2 1
d + m − 1 Z
R
du m dx Z
R
du m (∆p) 2 dx
− 2 Z
R
du m dx Z
R
du m
D 2 p − 1 d ∆p Id
2
dx
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Best matching
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Relative entropy and best matching
Consider the family of the Barenblatt profiles B σ (x) := σ −
d2C ⋆ + σ 1 | x | 2
m−11∀ x ∈ R d
The Barenblatt profile B σ plays the role of a local Gibbs state if C ⋆ is chosen so that R
R
dB σ dx = R
R
dv dx The relative entropy is defined by
F σ [v] := 1 m − 1
Z
R
dv m − B σ m − m B σ m− 1 (v − B σ ) dx To minimize F σ [v] with respect to σ is equivalent to fix σ such that
σ Z
R
d| x | 2 B 1 dx = Z
R
d| x | 2 B σ dx = Z
R
d| x | 2 v dx
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
A Csisz´ar-Kullback(-Pinsker) inequality
Let m ∈ ( d+2 d , 1) and consider the relative entropy F σ [u] := 1
m − 1 Z
R
du m − B σ m − m B σ m− 1 (u − B σ ) dx
Theorem
[J.D., Toscani] Assume that u is a nonnegative function in L 1 ( R d ) such that u m and x 7→ | x | 2 u are both integrable on R d . If k u k L
1( R
d) = M and R
R
d| x | 2 u dx = R
R
d| x | 2 B σ dx, then F σ [u]
σ
d2(1 −m) ≥ m 8 R
R
dB 1 m dx
C ⋆ k u − B σ k L
1( R
d) + 1 σ
Z
R
d| x | 2 | u − B σ | dx
2
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Temperature (fast diffusion case)
The second moment functional (temperature) is defined by Θ(t ) := 1
d Z
R
d| x | 2 u(t, x) dx and such that
Θ ′ = 2 E
0 t
τ(0)
τ(t) τ
∞= lim
s→+∞τ (s) s Θ
µ/2(s) s (s τ(0))Θ
s (s
+τ(t))Θ
µ/2µ/2
s (s
+τ
∞)Θ
µ/2s
Fast diffusion case
+J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
Temperature (porous medium case)
Let U ⋆ s be the best matching Barenblatt function, in the sense of relative entropy F [u | U ⋆ s ], among all Barenblatt functions ( U ⋆ s ) s>0 .
0
s τ(t) τ(0)
τ
∞= lim
s→+∞τ(s) t
s (s
+τ
(0)) Θ
µ/2s Θ
µ/2(s)
s (s
+τ (t))
Θ
µ/2s (s
+τ
∞) Θ
µ/2Porous medium case
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
Improved inequalities and scalings Scalings and a concavity property Best matching
A result on delays
Theorem
Assume that m ≥ 1 − d 1 and m 6 = 1. The best matching Barenblatt function of a solution u is (t, x) 7→ U ⋆ (t + τ(t), x) and the function t 7→ τ(t ) is nondecreasing if m > 1 and nonincreasing if 1 − d 1 ≤ m < 1 With G := Θ 1 −
η2, η = d (1 − m) = 2 − µ, the R´enyi entropy power functional H := Θ −
η2E is such that
G ′ = µ H with H := Θ −
η2E H ′
1 − m = Θ − 1 −
η2Θ I − d E 2
= d E 2
Θ
η2+1 (q − 1) with q := Θ I d E 2 ≥ 1 d E 2 = 1
d
− Z
R
dx · ∇ (u m ) dx 2
= 1 d
Z
R
dx · u ∇ p dx 2
≤ 1 d
Z
R
du | x | 2 dx Z
R
du |∇ p | 2 dx = Θ I
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
Fast diffusion equations on manifolds and sharp functional inequalities
1
The sphere
2
The line
3
Compact Riemannian manifolds
4
The Moser-Trudinger-Onofri inequality on
Riemannian manifolds
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
Interpolation inequalities on the sphere
Joint work with M.J. Esteban, M. Kowalczyk and M. Loss
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
A family of interpolation inequalities on the sphere
The following interpolation inequality holds on the sphere p − 2
d Z
S
d|∇ u | 2 d v g + Z
S
d| u | 2 d v g ≥ Z
S
d| u | p d v g
2/p
∀ u ∈ H 1 ( S d , dv g ) for any p ∈ (2, 2 ∗ ] with 2 ∗ = d 2d − 2 if d ≥ 3
for any p ∈ (2, ∞ ) if d = 2
Here dv g is the uniform probability measure: v g ( S d ) = 1
1 is the optimal constant, equality achieved by constants
p = 2 ∗ corresponds to Sobolev’s inequality...
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
Stereographic projection
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
Sobolev’s inequality
The stereographic projection of S d ⊂ R d × R ∋ (ρ φ, z) onto R d : to ρ 2 + z 2 = 1, z ∈ [ − 1, 1], ρ ≥ 0, φ ∈ S d− 1 we associate x ∈ R d such that r = | x | , φ = | x x |
z = r 2 − 1
r 2 + 1 = 1 − 2
r 2 + 1 , ρ = 2 r r 2 + 1
and transform any function u on S d into a function v on R d using u(y ) = ρ r
d−22v(x) = r
22 +1
d−22v(x) = (1 − z ) −
d−22v(x) p = 2 ∗ , S d = 1 4 d (d − 2) | S d | 2/d : Euclidean Sobolev inequality
Z
R
d|∇ v | 2 dx ≥ S d
Z
R
d| v |
d−2d2dx
d−2d∀ v ∈ D 1,2 ( R d )
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
Schwarz symmetrization and the ultraspherical setting
(ξ 0 , ξ 1 , . . . ξ d ) ∈ S d , ξ d = z, P d
i=0 | ξ i | 2 = 1 [Smets-Willem]
Lemma
Up to a rotation, any minimizer of Q depends only on ξ d = z
• Let dσ(θ) := (sin Z θ)
dd−1d θ, Z d := √ π Γ(
d 2 ) Γ( d+1 2 )
: ∀ v ∈ H 1 ([0, π], dσ) p − 2
d Z π
0 | v ′ (θ) | 2 dσ + Z π
0 | v (θ) | 2 d σ ≥ Z π
0 | v (θ) | p dσ
2p• Change of variables z = cos θ, v(θ) = f (z ) p − 2
d Z 1
− 1 | f ′ | 2 ν dν d + Z 1
− 1 | f | 2 dν d ≥ Z 1
− 1 | f | p dν d
2 p
where ν d (z ) dz = dν d (z) := Z d − 1 ν
d2− 1 dz , ν(z ) := 1 − z 2
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
The ultraspherical operator
With dν d = Z d − 1 ν
d2− 1 dz, ν(z) := 1 − z 2 , consider the space L 2 (( − 1, 1), dν d ) with scalar product
h f 1 , f 2 i = Z 1
− 1
f 1 f 2 dν d , k f k p = Z 1
− 1
f p dν d
p1The self-adjoint ultraspherical operator is
L f := (1 − z 2 ) f ′′ − d z f ′ = ν f ′′ + d 2 ν ′ f ′ which satisfies h f 1 , L f 2 i = − R 1
− 1 f 1 ′ f 2 ′ ν dν d
Proposition
Let p ∈ [1, 2) ∪ (2, 2 ∗ ], d ≥ 1
−h f , L f i = Z 1
| f ′ | 2 ν dν d ≥ d k f k 2 p − k f k 2 2
∀ f ∈ H 1 ([ − 1, 1], d ν d )
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
Flows on the sphere
Heat flow and the Bakry-Emery method
Fast diffusion (porous media) flow and the choice of the exponents
Joint work with M.J. Esteban, M. Kowalczyk and M. Loss
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
Heat flow and the Bakry-Emery method
With g = f p , i.e. f = g α with α = 1/p
(Ineq.) −h f , L f i = −h g α , L g α i =: I [g ] ≥ d k g k 2 1 α − k g 2 α k 1
p − 2 =: F [g ] Heat flow
∂g
∂t = L g d
dt k g k 1 = 0 , d
dt k g 2 α k 1 = − 2 (p − 2) h f , L f i = 2 (p − 2) Z 1
− 1 | f ′ | 2 ν dν d
which finally gives d
dt F [g (t , · )] = − d p − 2
d
dt k g 2 α k 1 = − 2 d I [g (t, · )]
Ineq. ⇐⇒ d
F [g (t , · )] ≤ − 2 d F [g (t, · )] ⇐ = d
I [g (t, · )] ≤ − 2 d I [g (t , · )]
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
The equation for g = f p can be rewritten in terms of f as
∂f
∂t = L f + (p − 1) | f ′ | 2 f ν
− 1 2
d dt
Z 1
− 1 | f ′ | 2 ν dν d = 1 2
d
dt h f , L f i = hL f , L f i + (p − 1) h | f ′ | 2 f ν, L f i d
dt I [g (t, · )] + 2 d I [g (t, · )] = d dt
Z 1
− 1 | f ′ | 2 ν dν d + 2 d Z 1
− 1 | f ′ | 2 ν dν d
= − 2 Z 1
− 1
| f ′′ | 2 + (p − 1) d d + 2
| f ′ | 4
f 2 − 2 (p − 1) d − 1 d + 2
| f ′ | 2 f ′′
f
ν 2 dν d
is nonpositive if
| f ′′ | 2 + (p − 1) d d + 2
| f ′ | 4
f 2 − 2 (p − 1) d − 1 d + 2
| f ′ | 2 f ′′
f is pointwise nonnegative, which is granted if
(p − 1) d − 1 d + 2
2
≤ (p − 1) d
d + 2 ⇐⇒ p ≤ 2 d 2 + 1
(d − 1) 2 = 2 # < 2 d d − 2 = 2 ∗
J. Dolbeault Entropy methods and sharp functional inequalities
Fast diffusion equations: new points of view Fast diffusion equations on manifolds and sharp functional inequalities Introduction to symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities Spectral estimates: Keller-Lieb-Thirring estimates on manifolds
The sphere The line
Compact Riemannian manifolds The Moser-Trudinger-Onofri inequality
... up to the critical exponent: a proof in two slides
d dz , L
u = ( L u) ′ − L u ′ = − 2 z u ′′ − d u ′ Z 1
− 1
( L u) 2 dν d = Z 1
− 1 | u ′′ | 2 ν 2 d ν d + d Z 1
− 1 | u ′ | 2 ν dν d
Z 1
− 1
( L u) | u ′ | 2
u ν dν d = d d + 2
Z 1
− 1
| u ′ | 4
u 2 ν 2 dν d − 2 d − 1 d + 2
Z 1
− 1
| u ′ | 2 u ′′
u ν 2 dν d
On ( − 1, 1), let us consider the porous medium (fast diffusion) flow u t = u 2 − 2β
L u + κ | u ′ | 2 u ν
If κ = β (p − 2) + 1, the L p norm is conserved d
dt Z 1
u βp dν d = β p (κ − β (p − 2) − 1) Z 1
u β(p − 2) | u ′ | 2 ν dν d = 0
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f = u β , k f ′ k 2 L
2( S
d) + p − d 2
k f k 2 L
2( S
d) − k f k 2 L
p( S
d)
≥ 0 ?
A :=
Z 1
− 1 | u ′′ | 2 ν 2 dν d − 2 d − 1
d + 2 (κ + β − 1) Z 1
− 1
u ′′ | u ′ | 2 u ν 2 dν d
+
κ (β − 1) + d
d + 2 (κ + β − 1) Z 1
− 1
| u ′ | 4 u 2 ν 2 dν d
A is nonnegative for some β if 8 d 2
(d + 2) 2 (p − 1) (2 ∗ − p) ≥ 0
A is a sum of squares if p ∈ (2, 2 ∗ ) for an arbitrary choice of β in a certain interval (depending on p and d)
A = Z 1
− 1
u ′′ − p + 2 6 − p
| u ′ | 2 u
2
ν 2 dν d ≥ 0 if p = 2 ∗ and β = 4 6 − p
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The rigidity point of view
Which computation have we done ? u t = u 2 − 2β
L u + κ |u u
′|
2ν
− L u − (β − 1) | u ′ | 2
u ν + λ
p − 2 u = λ p − 2 u κ Multiply by L u and integrate
...
Z 1
− 1 L u u κ dν d = − κ Z 1
− 1
u κ | u ′ | 2 u dν d
Multiply by κ | u u
′|
2and integrate ... = + κ
Z 1
− 1
u κ | u ′ | 2 u dν d
The two terms cancel and we are left only with the two-homogenous
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Improvements of the inequalities (subcritical range)
as long as the exponent is either in the range (1, 2) or in the range (2, 2 ∗ ), on can establish improved inequalities
An improvement automatically gives an explicit stability result of the optimal functions in the (non-improved) inequality
By duality, this provides a stability result for Keller-Lieb-Tirring inequalities
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What does “improvement” mean ?
An improved inequality is
d Φ (e) ≤ i ∀ u ∈ H 1 ( S d ) s.t. k u k 2 L
2( S
d) = 1 for some function Φ such that Φ(0) = 0, Φ ′ (0) = 1, Φ ′ > 0 and Φ(s) > s for any s. With Ψ(s) := s − Φ − 1 (s)
i − d e ≥ d (Ψ ◦ Φ)(e) ∀ u ∈ H 1 ( S d ) s.t. k u k 2 L
2( S
d) = 1 Lemma (Generalized Csisz´ar-Kullback inequalities)
k∇ u k 2 L
2( S
d) − d p − 2
h k u k 2 L
p( S
d) − k u k 2 L
2( S
d)
i
≥ d k u k 2 L
2( S
d) (Ψ ◦ Φ)
C k u k
2 (1−r) Ls(Sd)
k u k
2L2 (Sd)k u r − u ¯ r k 2 L
q( S
d)
∀ u ∈ H 1 ( S d )
s(p) := max { 2, p } and p ∈ (1, 2): q(p) := 2/p, r (p) := p; p ∈ (2, 4):
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Linear flow: improved Bakry-Emery method
Cf. [Arnold, JD]
w t = L w + κ | w ′ | 2 w ν With 2 ♯ := (d− 2 d
2+1 1)
2γ 1 :=
d − 1 d + 2
2
(p − 1) (2 # − p) if d > 1 , γ 1 := p − 1
3 if d = 1 If p ∈ [1, 2) ∪ (2, 2 ♯ ] and w is a solution, then
d
dt (i − d e) ≤ − γ 1
Z 1
− 1
| w ′ | 4
w 2 dν d ≤ − γ 1 | e ′ | 2 1 − (p − 2) e Recalling that e ′ = − i, we get a differential inequality
e ′′ + d e ′ ≥ γ 1 | e ′ | 2 1 − (p − 2) e After integration: d Φ(e(0)) ≤ i(0)
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Nonlinear flow: the H¨older estimate of J. Demange
w t = w 2 − 2β
L w + κ | w ′ | 2 w
For all p ∈ [1, 2 ∗ ], κ = β (p − 2) + 1, dt d R 1
− 1 w βp dν d = 0
− 2β 1
2dt d R 1
− 1
| (w β ) ′ | 2 ν + p − d 2 w 2β − w 2β
dν d ≥ γ R 1
− 1
|w
′|
4w
2ν 2 dν d
Lemma (Demange)
For all w ∈ H 1 ( − 1, 1), dν d
, such that R 1
− 1 w βp dν d = 1 Z 1
− 1
| w ′ | 4
w 2 ν 2 dν d ≥ 1 β 2
R 1
− 1 | (w β ) ′ | 2 ν d ν d R 1
− 1 | w ′ | 2 ν dν d
R 1
− 1 w 2β dν d
δ
.... but there are conditions on β
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Admissible ( p, β ) for d = 5
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1 2 3 4 5 6
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The line
A first example of a non-compact manifold
Joint work with M.J. Esteban, A. Laptev and M. Loss
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One-dimensional Gagliardo-Nirenberg-Sobolev inequalities
k f k L
p( R ) ≤ C GN (p) k f ′ k θ L
2( R ) k f k 1 L −
2( θ R ) if p ∈ (2, ∞ ) k f k L
2( R ) ≤ C GN (p) k f ′ k η L
2( R ) k f k 1 L −
p( η R ) if p ∈ (1, 2) with θ = p 2 − p 2 and η = 2 2+p − p
The threshold case corresponding to the limit as p → 2 is the logarithmic Sobolev inequality
R
R u 2 log
u
2k u k
2L2 (R)dx ≤ 1 2 k u k 2 L
2( R ) log
2 π e
ku
′k
2L2 (R)k u k
2L2 (R)If p > 2, u ⋆ (x) = (cosh x) −
p−22solves
− (p − 2) 2 u ′′ + 4 u − 2 p | u | p− 2 u = 0 If p ∈ (1, 2) consider u ∗ (x) = (cos x)
2−p2, x ∈ ( − π/2, π/2)
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Flow
Let us define on H 1 ( R ) the functional F [v] := k v ′ k 2 L
2( R ) + 4
(p − 2) 2 k v k 2 L
2( R ) − C k v k 2 L
p( R ) s.t. F [u ⋆ ] = 0 With z (x) := tanh x , consider the flow
v t = v 1 −
p2√ 1 − z 2
v ′′ + 2 p
p − 2 z v ′ + p 2
| v ′ | 2
v + 2
p − 2 v
Theorem (Dolbeault-Esteban-Laptev-Loss) Let p ∈ (2, ∞ ). Then
d
dt F [v(t)] ≤ 0 and lim
t →∞ F [v(t)] = 0
d
dt F [v (t )] = 0 ⇐⇒ v 0 (x) = u ⋆ (x − x 0 )
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The inequality ( p > 2) and the ultraspherical operator
The problem on the line is equivalent to the critical problem for the ultraspherical operator
Z
R
| v ′ | 2 dx + 4 (p − 2) 2
Z
R
| v | 2 dx ≥ C Z
R
| v | p dx
2pWith
z (x) = tanh x , v ⋆ = (1 − z 2 )
p−21and v (x) = v ⋆ (x ) f (z(x)) equality is achieved for f = 1 and, if we let ν(z ) := 1 − z 2 , then
Z 1
− 1 | f ′ | 2 ν d ν d + 2 p (p − 2) 2
Z 1
− 1 | f | 2 dν d ≥ 2 p (p − 2) 2
Z 1
− 1 | f | p dν d
2 p
where dν p denotes the probability measure dν p (z ) := ζ 1
p
ν
p−22dz d = p− 2 p 2 ⇐⇒ p = d− 2d 2
J. Dolbeault Entropy methods and sharp functional inequalities
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Change of variables = stereographic projection + Emden-Fowler
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Compact Riemannian manifolds
no sign is required on the Ricci tensor and an improved integral criterion is established
the flow explores the energy landscape... and shows the non-optimality of the improved criterion
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Riemannian manifolds with positive curvature
( M , g ) is a smooth closed compact connected Riemannian manifold dimension d, no boundary, ∆ g is the Laplace-Beltrami operator vol( M ) = 1, R is the Ricci tensor, λ 1 = λ 1 ( − ∆ g )
ρ := inf
M inf
ξ ∈ S
d−1R (ξ , ξ) Theorem (Licois-V´eron, Bakry-Ledoux)
Assume d ≥ 2 and ρ > 0. If λ ≤ (1 − θ) λ 1 + θ d ρ
d − 1 where θ = (d − 1) 2 (p − 1) d (d + 2) + p − 1 > 0 then for any p ∈ (2, 2 ∗ ), the equation
− ∆ g v + λ
p − 2 v − v p− 1
= 0
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Riemannian manifolds: first improvement
Theorem (Dolbeault-Esteban-Loss) For any p ∈ (1, 2) ∪ (2, 2 ∗ )
0 < λ < λ ⋆ = inf
u∈ H
2(M)
Z
M
h (1 − θ) (∆ g u) 2 + θ d
d − 1 R ( ∇ u, ∇ u) i d v g
R
M |∇ u | 2 d v g
there is a unique positive solution in C 2 ( M ): u ≡ 1 lim p → 1
+θ(p) = 0 = ⇒ lim p → 1
+λ ⋆ (p) = λ 1 if ρ is bounded λ ⋆ = λ 1 = d ρ/(d − 1) = d if M = S d since ρ = d − 1
(1 − θ) λ 1 + θ d ρ
d − 1 ≤ λ ⋆ ≤ λ 1
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Riemannian manifolds: second improvement
H g u denotes Hessian of u and θ = (d − 1) 2 (p − 1) d (d + 2) + p − 1 Q g u := H g u − g
d ∆ g u − (d − 1) (p − 1) θ (d + 3 − p)
∇ u ⊗ ∇ u
u − g
d
|∇ u | 2 u
Λ ⋆ := inf
u ∈ H
2( M ) \{ 0 }
(1 − θ) Z
M
(∆ g u) 2 d v g + θ d d − 1
Z
M
h k Q g u k 2 + R ( ∇ u, ∇ u) i Z
M |∇ u | 2 d v g
Theorem (Dolbeault-Esteban-Loss)
Assume that Λ ⋆ > 0. For any p ∈ (1, 2) ∪ (2, 2 ∗ ), the equation has a
unique positive solution in C 2 ( M ) if λ ∈ (0, Λ ⋆ ): u ≡ 1
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Optimal interpolation inequality
For any p ∈ (1, 2) ∪ (2, 2 ∗ ) or p = 2 ∗ if d ≥ 3 k∇ v k 2 L
2(M) ≥ λ
p − 2
h k v k 2 L
p(M) − k v k 2 L
2(M)
i ∀ v ∈ H 1 ( M ) Theorem (Dolbeault-Esteban-Loss)
Assume Λ ⋆ > 0. The above inequality holds for some λ = Λ ∈ [Λ ⋆ , λ 1 ] If Λ ⋆ < λ 1 , then the optimal constant Λ is such that
Λ ⋆ < Λ ≤ λ 1
If p = 1, then Λ = λ 1
Using u = 1 + ε ϕ as a test function where ϕ we get λ ≤ λ 1
A minimum of
v 7→ k∇ v k 2 L
2(M) − p − λ 2 h
k v k 2 L
p(M) − k v k 2 L
2(M)
i under the constraint k v k L
p( M ) = 1 is negative if λ > λ 1
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The flow
The key tool is the flow u t = u 2 − 2β
∆ g u + κ |∇ u | 2 u
, κ = 1 + β (p − 2) If v = u β , then dt d k v k L
p( M ) = 0 and the functional
F [u] :=
Z
M |∇ (u β ) | 2 d v g + λ p − 2
"
Z
M
u 2 β d v g − Z
M
u β p d v g
2/p #
is monotone decaying
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Elementary observations (1/2)
Let d ≥ 2, u ∈ C 2 ( M ), and consider the trace free Hessian L g u := H g u − g
d ∆ g u Lemma
Z
M
(∆ g u) 2 d v g = d d − 1
Z
M
k L g u k 2 d v g + d d − 1
Z
M
R ( ∇ u, ∇ u) d v g
Based on the Bochner-Lichnerovicz-Weitzenb¨ ock formula 1
2 ∆ |∇ u | 2 = k H g u k 2 + ∇ (∆ g u) · ∇ u + R ( ∇ u, ∇ u)
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Elementary observations (2/2)
Lemma
Z
M
∆ g u |∇ u | 2 u d v g
= d
d + 2 Z
M
|∇ u | 4
u 2 d v g − 2 d d + 2
Z
M
[L g u] :
∇ u ⊗ ∇ u u
d v g
Lemma
Z
M
(∆ g u) 2 d v g ≥ λ 1
Z
M
|∇ u | 2 d v g ∀ u ∈ H 2 ( M )
and λ 1 is the optimal constant in the above inequality
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The key estimates
G [u] := R
M
h θ (∆ g u) 2 + (κ + β − 1) ∆ g u |∇u| u
2+ κ (β − 1) |∇u| u
24i d v g
Lemma 1 2 β 2
d
dt F [u] = − (1 − θ) Z
M
(∆ g u) 2 d v g − G [u] + λ Z
M
|∇ u | 2 d v g
Q θ g u := L g u − 1 θ d d+2 − 1 (κ + β − 1) h
∇ u ⊗∇ u
u − g d |∇u| u
2i Lemma
G [u] = θ d d − 1
Z
M k Q θ g u k 2 d v g + Z
M
R ( ∇ u, ∇ u) d v g
− µ Z
M
|∇ u | 4 u 2 d v g
with µ := 1 θ
d − 1 d + 2
2
(κ + β − 1) 2 − κ (β − 1) − (κ + β − 1) d d + 2
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The end of the proof
Assume that d ≥ 2. If θ = 1, then µ is nonpositive if β − (p) ≤ β ≤ β + (p) ∀ p ∈ (1, 2 ∗ ) where β ± := b ± √ 2 a b
2− a with a = 2 − p + h
(d− 1) (p− 1) d+2
i 2
and b = d+3 d+2 − p Notice that β − (p) < β + (p) if p ∈ (1, 2 ∗ ) and β − (2 ∗ ) = β + (2 ∗ )
θ = (d − 1) 2 (p − 1)
d (d + 2) + p − 1 and β = d + 2 d + 3 − p Proposition
Let d ≥ 2, p ∈ (1, 2) ∪ (2, 2 ∗ ) (p 6 = 5 or d 6 = 2) 1
2 β 2 d
dt F [u] ≤ (λ − Λ ⋆ ) Z
|∇ u | 2 d v g
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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 by R the Ricci tensor, by H g u the Hessian of u and by
L g u := H g u − g d ∆ g u
the trace free Hessian. Let us denote by M g u the trace free tensor M g u := ∇ u ⊗ ∇ u − g
d |∇ u | 2 We define
λ ⋆ := inf
u ∈ H
2( M ) \{ 0 }
Z
M
h
k L g u − 1 2 M g u k 2 + R ( ∇ u, ∇ u) i
e − u/2 d v g
Z
M |∇ u | 2 e − u/2 d v g
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Theorem
Assume that d = 2 and λ ⋆ > 0. If u is a smooth solution to
− 1
2 ∆ g u + λ = e u then u is a constant function if λ ∈ (0, λ ⋆ ) The Moser-Trudinger-Onofri inequality on M
1
4 k∇ u k 2 L
2( M ) + λ Z
M
u d v g ≥ λ log Z
M
e u d v g
∀ u ∈ H 1 ( M ) for some constant λ > 0. Let us denote by λ 1 the first positive eigenvalue of − ∆ g
Corollary
If d = 2, then the MTO inequality holds with λ = Λ := min { 4 π, λ ⋆ } . Moreover, if Λ is strictly smaller than λ 1 /2, then the optimal constant in the MTO inequality is strictly larger than Λ
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The flow
∂f
∂t = ∆ g (e −f /2 ) − 1 2 |∇ f | 2 e −f /2 G λ [f ] :=
Z
M k L g f − 1 2 M g f 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 any λ ≤ λ ⋆ we have
d
dt F λ [f (t, · )] = Z
M − 1 2 ∆ g f + λ
∆ g (e − f /2 ) − 1 2 |∇ f | 2 e − f /2 d v g
= − G λ [f (t , · )]
Since F λ is nonnegative and lim t→∞ F λ [f (t, · )] = 0, we obtain that F λ [u] ≥
Z ∞
G λ [f (t , · )] dt
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Weighted Moser-Trudinger-Onofri inequalities on the two-dimensional Euclidean space
On the Euclidean space R 2 , given a general probability measure µ does the inequality
1 16 π
Z
R
2|∇ u | 2 dx ≥ λ
log Z
R
de u dµ
− Z
R
du dµ
hold for some λ > 0 ? Let
Λ ⋆ := inf
x ∈ R
2− ∆ log µ 8 π µ Theorem
Assume that µ is a radially symmetric function. Then any radially symmetric solution to the EL equation is a constant if λ < Λ ⋆ and the inequality holds with λ = Λ ⋆ if equality is achieved among radial functions
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