Des équations de dérive-diffusion avec champ moyen aux inégalités de Hardy-Littlewood-Sobolev inversées

moyen aux inégalités de

Hardy-Littlewood-Sobolev inversées

Jean Dolbeault

http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Université Paris-Dauphine

7 Février 2019

Séminaire du Département de Mathématiques, Faculté des Sciences de Tunis

Reverse HLS: joint work with J. A. Carrillo, M. G. Delgadino,

Outline

Sharp asymptotics for the subcritical Keller-Segel model B An introduction to the Keller-Segel model

B Functional framework and sharp asymptotics Reverse HLS inequality

B The inequality and the conformally invariant case B A proof based on Carlson’s inequality, the case λ = 2 B Concentration and a relaxed inequality

Existence of minimizers and relaxation B Existence minimizers if q > 2N/(2N + λ) B Rlaxation and measure valued minimizers

Regions of no concentration and regularity of measure valued minimizers

B No concentration results B Regularity issues Free Energy

B Free energy: toy model, equivalence with reverse HLS ineq.

B Relaxed free energy, uniqueness

Sharp asymptotics for the subcritical Keller-Segel model

1

Literature is huge

2

Physics can be addressed in various ways: gravitation (Smoluchowski-Poisson) and statistics of gravitating systems, aggregation dynamics (sticky systems), biology (Patlak, Keller-Segel)

3

Standard techniques have been reinvented many times: virial estimates, cumulated mass densities, matched asymptotics do not specialize to radial solutions

put emphasis on functional analysis

insist on nonlinear evolution

deal with the subcritical case

The parabolic-elliptic Keller – Segel system



 

 



 

 



∂u

∂t = ∆u − ∇ · (u ∇v) x ∈ R ² , t > 0

−∆v = u x ∈ R ² , t > 0 u(·, t = 0) = n 0 ≥ 0 x ∈ R ² We make the choice:

v(t, x) = − 1 2π

Z

R

log |x − y| u(t, y) dy and observe that

∇v(t, x) = − 1 2π

Z

R

x − y

|x − y| ² u(t, y) dy

Mass conservation: d dt

Z

R

u(t, x) dx = 0

Blow-up: the virial computation

Collapse (S. Childress, J.K. Percus 81) M = R

R

n ₀ dx > 8π and R

R

|x| ² n ₀ dx < ∞: blow-up in finite time A solution u of

∂u

∂t = ∇ · (∇u − u ∇v) satisfies

d dt

Z

R

|x| ² u(t, x) dx

= − Z

R

2 x · ∇u dx

| {z }

−4 M

+ 1 2π

Z Z

R

×R

2x·(y−x)

|x−y|

u(t, x) u(t, y) dx dy

| {z }

(x−y)·(y−x)

|x−y|

u(t,x)u(t,y) dx dy

= 4 M − M ²

2π < 0 if M > 8π

The super-critical range: regularization & life after blow-up

Regularize the Poisson kernel (−∆) ⁻¹ _ε ∗ ρ (x) = − 1

2π Z

R

log(|x − y| + ε) ρ(y) dy

[F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equations, Meth. Appl. Anal. 9 (2002), pp. 533–561]

Proposition (JD, C. Schmeiser 2009)

For every ε > 0, the regularized problem has a global solution satisfying

kρ ^ε (·, t)k L

( R

) = kρ 0 k L

( R

) := M kρ ^ε (t, ·)k L

( R

) ≤ c

1 + 1

ε ²

with an ε-independent constant c

The nonlinear term

m ^ε (t, x) :=

Z

R

K ^ε (x − y) ρ ^ε (t, x) ρ ^ε (t, y)dy with K ^ε (x) = x ^⊗2

|x|(|x| + ε) Lemma (Poupaud)

The families {ρ ^ε (t)} _ε>0 and {m ^ε (t)} _ε>0 are tightly bounded locally uniformly in t, and {ρ ^ε (t)} ε>0 is tightly equicontinuous in t Tight boundedness and equicontinuity of ρ ^ε (t) = ⇒ compactness

R

R

R

R

ϕ(x, y) ρ ^ε (t, x) ρ ^ε (t, y) dx dy → R

R

R

R

ϕ(x, y) ρ(t, x) ρ(t, y) dx dy R t

t

R

R

ϕ(t, x) m ^ε (t, x) dx dt → R t

t

R

R

ϕ(t, x) m(t, x) dx dt

for all ϕ ∈ C b ([t 1 , t 2 ] × R ² ) Defect measure

ν(t, x) = m(t, x) − Z

R

K (x − y) ρ(t, x) ρ(t, y) dy , K (x) = x ^⊗2

|x| ²

Atomic support

The limit is characterized by the pair (ρ, ν), the atomic support of ρ is an at most countable set

Lemma (Poupaud 2002)

ν is symmetric, nonnegative, and satisfies tr(ν (t, x)) ≤ X

a∈S

(ρ(t))

(ρ(t)({a})) ² δ(x − a) M : Radon measures, M ⁺ 1 : nonnegative bounded measures DM ⁺ (I; R ² ) = n

(ρ, ν) : ρ(t) ∈ M ⁺ 1 ( R ² ) ∀t ∈ I, ν ∈ M (I × R ² ) ^2×2 ρ is tightly continuous with respect to t

ν is a nonnegative, symmetric, matrix valued measure tr(ν(t, x)) ≤ X

a∈S

(ρ(t))

(ρ(t)({a})) ² δ(x − a) o

Limiting problem

∀ϕ ∈ C _b ¹ ((0, T ) , × R ² ) R T 0

R

R

ϕ(t, x) j[ρ, ν](t, x) dx dt

= − _4π ¹ R T 0

R

R

(ϕ(t, x) − ϕ(t, y)) K(x − y) ρ(t, x) ρ(t, y) dx dy dt

− _4π ¹ R T 0

R

R

ν(t, x)∇ϕ(t, x) dx dt Theorem (JD, C. Schmeiser 2009)

For every T > 0, ρ ^ε converges tightly and uniformly in time to ρ(t) and there exists ν(t) such that (ρ, ν) ∈ DM ⁺ ((0, T ); R ² ) is a generalized solution of

∂ _t ρ + ∇ · (j[ρ, ν] − ∇ρ) = 0

ρ(t = 0) = ρ ₀ holds in the sense of tight continuity

Strong formulation (formal) : an ansatz

ρ = ρ + ˆ ρ, ˆ ρ(t, x) = P

n∈N M n (t) δ n (t, x), δ n (t, x) = δ(x − x n (t)) (ρ, ν) ∈ DM ⁺ ((0, T ); R ² )

= ⇒ ν (t, x) = P

n∈N ν n (t) δ n (t, x), tr(ν n ) ≤ M _n ² j[ρ, ν] = ρ ∇S 0 [ρ+ ˆ ρ]+ X

n

M n δ n ∇S 0



ρ + X

m6=n

M m δ m



+ 1 4π

X

n

M n ν n ∇δ n

∂ t ρ + ∇ · (ρ ∇S 0 [ρ] − ∇ρ) + ∇ρ · ∇S 0 [ ˆ ρ]

+ X

n

δ n ( ˙ M n − ρ M n )

− X

n

M n ∇δ n

˙

x n − ∇S 0

h ρ + P

m6=n M m δ m

i

+ X

n 1

4π ν n : ∇ ² δ n − M n ∆δ n

= 0

Keller-Segel model: the subcritical range

M = R

R

n 0 dx ≤ 8π: global existence (W. Jäger, S. Luckhaus 1992), (JD, B. Perthame 2004), (A. Blanchet, JD, B. Perthame 2006) If u solves

∂u

∂t = ∇ · [u (∇ (log u) − ∇v)]

the free energy

F [u] :=

Z

R

u log u dx − 1 2

Z

R

u v dx satisfies

d

dt F [u(t, ·)] = − Z

R

u |∇ (log u) − ∇v| ² dx (log HLS) inequality (E. Carlen, M. Loss 1992):

F is bounded from below if M ≤ 8π

... M = 8π the critical case (A. Blanchet, J.A. Carrillo, N. Masmoudi

2008), (A. Blanchet et al.)

The existence setting for the subcritical regime



 

 



 

 



∂u

∂t = ∆u − ∇ · (u ∇v) x ∈ R ² , t > 0

−∆v = u x ∈ R ² , t > 0 u(·, t = 0) = n ₀ ≥ 0 x ∈ R ² Initial conditions

n ₀ ∈ L ¹ ₊ ( R ² , (1+|x| ² ) dx) , n ₀ log n ₀ ∈ L ¹ ( R ² , dx) , M :=

Z

R

n ₀ (x) dx < 8 π Global existence and mass conservation: M = R

R

u(x, t) dx ∀ t ≥ 0

v = − _2π ¹ log | · | ∗ u

Time-dependent rescaling

u(x, t) = 1 R ² (t) n

x R(t) , τ (t)

and v(x, t) = c x

R(t) , τ(t)

with R(t) = √

1 + 2t and τ(t) = log R(t)



 

 

 

 

∂n

∂t = ∆n − ∇ · (n (∇c − x)) x ∈ R ² , t > 0 c = − 1

2π log | · | ∗ n x ∈ R ² , t > 0 n(·, t = 0) = n 0 ≥ 0 x ∈ R ²

(A. Blanchet, JD, B. Perthame) Convergence in self-similar variables

t→∞ lim kn(·, · + t) − n ∞ k _L

( R

) = 0 and lim

t→∞ k∇c(·, · + t) − ∇c ∞ k _L

( R

) = 0 means intermediate asymptotics in original variables:

ku(x, t) − _R

¹ (t) n ∞

x

R(t) , τ (t)

k L

( R

) & 0

The stationary solution in self-similar variables

n _∞ = M e ^c

^−|x|

^/2 R

R

e ^c

^−|x|

^/2 dx = −∆c _∞ , c _∞ = − 1

2π log | · | ∗ n _∞ Radial symmetry (Y. Naito)

Uniqueness (P. Biler, G. Karch, P. Laurençot, T. Nadzieja) As |x| → +∞, n _∞ is dominated by e ^{−(1−ε)|x|}

^/2 for any ε ∈ (0, 1) (A. Blanchet, JD, B. Perthame)

Bifurcation diagram of kn _∞ k _L

( R

) as a function of M lim

M →0

kn _∞ k _L

( R

) = 0

(D.D. Joseph, T.S. Lundgren) (JD, R. Stańczy)

The stationary solution when mass varies

0.5 1.0 1.5 2.0

2 4 6 8

Figure: Representation of the solution appropriately scaled so that the 8π

case appears as a limit (in red)

The free energy in self-similar variables

∂n

∂t = ∇ h

n (log n − x + ∇c) i F [n] :=

Z

R

n log n dx + Z

R

1

2 |x| ² n dx − 1 2

Z

R

n c dx satisfies

d

dt F [n(t, ·)] = − Z

R

n |∇ (log n) + x − ∇c| ² dx A last remark on 8π and scalings: n ^λ (x) = λ ² n(λ x) F[n ^λ ] = F [n]+

Z

R

n log(λ ² ) dx+

Z

R

λ

−1

2 |x| ² n dx+ 1 4π

Z

R

× R

n(x) n(y) log 1 λ dx dy F[n ^λ ] − F[n] =

2M − M ² 4π

| {z }

>0 if M <8π

log λ + λ ⁻² − 1 2

Z

R

|x| ² n dx

Keller-Segel with subcritical mass in self-similar variables



 

 

 

 

∂n

∂t = ∆n − ∇ · (n (∇c − x)) x ∈ R ² , t > 0 c = − 1

2π log | · | ∗ n x ∈ R ² , t > 0 n(·, t = 0) = n 0 ≥ 0 x ∈ R ²

t→∞ lim kn(·, · + t) − n ∞ k _L

( R

) = 0 and lim

t→∞ k∇c(·, · + t) − ∇c ∞ k _L

( R

) = 0 n _∞ = M e ^c

^−|x|

^/2

R

R

e ^c

^−|x|

^/2 dx = −∆c ∞ , c _∞ = − 1

2π log | · | ∗ n _∞

A parametrization of the solutions and the linearized operator

(J. Campos, JD)

−∆c = M e ⁻

^|x|

^+c R

R

e ⁻

^|x|

^+c dx Solve

−ϕ ⁰⁰ − 1

r ϕ ⁰ = e ⁻

^r

^+ϕ , r > 0

with initial conditions ϕ(0) = a, ϕ ⁰ (0) = 0 and get with r = |x|

M (a) := 2π Z

R

e ⁻

^r

^+ϕ

dx

n _a (x) = M (a) e ⁻

^r

^+ϕ

^(r) 2π R

R

r e ⁻

^r

^+ϕ

dx = e ⁻

^r

^+ϕ

^(r)

Mass

-4 -2 2 4 6 8 10

0.2 0.4 0.6 0.8 1.0

Figure: The mass can be computed as M (a) = 2π R

0

n

(r) r dr. Plot of

a 7→ M (a)/8π

Bifurcation diagram

1.5 2.0 2.5 3.0 3.5 4.0

5 10 15

Figure: The bifurcation diagram can be parametrized by

a 7→ (

M (a), kc

k

) with kc

k

= c

(0) = a − b(a) (cf. Keller-Segel

system in a ball with no flux boundary conditions)

Linearization

We can introduce two functions f and g such that n = n _∞ (1 + f ) and c = c _∞ (1 + g) and rewrite the Keller-Segel model as

∂f

∂t = L f + 1

n _∞ ∇(f n _∞ ∇(c _∞ g)) where the linearized operator is

L f = 1

n _∞ ∇ · n _∞ ∇(f − c _∞ g) and

−∆(c ∞ g) = n _∞ f

Spectrum of L (lowest eigenvalues only)

5 10 15 20 25

1 2 3 4 5 6 7

Figure: The lowest eigenvalues of − L = (−∆)

(n

f) (shown as a function of the mass) are 0, 1 and 2, thus establishing that the spectral gap of −L is 1

(A. Blanchet, JD, M. Escobedo, J. Fernández), (J. Campos, JD),

(V. Calvez, J.A. Carrillo), (J. Bedrossian, N. Masmoudi)

Functional framework and sharp asymptotics

Lemma (A. Blanchet, JD, B. Perthame)

Sub-critical HLS inequality (A. Blanchet, JD, B. Perthame) F [n] :=

Z

R

n log n

n _∞

dx − 1 2

Z

R

(n − n ∞ ) (c − c ∞ ) dx ≥ 0 achieves its minimum for n = n _∞

Q 1 [f ] = lim

ε→0

1

ε ² F [n ∞ (1 + ε f )] ≥ 0 if R

R

f n _∞ dx = 0. Notice that f 0 generates the kernel of Q 1

hf, f i :=

Z

R

|f | ² n _∞ dx − Z

R

f n _∞ (g c _∞ ) dx

Eigenvalues

With g such that −∆(g c _∞ ) = f n _∞ , Q 1 determines a scalar product hf ₁ , f ₂ i :=

Z

R

f ₁ f ₂ n _∞ dx − Z

R

f ₁ n _∞ (g ₂ c _∞ ) dx on the orthogonal space to f 0 in L ² (n ∞ dx)

Q 2 [f ] :=

Z

R

|∇(f − g c _∞ )| ² n _∞ dx with g = − 1 c _∞

1

2π log |·|∗(f n _∞ ) is a positive quadratic form, whose polar operator is the self-adjoint operator L

hf, L f i = Q ₂ [f ] ∀ f ∈ D (L ₂ ) Lemma (J. Campos, JD)

L has pure discrete spectrum and its lowest eigenvalue is 1

Linearized Keller-Segel theory

L f = 1

n _∞ ∇ · n _∞ ∇(f − c _∞ g) Corollary (J. Campos, JD)

hf, f i ≤ h L f, f i The linearized problem takes the form

∂f

∂t = L f

where L is a self-adjoint operator on the orthogonal of f ₀ equipped with h·, ·i. Exponential decay:

d

dt hf, fi = − 2 h L f, f i

(J. Campos, JD, 2014) (G. E. Fernández, S. Mischler, 2016)

Reverse

Hardy-Littlewood-Sobolev

inequality

The reverse HLS inequality

For any λ > 0 and any measurable function ρ ≥ 0 on R ^N , let I λ [ρ] :=

Z Z

R

× R

|x − y| ^λ ρ(x) ρ(y) dx dy

N ≥ 1 , 0 < q < 1 , α := 2 N − q (2 N + λ) N (1 − q) Convention: ρ ∈ L ^p ( R ^N ) if R

R

|ρ(x)| ^p dx for any p > 0 Theorem

The inequality

I λ [ρ] ≥ C N,λ,q

Z

R

ρ dx α Z

R

ρ ^q dx

(2−α)/q

(1) holds for any ρ ∈ L ¹ ₊ ∩ L ^q ( R ^N ) with C N,λ,q > 0 if and only if

q > N/(N + λ)

If either N = 1, 2 or if N ≥ 3 and q ≥ min

1 − 2/N , 2N/(2N + λ) ,

then there is a radial nonnegative optimizer ρ ∈ L ¹ ∩ L ^q ( R ^N )

0 2 4 6 8 10 12 0.0

0.2 0.4 0.6 0.8 1.0

N = 4, region of the parameters (λ, q) for which C N,λ,q > 0

Optimal functions exist in the light grey area

The conformally invariant case q = 2N/(2N + λ)

I _λ [ρ] = Z Z

R

× R

|x − y| ^λ ρ(x) ρ(y) dx dy ≥ C N,λ,q

Z

R

ρ ^q dx ^2/q

2N/(2N + λ) ⇐⇒ α = 0 (Dou, Zhu 2015) (Ngô, Nguyen 2017)

The optimizers are given, up to translations, dilations and multiplications by constants, by

ρ(x) = 1 + |x| ² −N/q

∀ x ∈ R ^N and the value of the optimal constant is

C N,λ,q(λ) = 1 π

Γ ^N ₂ + ^λ ₂ Γ N + ^λ ₂

Γ(N ) Γ ^N ₂

! ¹⁺

0 2 4 6 8 10 12 0.2

0.4 0.6 0.8 1.0

N = 4, region of the parameters (λ, q) for which C N,λ,q > 0

The plain, red curve is the conformally invariant case α = 0

Z Z

R

× R

|x−y| ^λ ρ(x) ρ(y) dx dy ≥ C N,λ,q

Z

R

ρ dx ^α Z

R

ρ ^q dx

^(2−α)/q

0 2 4 6 8 10 12

0.2 0.4 0.6 0.8 1.0

α < 0

0 < α < 1

α > 1

A Carlson type inequality

Lemma

Let λ > 0 and N/(N + λ) < q < 1 Z

R

ρ dx

1−

Z

R

|x| ^λ ρ dx

≥ c _N,λ,q Z

R

ρ ^q dx

c N,λ,q = _λ ¹ _{(N +λ) q−N}

q

_{N (1−q)}

(N+λ)q−N

Γ (

) ^Γ (

)

2 π

Γ (

−

) ^Γ (

)

Equality is achieved if and only if

ρ(x) = 1 + |x| ^{λ −}

up to dilations and constant multiples

(Carlson 1934) (Levine 1948)

An elementary proof of Carlson’s inequality

Z

{|x|<R}

ρ ^q dx ≤ Z

R

ρ dx ^q

|B R | ^1−q = C ₁ Z

R

ρ dx ^q

R ^{N (1−q)}

and Z

{|x|≥R}

ρ ^q dx ≤ Z

R

|x| ^λ ρ dx q Z

{|x|≥R}

|x| ⁻

dx

! 1−q

= C 2

Z

R

|x| ^λ ρ dx ^q

R ^{−λq+N (1−q)}

and optimize over R > 0

... existence of a radial monotone non-increasing optimal function;

rearrangement; Euler-Lagrange equations

Proposition

Let λ > 0. If N/(N + λ) < q < 1, then C N,λ,q > 0

By rearrangement inequalities: prove the reverse HLS inequality for symmetric non-increasing ρ’s so that

Z

R

|x − y| ^λ ρ(y) dx ≥ Z

R

|x| ^λ ρ dx for all x ∈ R ^N implies

I λ [ρ] ≥ Z

R

|x| ^λ ρ dx Z

R

ρ dx In the range _N ^N _+λ < q < 1

I λ [ρ]

R

R

ρ(x) dx ^α ≥ Z

R

ρ dx dx 1−α Z

R

|x| ^λ ρ dx ≥ c ^2−α _N,λ,q Z

R

ρ ^q dx

and conclude with Carlson’s inequality

The case λ = 2

Corollary

Let λ = 2 and N/(N + 2) < q < 1. Then the optimizers for (1) are given by translations, dilations and constant multiples of

ρ(x) = 1 + |x| ^{2 −}

and the optimal constant is

C N,2,q = ¹ ₂ c

2q N(1−q)

N,2,q

By rearrangement inequalities it is enough to prove (7) for symmetric non-increasing ρ’s, and so R

R

x ρ dx = 0. Therefore I 2 [ρ] = 2

Z

R

ρ dx Z

R

|x| ² ρ dx

and the optimal function is optimal for Carlson’s inequality

0 2 4 6 8 10 12 0.2

0.4 0.6 0.8 1.0

N = 4, region of the parameters (λ, q) for which C N,λ,q > 0. The

dashed, red curve is the threshold case q = N/(N + λ)

The threshold case q = N/(N + λ) and below

Proposition

If 0 < q ≤ N/(N + λ), then C N,λ,q = 0 = lim _{q→N/(N +λ)}

C N,λ,q

Let ρ, σ ≥ 0 such that R

R

σ dx = 1, smooth (+ compact support) ρ ε (x) := ρ(x) + M ε ^−N σ(x/ε)

Then R

R

ρ _ε dx = R

R

ρ dx + M and, by simple estimates, Z

R

ρ ^q _ε dx → Z

R

ρ ^q dx as ε → 0 +

and

I λ [ρ ε ] → I λ [ρ] + 2M Z

R

|x| ^λ ρ dx as ε → 0 +

If 0 < q < N/(N + λ), i.e., α > 1, take ρ _ε as a trial function, C N,λ,q ≤ I λ [ρ] + 2M R

R

|x| ^λ ρ dx R

R

ρ dx + M ^α R

R

ρ ^q dx (2−α)/q =: Q [ρ, M]

J. Dolbeault Drift-Diffusion and reverse HLS inequalities

The threshold case: If α = 1, i.e., q = N/(N + λ), by taking the limit as M → + ∞, we obtain

C N,λ,q ≤ 2 R

R

|x| ^λ ρ dx R

R

ρ ^q dx (2−α)/q

For any R > 1, we take

ρ R (x) := |x| ^{−(N +λ)} 1 1≤|x|≤R (x)

Then Z

R

|x| ^λ ρ _R dx = Z

R

ρ ^q _R dx = S ^{N −1}

log R and, as a consequence,

R

R

|x| ^λ ρ _R dx R

R

ρ ^N/(N _R ^+λ) dx (N+λ)/N = S ^{N −1}

log R ^−λ/N

→ 0 as R → ∞

This proves that C N,λ,q = 0 for q = N/(N + λ)

A relaxed inequality

I λ [ρ]+2M Z

R

|x| ^λ ρ dx ≥ C N,λ,q

Z

R

ρ dx + M ^α Z

R

ρ ^q dx

(2−α)/q

(2) Proposition

If q > N/(N + λ), the relaxed inequality (2) holds with the same optimal constant C N,λ,q as (1) and admits an optimizer (ρ, M )

Heuristically, this is the extension of the reverse HLS inequality (1) I λ [ρ] ≥ C N,λ,q

Z

R

ρ dx α Z

R

ρ ^q dx

(2−α)/q

to measures of the form ρ + M δ

Existence of minimizers and

relaxation

Existence of a minimizer: first case

0 2 4 6 8 10 12

0.2 0.4 0.6 0.8 1.0

The α < 0 case: dark grey region Proposition

If λ > 0 and _2N ^2N _+λ < q < 1, there is a minimizer ρ for C N,λ,q

The limit case α = 0, q = _2N+λ ^2N is the conformally invariant case: see

(Dou, Zhu 2015) and (Ngô, Nguyen 2017)

A minimizing sequence ρ _j can be taken radially symmetric non-increasing by rearrangement, and such that

Z

R

ρ _j (x) dx = Z

R

ρ _j (x) ^q dx = 1 for all j ∈ N Since ρ _j (x) ≤ C min

|x| ^−N , |x| ^−N/q by Helly’s selection theorem we may assume that ρ _j → ρ a.e., so that

lim inf

j→∞ I λ [ρ j ] ≥ I λ [ρ] and 1 ≥ Z

R

ρ(x) dx

by Fatou’s lemma. Pick p ∈ (N/(N + λ), q) and apply (1) with the same λ and α = α(p):

I λ [ρ j ] ≥ C N,λ,p

Z

R

ρ ^p _j dx

(2−α(p))/p

Hence the ρ j are uniformly bounded in L ^p ( R ^N ): ρ j (x) ≤ C ⁰ |x| ^−N/p , Z

R

ρ ^q _j dx → Z

R

ρ ^q dx = 1

by dominated convergence

Existence of a minimizer: second case

If N/(N + λ) < q < 2N/(2N + λ) we consider the relaxed inequality I λ [ρ] + 2M R

R

|x| ^λ ρ dx ≥ C N,λ,q

R

R

ρ dx + M α R

R

ρ ^q dx (2−α)/q

0 2 4 6 8 10 12

0.2 0.4 0.6 0.8 1.0

The 0 < α < 1 case: dark grey region Proposition

If q > N/(N + λ), the relaxed inequality holds with the same optimal

constant C N,λ,q as (1) and admits an optimizer (ρ, M )

Let (ρ _j , M _j ) be a minimizing sequence with ρ _j radially symmetric non-increasing by rearrangement, such that

Z

R

ρ j dx + M j = Z

R

ρ ^q _j = 1

Local estimates + Helly’s selection theorem: ρ j → ρ almost everywhere and M j → M := L + lim _j→∞ M j , so that R

R

ρ dx + M = 1, and R

R

ρ(x) ^q dx = 1

We cannot invoke Fatou’s lemma because α ∈ (0, 1): let dµ j := ρ j dx µ j R ^N \ B R (0)

= Z

{|x|≥R}

ρ j dx ≤ C Z

{|x|≥R}

dx

|x| ^N/q = C ⁰ R ^{−N (1−q)/q} µ j are tight: up to a subsequence, µ j → µ weak * and dµ = ρ dx + L δ

lim inf

j→∞ I λ [ρ j ] ≥ I λ [ρ] + 2M Z

R

|x| ^λ ρ dx ,

lim inf

j→∞

Z

R

|x| ^λ ρ j dx ≥ Z

R

|x| ^λ ρ dx

Conclusion: lim inf _j→∞ Q [ρ j , M j ] ≥ Q [ρ, M]

Optimizers are positive

Q [ρ, M] := I λ [ρ] + 2M R

R

|x| ^λ ρ dx R

R

ρ dx + M α R

R

ρ ^q dx (2−α)/q

Lemma

Let λ > 0 and N/(N + λ) < q < 1. If ρ ≥ 0 is an optimal function for some M > 0, then ρ is radial (up to a translation), monotone

non-increasing and positive a.e. on R ^N

If ρ vanishes on a set E ⊂ R ^N of finite, positive measure, then Q

ρ, M + ε 1 E

= Q [ρ, M ]

1 − 2 − α q

|E|

R

R

ρ(x) ^q dx ε ^q + o(ε ^q )

as ε → 0 + , a contradiction if (ρ, M) is a minimizer of Q

Euler–Lagrange equation

Euler–Lagrange equation for a minimizer (ρ _∗ , M _∗ ) 2 R

R

|x − y| ^λ ρ _∗ (y) dy + M _∗ |x| ^λ I λ [ρ _∗ ] + 2M _∗ R

R

|y| ^λ ρ _∗ dy − α R

R

ρ _∗ dy + M _∗ − (2 − α) ρ _∗ (x) ^−1+q R

R

ρ _∗ (y) ^q dy = 0 We can reformulate the question of the optimizers of (1) as: when is it true that M ∗ = 0 ? We already know that M ∗ = 0 if

2N

2N + λ < q < 1

Regions of no concentration and regularity of measure

valued minimizers

0 2 4 6 8 10 12 0.0

0.2 0.4 0.6 0.8 1.0

λ q

q = ^N _N ⁻²

q=¯ q(λ,N )

q = _2N ^2N _+λ

q = _N+λ ^N

No concentration 1

0 2 4 6 8 10 12

0.0 0.2 0.4 0.6 0.8 1.0

Proposition

Let N ≥ 1, λ > 0 and N

N + λ < q < 2N 2N + λ

If N ≥ 3 and λ > 2N/(N − 2), assume further that q ≥ N − 2

N

If (ρ _∗ , M _∗ ) is a minimizer, then M _∗ = 0

Two ingredients of the proof

Based on the Brézis–Lieb lemma Lemma

Let 0 < q < p, let f ∈ L ^p ∩ L ^q ( R ^N ) be a symmetric non-increasing function and let g ∈ L ^q ( R ^N ). Then, for any τ > 0, as ε → 0 + ,

Z

R

f (x) + ε ^−N/p τ g(x/ε)

q

dx = Z

R

f ^q dx + ε ^N(1−q/p) τ ^q

Z

R

|g| ^q dx + o

ε ^{N (1−q/p)} τ ^q

I λ

ρ ∗ + ε ^−N τ σ(·/ε)

+ 2 (M ∗ − τ) Z

R

|x| ^λ ρ ∗ (x) + ε ^−N τ σ(x/ε) dx

=I _λ [ρ _∗ ]+2 M _∗ Z

R

|x| ^λ ρ _? dx + (

2 τ RR

R

×R

ρ _∗ (x) |x − y| ^λ − |x| ^{λ σ} (

)

ε

dx dy + ε ^λ τ ² I λ [σ] + 2 (M _∗ − τ ) τ ε ^λ R

R

|x| ^λ σ dx

| {z }

=O(ε

τ) with β:=min{2,λ}

Regularity and concentration

0 2 4 6 8 10 12

0.0 0.2 0.4 0.6 0.8 1.0

Proposition

If N ≥ 3, λ > 2N/(N − 2) and N

N + λ < q < min

N − 2

N , 2N

2N + λ

,

and (ρ _∗ , M _∗ ) ∈ L ^{N (1−q)/2} ( R ^N ) × [0, + ∞) is a minimizer, then M _∗ = 0

Regularity

Proposition

Let N ≥ 1, λ > 0 and N/(N + λ) < q < 2N/(2N + λ) Let (ρ _∗ , M _∗ ) be a minimizer

1

If R

R

ρ _∗ dx > ^α ₂ R ^I

^[ρ

^]

RN

|x|

ρ

dx , then M _∗ = 0 and ρ _∗ , bounded and ρ _∗ (0) = (2 − α)I λ [ρ _∗ ] R

R

ρ _∗ dx R

R

ρ ^q _∗ dx 2 R

R

|x| ^λ ρ _∗ dx R

R

ρ _∗ dx − αI λ [ρ _∗ ]

!

2

If R

R

ρ _∗ dx = ^α ₂ R ^I

^[ρ

^]

RN

|x|

ρ

dx , then M _∗ = 0 and ρ _∗ is unbounded

3

If R

R

ρ _∗ dx < ^α ₂ R ^I

^[ρ

^]

RN

|x|

ρ

dx , then ρ _∗ is unbounded and M ∗ = αI _λ [ρ _∗ ] − 2 R

R

|x| ^λ ρ _∗ dx R

R

ρ _∗ dx 2 (1 − α) R

R

|x| ^λ ρ _∗ dx > 0

An ingredient of the proof

Lemma

For constants A, B > 0 and 0 < α < 1, define f (M ) = A + M

(B + M ) ^α for M ≥ 0

Then f attains its minimum on [0, ∞) at M = 0 if α A ≤ B and at

M = (α A − B)/(1 − α) > 0 if α A > B

No concentration 2

For any λ ≥ 1 we deduce from

|x − y| ^λ ≤ |x| + |y| ^λ

≤ 2 ^λ−1 |x| ^λ + |y| ^λ that

I λ [ρ] < 2 ^λ Z

R

|x| ^λ ρ dx Z

R

ρ(x) dx For all α ≤ 2 ^−λ+1 , we infer that M _∗ = 0 if

q ≥ 2N 1 − 2 ^−λ 2N 1 − 2 ^−λ

+ λ

0 2 4 6 8 10 12

0.0 0.2 0.4 0.6 0.8 1.0

No concentration 3

Layer cake representation (superlevel sets are balls) I λ [ρ] ≤ 2 A N,λ

Z

R

|x| ^λ ρ dx Z

R

ρ(x) dx

A N,λ := sup

0≤R,S<∞

RR

B

×B

|x − y| ^λ dx dy

|B R | R

B

|x| ^λ dx + |B S | R

B

|y| ^λ dy

0 2 4 6 8 10 12

0.0 0.2 0.4 0.6 0.8 1.0

Proposition

Assume that N ≥ 3 and λ > 2N/(N − 2) and observe that N

N + λ < q(λ, N ¯ ) ≤ 2N 1 − 2 ^−λ 2N 1 − 2 ^−λ

+ λ < 2 N 2 N + λ for λ > 2 large enough. If

max

q(λ, N ¯ ), N N + λ

< q < N − 2 N

and if (ρ ∗ , M ∗ ) is a minimizer, then M ∗ = 0 and ρ ∗ ∈ L ^∞ ( R ^N )

More on regularity

Lemma

Assume that ρ _∗ is an unbounded minimizer if λ < 2, there is a constant c > 0 such that

ρ _∗ (x) ≥ c |x| ^{−λ/(1−q)} as x → 0 if λ ≥ 2, there is a constant C > 0 such that

ρ _∗ (x) = C |x| ^−2/(1−q) 1 + o(1)

as x → 0 Corollary

q 6= 2N

2N + λ , N

N + λ < q < 1 and q ≥ N − 2

N if N ≥ 3

If ρ ∗ is a minimizer for C N,λ,q , then ρ ∗ ∈ L ^∞ ( R ^N )

0 2 4 6 8 10 12 0.0

0.2 0.4 0.6 0.8 1.0

λ q

q = ^N _N ⁻²

q=¯ q(λ,N )

q = _2N ^2N _+λ

q = _N+λ ^N

Free energy point of view

A toy model

Assume that u solves the fast diffusion with external drift V given by

∂u

∂t = ∆u ^q + ∇ · u ∇V

To fix ideas: V (x) = 1 + ¹ ₂ |x| ² + _λ ¹ |x| ^λ . Free energy functional F [u] :=

Z

R

V u dx − 1 1 − q

Z

R

u ^q dx Under the mass constraint M = R

R

u dx, smooth minimizers are u µ (x) = µ + V (x) −

The equation can be seen as a gradient flow d

dt F [u(t, ·)] = − Z

R

u

q

1−q ∇u ^q−1 − ∇V

2

dx

A toy model (continued)

If λ = 2, the so-called Barenblatt profile u µ has finite mass if and only if

q > q c := N − 2 N

For λ > 2, the integrability condition is q > 1 − λ/N but q = q c is a threshold for the regularity: the mass of u µ = (µ + V ) ^1/(1−q) is

M (µ) :=

Z

R

u µ dx ≤ M ? = Z

R

1

2 |x| ² + ¹ _λ |x| ^{λ −}

dx If one tries to minimize the free energy under the mass contraint R

R

u dx = M for an arbitrary M > M ? , the limit of a minimizing sequence is the measure

M − M ?

δ + u −1

A model for nonlinear springs: heuristics

V = ρ ∗ W λ , W λ (x) := _λ ¹ |x| ^λ

is motivated by the study of the nonnegative solutions of the evolution equation

∂ρ

∂t = ∆ρ ^q + ∇ · (ρ ∇W _λ ∗ ρ)

Optimal functions for (1) are energy minimizers (eventually measure valued) for the free energy functional

F [ρ] := 1 2 Z

R

ρ (W λ ∗ρ) dx− 1 1 − q

Z

R

ρ ^q dx = 1

2λ I λ [ρ]− 1 1 − q

Z

R

ρ ^q dx under a mass constraint M = R

R

ρ dx while smooth solutions obey to d

dt F [ρ(t, ·)] = − Z

R

ρ

q

1−q ∇ρ ^q−1 − ∇W λ ∗ ρ

2

dx

Free energy or minimization of the quotient

F [ρ] = − 1 1 − q

Z

R

ρ ^q dx + 1 2 λ I λ [ρ]

If 0 < q ≤ N/(N + λ), then C N,λ,q = 0: take test functions ρ n ∈ L ¹ ₊ ∩ L ^q ( R ^N ) such that kρ n k _L

( R

) = I λ [ρ n ] = 1 and R

R

ρ ^q _n dx = n ∈ N

n→+∞ lim F [ρ n ] = − ∞

If N/(N + λ) < q < 1, ρ <sub>`</sub> (x) := ` ^−N ρ(x/`)/kρk _L

( R

)

F [ρ ` ] = − ` ^(1−q)N A + ` <sup>λ</sup> B has a minimum at ` = ` ? and

F [ρ] ≥ F [ρ `

] = − κ ? (Q q,λ [ρ]) ⁻

N(1−q) λ−N(1−q)

Proposition

F is bounded from below if and only if C N,λ,q > 0

Relaxed free energy F ^rel [ρ, M ] := − 1

1 − q Z

R

ρ ^q dx + 1

2 λ I λ [ρ] + M λ

Z

R

|x| ^λ ρ dx

Corollary

Let q ∈ (0, 1) and N/(N + λ) < q < 1 inf

F ^rel [ρ, M ] : 0 ≤ ρ ∈ L ¹ ∩ L ^q ( R ^N ) , M ≥ 0 , Z

R

ρ dx + M = 1

is achieved by a minimizer of (2) such that R

R

ρ _∗ dx + M _∗ = 1 and I λ [ρ ∗ ] + 2 M ∗

Z

R

|x| ^λ ρ ∗ dx = 2 N Z

R

ρ ^q _∗ dx

Uniqueness

Proposition

Let N/(N + λ) < q < 1 and assume either that (N − 1)/N < q < 1 and λ ≥ 1, or 2 ≤ λ ≤ 4. Then the minimizer of

F ^rel [ρ, M ] := 1

2 λ I _λ [ρ] + M λ

Z

R

|x| ^λ ρ dx − 1 1 − q

Z

R

ρ ^q dx

is unique up to translation, dilation and multiplication by a positive

constant

If (N − 1)/N < q < 1 and λ ≥ 1, the lower semi-continuous extension of F to probability measures is strictly geodesically convex in the Wasserstein-p metric for p ∈ (1, 2)

By strict rearrangement inequalities a minimizer (ρ, M ) such that M ∈ [0, 1) of the relaxed free energy F ^rel is (up to a translation) such that ρ is radially symmetric and R

R

x ρ dx = 0 Let (ρ, M ) and (ρ ⁰ , M ⁰ ) be two minimizers and

[0, 1] 3 t 7→ f (t) := F ^rel

(1 − t) ρ + t ρ ⁰ , (1 − t) M + t M ⁰

f ⁰⁰ (t) = 1

λ I λ [ρ ⁰ − ρ] + 2

λ (M ⁰ − M ) Z

R

|x| ^λ (ρ ⁰ − ρ) dx + q

Z

R

(1 − t) ρ + t ρ ^{0 q−2}

(ρ ⁰ − ρ) ² dx (Lopes, 2017) I _λ [h] ≥ 0 if 2 ≤ λ ≤ 4, for all h such that

R

R

1 + |x| ^λ

|h| dx < ∞ with R

R

h dx = 0 and R

R

x h dx = 0

N = 4

0 2 4 6 8 10 12

0.0 0.2 0.4 0.6 0.8 1.0

N = 10

2 4 6 8 10 12

0.0 0.2 0.4 0.6 0.8 1.0

References

J. Dou and M. Zhu. Reversed Hardy-Littlewood-Sobolev inequality. Int. Math. Res. Not. IMRN, 2015(19):9696-9726, 2015

Q.A. Ngô and V. Nguyen. Sharp reversed

Hardy-Littlewood-Sobolev inequality on R ⁿ . Israel J. Math., 220 (1):189-223, 2017

J. A. Carrillo and M. Delgadino. Free energies and the reversed HLS inequality. ArXiv e-prints, Mar. 2018 # 1803.06232

J. Dolbeault, R. Frank, and F. Hoffmann. Reverse

Hardy-Littlewood-Sobolev inequalities. ArXiv e-prints, Mar. 2018 # 1803.06151

J. A. Carrillo, M. Delgadino, J. Dolbeault, R. Frank, and F.

Hoffmann. Reverse Hardy-Littlewood-Sobolev inequalities. In

preparation.

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