Stability in Sobolev and related inequalities

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Stability in Sobolev and related inequalities

Jean Dolbeault

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

June 15, 2018

CIMI – EFI workshop onStability of functional inequalities and applications(13-15/6/2018)

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Critical Sobolev and HLS inequalities Subcritical interpolation inequalities Reverse Hardy-Littewood-Sobolev inequality

Outline

The stability issue in critical Sobolev and related inequalities Sobolev and Hardy-Littlewood-Sobolev inequalities Joint work with G. Jankowiak

Subcritical interpolation inequalities

BOn the Euclidean space: joint work with G. Toscani BOn the sphere: joint work with M.J. Esteban and M. Loss Reverse HLS inequality

BA quick introduction to a new family of inequalities for mean-field diffusion equations

J. Dolbeault Sobolev and related inequalities

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A question by H. Brezis and E. Lieb

(Brezis, Lieb (1985))Is there a natural way to bound

Sdk∇uk²_L2(R^d)− kuk²_L2∗

(R^d)

from below in terms of the“distance” off from the set of optimal [Aubin-Talenti] functionswhend≥3 ?

(Bianchi, Egnell 1990)There is a positive constantαsuch that Sdk∇uk²_L2(R^d)− kuk²_L2∗(R^d)≥α inf

ϕ∈Mk∇u− ∇ϕk²_L2(R^d)

(Cianchi, Fusco, Maggi, Pratelli 2009)(also a version for k∇uk^p_Lp(R^d)) There are constantsαandκsuch that

Sdk∇uk²_L2(R^d)≥(1 +κ λ(u)^α)kuk²_L2∗

(R^d)

where λ(u) = inf_ϕ∈_M

_ku−ϕk2∗ L2∗

(Rd)

kuk²^∗

L2∗ (Rd)

: kuk²_L^∗₂∗

(R^d)=kϕk²_L^∗₂∗

(R^d)

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Duality Yamabe flow

Sobolev and Hardy-Littlewood-Sobolev inequalities

BStability in a weaker norm but with explicit constants BFrom duality to improved estimates based on Yamabe’s flow

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Duality Yamabe flow

Sobolev and HLS

As it has been noticed by E. Lieb, Sobolev’s inequality inR^d,d≥3, kuk²_L2∗(R^d)≤S_dk∇uk²_L2(R^d) ∀u∈D^1,2(R^d) (1) and the Hardy-Littlewood-Sobolev inequality

Sdkvk²

L²

d d+2(R^d)

≥ Z

R^d

v(−∆)⁻¹v dx ∀v∈L^d+2²^d(R^d) (2) aredual of each other. HereS_d is the Aubin-Talenti constant and 2^∗= _d−2²^d

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Duality Yamabe flow

Improved Sobolev inequality by duality

Theorem

(JD, G. Jankowiak)Assume thatd≥3 and letq= ^d+2_d−2. There exists a positive constantC≤1 such that

S_dkw^qk²

L^d+2²^d(R^d)

− Z

R^d

w^q(−∆)⁻¹w^qdx

≤CS_dkwk

8 d−2

L²^∗(R^d)

hk∇wk²_L2(R^d)−S_dkwk²_L2∗

(R^d)

i

for anyw∈D^1,2(R^d)

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Duality Yamabe flow

Proof: the completion of a square

Integrations by parts show that Z

R^d

|∇(−∆)⁻¹v|²dx= Z

R^d

v(−∆)⁻¹v dx and, ifv=u^q withq=^d+2_d−2,

Z

R^d

∇u· ∇(−∆)⁻¹v dx= Z

R^d

u v dx= Z

R^d

u²^∗dx

Hence the expansion of the square 0≤

Z

R^d

Sdkuk

4 d−2

L²^∗(R^d)∇u− ∇(−∆)⁻¹v

2

dx

shows that 0≤Sdkuk

8 d−2

L²^∗(R^d)

h

(R^d)

i

−h

Sdku^qk²

L

2d d+2(R^d)

− Z

R^d

u^q(−∆)⁻¹u^qdxi

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Duality Yamabe flow

Using a nonlinear flow to relate Sobolev and HLS

Consider thefast diffusionequation

∂v

∂t = ∆v^m t >0, x∈R^d (3) If we defineH(t) :=Hd[v(t,·)], with

Hd[v] :=

Z

R^d

v(−∆)⁻¹v dx−Sdkvk²

L

2d d+2(R^d)

then we observe that 1

2H⁰ =− Z

R^d

v^m+1dx+Sd

Z

R^d

v^d+2²^d dx ²_dZ

R^d

∇v^m· ∇v^d−2^d+2dx wherev=v(t,·) is a solution of (3). With the choicem= ^d−2_d+2, we find thatm+ 1 = _d+2²^d

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Duality Yamabe flow

A preliminary observation

Proposition

(JD)Assume thatd≥3 andm= ^d−2_d+2. Ifv is a solution of (3)with nonnegative initial datum inL^2d/(d+2)(R^d), then

1 2

d dt

Z

R^d

v(−∆)⁻¹v dx−S_dkvk²

L^d+2²^d(R^d)

= Z

R^d

v^m+1dx ²_dh

Sdk∇uk²_L2(R^d)− kuk²_L2∗(R^d)

i≥0

The HLS inequality amounts toH≤0 and appears as a consequence of Sobolev, that isH⁰ ≥0 if we show that lim sup_t>0H(t) = 0

Notice thatu=v^mis an optimal function for (1) ifvis optimal for (2)

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Duality Yamabe flow

Improved Sobolev inequality

By integrating along the flow defined by (3), we can actually obtain optimal integral remainder terms which improve on the usual Sobolev inequality (1), but only whend≥5 for integrability reasons

Theorem

(JD)Assume thatd≥5 and letq= ^d+2_d−2. There exists a positive constantC≤ 1 +_d²

1−e^−d/2

Sd such that

Sdkw^qk²

L^d+2²^d(R^d)

− Z

R^d

≤Ckwk_L^d−2₂⁸∗(R^d)

hk∇wk²_L2(R^d)−S_dkwk²_L2∗(R^d)

i

for anyw∈D^1,2(R^d)

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Duality Yamabe flow

Solutions with separation of variables

Consider the solution of ^∂v_∂t = ∆v^m vanishing att=T: vT(t, x) =c(T−t)^α(F(x))^d+2^d−2 whereF is the Aubin-Talenti solution of

−∆F =d(d−2)F(d+2)/(d−2)

Letkvk_∗:= sup_x∈_Rd(1 +|x|²)^d+2|v(x)|

Lemma

(M. del Pino, M. Saez),(J. L. Vázquez, J. R. Esteban, A. Rodriguez) For any solutionv with initial datumv0∈L^2d/(d+2)(R^d),v0>0, there existsT >0,λ >0 andx0∈R^d such that

t→Tlim−

(T−t)⁻^1−m¹ kv(t,·)/v(t,·)−1k∗= 0

withv(t, x) =λ^(d+2)/2vT(t,(x−x₀)/λ)

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Duality Yamabe flow

Improved inequality: proof (1/2)

The functionJ(t) :=R

R^dv(t, x)^m+1dxsatisfies J⁰=−(m+ 1)k∇v^mk²_L2(R^d)≤ −m+ 1

Sd

J¹⁻²^d

Ifd≥5, then we also have J⁰⁰= 2m(m+ 1)

Z

R^d

v^m−1(∆v^m)²dx≥0 Notice that

J⁰

J ≤ −m+ 1

S_d J⁻²^d ≤ −κ with κ T = 2d d+ 2

T S_d

Z

R^d

v^m+1₀ dx −²_d

≤ d 2

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Duality Yamabe flow

Improved inequality: proof (2/2)

By theCauchy-Schwarz inequality, we have J⁰²

(m+ 1)² =k∇v^mk⁴_L2(R^d)= Z

R^d

v^(m−1)/2∆v^m·v^(m+1)/2dx 2

≤ Z

R^d

v^m−1(∆v^m)²dx Z

R^d

v^m+1dx=CstJ⁰⁰J so thatQ(t) :=k∇v^m(t,·)k²_L2(R^d)

R

R^dv^m+1(t, x)dx^−(d−2)/d is monotone decreasing, and

H⁰ = 2J(SdQ−1), H⁰⁰= J⁰

J H⁰+ 2J SdQ⁰≤ J⁰ J H⁰≤0 H⁰⁰≤ −κH⁰ with κ= 2d

d+ 2 1 Sd

Z

R^d

v₀^m+1dx −2/d

By writing that−H(0) =H(T)−H(0)≤H⁰(0) (1−e^{−κ T})/κand using the estimateκ T ≤d/2, the proof is completed

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Duality Yamabe flow

d = 2: Onofri’s and log HLS inequalities

H2[v] :=

Z

R²

(v−µ) (−∆)⁻¹(v−µ)dx− 1 4π

Z

R²

vlog v

µ

dx Withµ(x) := _π¹(1 +|x|²)⁻². Assume thatv is a positive solution of

∂v

∂t = ∆ log (v/µ) t >0, x∈R² Proposition

Ifv=µ e^u/2 is a solution with nonnegative initial datumv₀ in L¹(R²) such thatR

R²v0dx= 1,v0logv0∈L¹(R²)andv0 logµ∈L¹(R²), then

d

dtH₂[v(t,·)] = 1 16π

Z

R²

|∇u|²dx− Z

R²

e^u² −1 u dµ

≥ ₁₆¹_πR

R²|∇u|²dx+R

R²u dµ−log R

R²e^udµ

≥0

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Duality Yamabe flow

Another improvement

J_d[v] :=

Z

R^d

v^d+2²^d dx and H_d[v] :=

Z

R^d

v(−∆)⁻¹v dx−Sdkvk²

L^d+2²^d(R^d)

Theorem (J.D., G. Jankowiak)

Assume thatd≥3. Then we have 0≤Hd[v] +SdJd[v]¹⁺²^dϕ

Jd[v]^d²⁻¹ h

(R^d)

i

∀u∈D, v=u^d+2^d−2 whereϕ(x) :=√

C²+ 2Cx−C for anyx≥0 Proof: H(t) =−Y(J(t))∀t∈[0, T),κ0:= ^H

0 0

J₀ and consider the differential inequality

Y⁰

CSds¹⁺²^d+Y

≤ d+ 2

2d Cκ0S²_ds¹⁺⁴^d, Y(0) = 0, Y(J0) =−H0

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Duality Yamabe flow

... but C = 1 is not optimal

Theorem

(JD, G. Jankowiak)In the inequality

Sdkw^qk²

L^d+2²^d(R^d)

− Z

R^d

≤C_dS_dkwk

8 d−2

L²^∗(R^d)

hk∇wk²_L2(R^d)−S_dkwk²_L2∗(R^d)

i

we have

d

d+ 4 ≤Cd<1

based on a (painful) linearization Extensions:

fractional Laplacian operator (Jankowiak, Nguyen) Moser-Trudinger-Onofri inequality

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Fast diffusion equations and best matching onRd Improved interpolation inequalities on the sphere

Subcritical interpolation inequalities

BEuclidean space: fast diffusion, entropies and improved asymptotic expansions

Based on papers with A. Blanchet, M. Bonforte, G. Grillo, J.L. Vázquez and papers with G. Toscani

BSphere: explicit remainder terms based on nonlinear diffusions Joint work with MJ. Esteban and M. Loss

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Higher order matching asymptotics

(J.D., G. Toscani)For somem∈(mc,1) with mc:= (d−2)/d, we consider onR^d the fast diffusion equation

∂u

∂τ +∇ · u∇u^m−1

= 0

The strategy is easy to understand using a time-dependent rescaling and the relative entropy formalism. Define the functionvsuch that

u(τ, y+x0) =R^−dv(t, x), R=R(τ), t= ¹₂ logR , x= y R Thenvhas to be a solution of

∂v

∂t +∇ ·h v

σ^d²^(m−m^c⁾∇v^m−1−2xi

= 0 t >0, x∈R^d with (as long as we make no assumption onR)

2σ⁻^d²^(m−m^c⁾=R^1−d^(1−m)dR dτ

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Refined relative entropy

Consider the family of the Barenblatt profiles B_σ(x) :=σ⁻^d² C_M+_σ¹|x|²_m−1¹

∀x∈R^d (4) Note thatσis a function oft: as long as ^dσ_dt 6= 0, the Barenblatt profile Bσ is nota solution but we may still consider the relative entropy

Fσ[v] := 1 m−1

Z

R^d

v^m−B_σ^m−m B_σ^m−1(v−Bσ) dx Let us briefly sketch the strategy of our method before giving all details

The time derivative of this relative entropy is d

dtFσ(t)[v(t,·)] = dσ dt

d dσFσ[v]

|σ=σ(t)

| {z } choose it = 0

⇐⇒ MinimizeFσ[v] w.r.t.σ ⇐⇒R

R^d|x|²B_σdx=R

R^d|x|²v dx

+ m

m−1 Z

R^d

v^m−1−B_σ(t)^m−1∂v

∂tdx

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The entropy / entropy production estimate

According to the definition ofBσ, we know that 2x=σ^d²^(m−m^c⁾∇B^m−1_σ

Using the new change of variables, we know that d

dtFσ(t)[v(t,·)] =−m σ(t)^d²^(m−m^c⁾ 1−m

Z

R^d

v ∇h

v^m−1−B_σ(t)^m−1i

2

dx

Letw:=v/B_σand observe that the relative entropy can be written as Fσ[v] = m

1−m Z

R^d

h

w−1− 1

m w^m−1i B_σ^mdx

(Repeating) define therelative Fisher informationby Iσ[v] :=

Z

R^d

1 m−1∇

(w^m−1−1)B_σ^m−1

2

Bσw dx

so that d

dtFσ(t)[v(t,·)] =−m(1−m)σ(t)Iσ(t)[v(t,·)] ∀t >0 When linearizing, one more mode is killed andσ(t)scales out

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Improved rates of convergence

Theorem (J.D., G. Toscani)

Letm∈(me₁,1),d≥2,v₀∈L¹₊(R^d)such that v₀^m,|y|²v₀∈L¹(R^d) E[v(t,·)]≤C e⁻²^γ(m)t ∀t≥0

where

γ(m)=











((d−2)m−(d−4))²

4 (1−m) ifm∈(me₁,me₂] 4 (d+ 2)m−4d ifm∈[me2, m2]

4 ifm∈[m2,1)

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Spectral gaps and best constants

mc=^d−2_d 0

m1=^d−1_d

m2=^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

(A. Blanchet, M. Bonforte, J.D., G. Grillo, J.-L. Vázquez)

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Best matching Barenblatt profiles

(Repeating) Consider thefast diffusion equation

∂u

∂t +∇ ·h u

σ^d²^(m−m^c⁾∇u^m−1−2xi

= 0 t >0, x∈R^d with a nonlocal, time-dependent diffusion coefficient

σ(t)= 1 K_M

Z

R^d

|x|²u(x, t)dx , KM :=

Z

R^d

|x|²B1(x)dx where

Bλ(x) :=λ⁻^d² CM+_λ¹|x|²_m−1¹

∀x∈R^d and define the relative entropy

Fλ[u] := 1 m−1

Z

R^d

u^m−B_λ^m−m B^m−1_λ (u−B_λ) dx

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Three ingredients for global improvements

1 infλ>0Fλ[u(x, t)] =Fσ(t)[u(x, t)] so that d

dtFσ(t)[u(x, t)] =−Jσ(t)[u(·, t)]

where the relative Fisher information is Jλ[u] :=λ^d²^(m−m^c⁾ m

1−m Z

R^d

u

∇u^m−1− ∇B_λ^m−1

2dx

2 In the Bakry-Emery method, there isan additional (good) term 4

"

1 + 2Cm,d

Fσ(t)[u(·, t)]

M^γσ₀^d²^(1−m)

# d

dt Fσ(t)[u(·, t)]

≥ d

dt Jσ(t)[u(·, t)]

3 TheCsiszár-Kullback inequalityis also improved Fσ[u]≥ m

8R

R^dB^m₁ dxC_M² ku−Bσk²_L1(R^d)

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improved decay for the relative entropy

0.0 0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

t!→f(t)e^4t f(0)

t (a)

(b) (c)

Figure:Upper bounds on the decay of the relative entropy:

t7→f(t)e^4t/f(0) (a): estimate given by the entropy-entropy production method

(b): exact solution of a simplified equation

(c): numerical solution (found by a shooting method)

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A Csiszár-Kullback(-Pinsker) inequality

Letm∈(me₁,1) withme₁= _d+2^d and consider the relative entropy Fσ[u] := 1

m−1 Z

R^d

u^m−B_σ^m−m B^m−1_σ (u−Bσ) dx

Theorem

Letd≥1,m∈(me1,1)and assume thatuis a nonnegative function in L¹(R^d)such that u^m andx7→ |x|²uare both integrable onR^d. If kuk_L1(R^d)=M andR

R^d|x|²u dx=R

R^d|x|²Bσdx, then Fσ[u]

σ^d²^(1−m) ≥ m 8R

R^dB₁^mdx

C_Mku−B_σk_L1(R^d)+1 σ Z

R^d

|x|²|u−B_σ|dx 2

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An improved Gagliardo-Nirenberg inequality: setting

The inequality

kfk_L2p(R^d)≤C^GNp,dk∇fk^θ_L2(R^d)kfk^1−θ_L_p+1₍

R^d)

withθ=θ(p) :=^p−1_p _d+2−p^d_(d−2), 1< p≤ _d−2^d ifd≥3 and 1< p <∞ ifd= 2, can be rewritten, in a non-scale invariant form, as

Z

R^d

|∇f|²dx+ Z

R^d

|f|^p+1dx≥K_p,d Z

R^d

|f|²^pdx ^γ

withγ=γ(p, d) := ^d+2−p_d−p_(d−4)^(d−2). Optimal function are given by f_M,y,σ(x) = 1

σ^d²

C_M+|x−y|² σ

⁻p−1¹

∀x∈R^d whereCM is determined byR

R^df_M,y,σ²^p dx=M Md:=

fM,y,σ : (M, y, σ)∈Md:= (0,∞)×R^d×(0,∞)

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An improved Gagliardo-Nirenberg inequality

Relative entropyfunctional R^(p)[f] := inf

g∈M^(p)_d

Z

R^d

hg^1−p |f|²^p−g²^p

−_p+1²^p |f|^p+1−g^p+1i dx

Theorem

Letd≥2,p >1 and assume thatp < d/(d−2) ifd≥3. If R

R^d|x|²|f|²^pdx R

R^d|f|²^pdxγ =^d^(p−1)^σ^∗^M

γ−1

∗

d+2−p(d−2) , σ∗(p) :=

4^d+2−p_(p−1)2^(d−2)(p+1)

_d−p(d−4)⁴^p

for anyf ∈L^p+1∩D^1,2(R^d), then we have Z

R^d

|∇f|²dx+

Z

R^d

|f|^p+1dx−Kp,d

Z

R^d

|f|²^pdx γ

≥Cp,d

R^(p)[f]² R

R^d|f|²^pdx^γ By Csiszár-Kullback: control of

|f|²^p−g²^p

4 L¹(R^d)

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Best matching Barenblatt profiles are delayed

Letube such that

v(τ, x) = µ^d R(D τ)^du

1

2logR(D τ), µ x R(D τ)

withτ7→R(τ) given as the solution to 1

R dR dτ =

µ² KM

Z

R^d

|x|²v(τ, x)dx

⁻^d₂^(m−mc)

, R(0) = 1 Then

1 R

dR dτ =

R²(τ)σ

1

2logR(D τ)

⁻^d₂^(m−mc)

that isR(τ) =R0(τ)≤R0(τ) where _R¹ ^dR_dτ⁰ = R²₀(τ)σ(0)−^d₂(m−mc)

and asymptotically asτ → ∞,R(τ) =R0(τ−δ) for somedelayδ >0

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The interpolation inequalities on S

^d

On thed-dimensional sphere, let us consider the interpolation inequality

k∇uk²_L2(S^d)+ d

p−2kuk²_L2(S^d)≥ d

p−2kuk²_Lp(S^d) ∀u∈H¹(S^d, dµ) where the measuredµis the uniform probability measure on S^d⊂R^d+1 corresponding to the measure induced by the Lebesgue measure onR^d+1, and the exposantp≥1,p6= 2, is such that

p≤2^∗:= 2d d−2

ifd≥3. We adopt the convention that 2^∗=∞ifd= 1 ord= 2. The casep= 2 corresponds to the logarithmic Sobolev inequality

k∇uk²_L2(S^d)≥d 2

Z

S^d

|u|² log |u|² kuk²_L₂₍

S^d)

!

dµ ∀u∈H¹(S^d, dµ)\ {0}

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The Bakry-Emery method

Entropy functional

Ep[ρ] := _p−2¹ h R

S^dρ²^p dµ− R

S^dρ dµ²_pi

if p6= 2 E2[ρ] :=R

S^dρlog

ρ kρk_{L1 (S}d)

dµ Fisher informationfunctional

Ip[ρ] :=R

S^d|∇ρ¹^p|²dµ

Bakry-Emery (carré du champ) method: use the heat flow

∂ρ

∂t = ∆ρ

and compute _dt^dEp[ρ] =−Ip[ρ] and _dt^dIp[ρ]≤ −dIp[ρ] to get d

dt(Ip[ρ]−dEp[ρ])≤0 =⇒ Ip[ρ]≥dEp[ρ]

withρ=|u|^p, ifp≤2^#:=_(d−1)²^d²⁺¹₂

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The evolution under the fast diffusion flow

To overcome the limitationp≤2^#, one can consider a nonlinear diffusion of fast diffusion / porous medium type

∂ρ

∂t = ∆ρ^m. (5)

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

dt

Ip[ρ]−dEp[ρ]

≤0

1.0 1.5 2.5 3.0

0.0 0.5 1.5 2.0

(p, m)admissible region,d= 5

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Improved interpolation inequalities in the sphere

Let

λ^? := inf

v∈H¹₊(S^d, dµ) R

S^dv dµ= 1 R

S^dx|v|^pdµ= 0 R

S^d(∆v)²dµ R

S^d|∇v|²ν dµ > d

and consider the inequality Z

S^d

|∇f|²ν dµ+ λ

p−2kfk²₂≥ λ p−2kfk²_p

∀f ∈H¹(S^d, dµ) s.t.

Z

S^d

x|f|^pdµ= 0 Proposition

For anyp∈(2,2^#), the inequality holds with λ≥d+ (d−1)²

d(d+ 2)(2^#−p) (λ^?−d)

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p = 2: the logarithmic Sobolev case

λ^?=d+ 2 (d+ 2) 2 (d+ 3) +p

2 (d+ 3) (2d+ 3) Proposition

Letd≥2. For anyu∈H¹(S^d, dµ)\ {0}such that R

S^dx|u|²dµ= 0, we have

Z

S^d

|∇u|²dµ≥ δ 2

Z

S^d

|u|²log |u|²

kuk²₂

dµ withδ:=d+²_d ⁴^d−1

2 (d+3)+√

2 (d+3) (2d+3)

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Stability under antipodal symmetry

With the additional restriction ofantipodal symmetry, that is u(−x) =u(x) ∀x∈S^d

Theorem

Ifp∈(1,2)∪(2,2^∗), we have Z

S^d

|∇u|²dµ≥ d p−2

1 + (d²−4) (2^∗−p) d(d+ 2) +p−1

kuk²_Lp(S^d)− kuk²_L2(S^d)

for anyu∈H¹(S^d, dµ)with antipodal symmetry. The limit casep= 2 corresponds to the improved logarithmic Sobolev inequality

Z

S^d

|∇u|²dµ≥ d 2

(d+ 3)² (d+ 1)²

Z

S^d

|u|² log |u|² kuk²_L₂₍

S^d)

! dµ

(37)

The optimal constant in the antipodal framework

æ

æ æ

1.5 2.0 2.5 3.0

6 7 8 9 10 11 12

Numerical computation of the optimal constant when d= 5and 1≤p≤10/3≈3.33. The limiting value of the constant is numerically

found to be equal toλ_?= 2^1−2/pd≈6.59754 with d= 5andp= 10/3

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Basic properties Relaxation Free energy

Reverse Hardy-Littewood-Sobolev inequality

Joint work with J. A. Carrillo, M. G. Delgadino, R. Frank, F. Hoffmann

BA family of inequalities

BExistence of minimizers and relaxation

BNo concentration and regularity of measure valued minimizers BFree Energy

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The reverse HLS inequality

For anyλ >0 and any measurable function ρ≥0 onR^N, let Iλ[ρ] :=

Z Z

R^N×R^N

|x−y|^λρ(x)ρ(y)dx dy N ≥1, 0< q <1, α:= 2N−q(2N+λ)

N(1−q) Convention: ρ∈L^p(R^N) ifR

R^N|ρ(x)|^pdxfor anyp >0 Theorem

Iλ[ρ] ≥CN,λ,q

Z

R^N

ρ(x)dx ^αZ

R^N

ρ(x)^qdx

(2−α)/q

(6) holds for anyρ∈L¹₊∩L^q(R^N)with CN,λ,q>0 if and only if

q > N/(N+λ)

If eitherN= 1,2or if N ≥3andq≥min

1−2/N ,2N/(2N+λ) , then there is a radial nonnegative optimizerρ∈L¹∩L^q(R^N)

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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 whichCN,λ,q>0