[PPT] - Nonlinear flows and optimality for functional inequalities Maria J. PowerPoint Presentation

SLIDE 1

Nonlinear flows and optimality for functional inequalities

Maria J. Esteban CEREMADE CNRS & Universit´ e Paris-Dauphine

IN COLLABORATION WITH

J. DOLBEAULT, M. LOSS

Dedicated to Jean-Michel Coron, on his 60th birthday

Nonlinear flows and optimality for functional inequalities – p.1/19

SLIDE 2

OUTLINE

Use of linear and nonlinear flows to prove Sobolev-like inequalities on manifolds
Generalities of functional inequalities
The Caffarelli-Kohn-Nirenberg inequalities
Symmetry and symmetry breaking for extremals of Caffarelli-Kohn-Nirenberg inequalities

Nonlinear flows and optimality for functional inequalities – p.2/19

SLIDE 3

Sobolev-like inequalities on the sphere

On the d-dimensional sphere, let us consider the interpolation inequality ∇u2

L2(Sd) +

d p − 2 u2

L2(Sd) ≥

d p − 2 u2

Lp(Sd)

∀ u ∈ H1(Sd, dµ) , (1) where the measure dµ is the uniform probability measure on Sd ⊂ Rd+1 corresponding to the measure induced by the Lebesgue measure on Rd+1, and the exposant p ≥ 1, p = 2, is such that p ≤ 2∗ :=

2 d d−2 if d ≥ 3.

The case p =

2 d d−2 corresponds to the Sobolev inequality (equivalent via the stereographic

projection).

Rd |∇v|2 dx ≥ S
Rd |v|

2 d d−2 dx

d−2

d

∀ u ∈ H1(Rd) , PROOFS OF (1) + MINIMIZERS ARE CONSTANTS BY: Bidaut-Véron – Véron (PDE, rigidity methods, 1991); Beckner (harmonic analysis methods, 1993); Bakry-Ledoux et al (“carré du champ" method, linked to a flow method, 1996 +; only for 2 < p ≤ 2# := 2 d2+1

(d−1)2 < 2∗).

Nonlinear flows and optimality for functional inequalities – p.3/19

SLIDE 4

Linear flow method

Let us define ρ = |u|p. The two inequalities below are equivalent ∇u2

L2(Sd) +

d p − 2 u2

L2(Sd) ≥

d p − 2 u2

Lp(Sd) .

Sd |∇ρ

1 p |2 dω ≥

d p − 2

Sd ρ dω

2

p

−

Sd ρ

2 p dω

.

If we define the functionals Ep and Ip respectively by Ip[ρ] :=

Sd |∇ρ

1 p |2 dω ,

Ep[ρ] := 1 p − 2

Sd ρ dω

2

p

−

Sd ρ

2 p dω

if

p = 2 , then the above inequalities amount to Ip[ρ] ≥ d Ep[ρ]. To establish such inequalities, one can use the heat flow ∂ρ ∂t = ∆ρ where ∆ denotes the Laplace-Beltrami operator on Sd. We have

d dt

Sd ρ dω
= 0

If p ≤ 2#, d dtEp[ρ] = − Ip[ρ] and d dt Ip[ρ] ≤ − d Ip[ρ] . Details of the computation based on the carré du champ will be given below. However, there is a

Nonlinear flows and optimality for functional inequalities – p.4/19

SLIDE 5

Nonlinear versus linear flow

We want to prove Ip[ρ] − d Ep[ρ] ≥ 0. For p ≤ 2#, d dt

Ip[ρ] − d Ep[ρ]
≤ (−d + d) Ip[ρ] = 0 .

Not difficult to prove that ρ converges to a constant as t → +∞ and lim

t→+∞

Ip[ρ] − d Ep[ρ]
= 0 .

What if 2# < p < 2∗? LEMMA [Dolbeault, E., Loss]. When 2# < p < 2∗, we can find a function ρ0 such that ρ solution

f

∂ρ ∂t = ∆ρ , ρ(t = 0) = ρ0 , and

d dt

Ip[ρ] − d Ep[ρ]
t=0 > 0 .

Then, we can get the same result by considering the flow

d ρ dt = ∆ρm , for a well-chosen m = 1.

The computations are much more involved, but the idea is “more or less" the same. And it covers also the case p ∈ (1, 2).

Nonlinear flows and optimality for functional inequalities – p.5/19

SLIDE 6

Prove rigidity directly, “without the flow": heuristics

1) AIM: On Rd, show that minimizers of E[v] does not depend on the angles ω, only on r. 2) Define flow (linear, nonlinear), d

dt v = H[v]. show that it is well defined for all times.

3)

d dt E[v(t)] = −|A[v(t)]| − |C| |∇ωv(t)|2 ≤ 0.

4) If E bounded below, for any initial value v0, when t → +∞, v(t) → w, minimizer. And |∇ωw| = 0. To carry out this program, we need to prove a lot of things about the flow, and this not always easy for nonlinear flows... or very technical at least. Way out? ALTERNATIVE: Consider any minimizer of E or even any critical point of E, that is, a function w that satisfies E′(w) = 0. Consider the same flow as above, with initial datum v0 = w. d dt E[v(t)]|t=0 = −|A[w]| − |C| |∇ωw|2 = E′[w] · H[w] = 0. So, ∇ωw ≡ 0 .

Nonlinear flows and optimality for functional inequalities – p.6/19

SLIDE 7

Attainability and value of best constants in functional inequalities

F(Dv, v, x) ≤ C G(D2v, Dv, v, x) ∀v ∈ X . Functional inequalities play an important role in obtaining a priori estimates for solutions of PDEs, in analyzing the long time behavior of solutions of evolution problems, in describing the blow-up profile for finite time blow-up phenomena, etc Important questions :

Is C attained in X?

What is its value??

If yes, how do the optimal functions v look like?

If we know a priori that the optimal solutions have some symmetry properties, for instance, that they are radially symmetric, then it might be easier to compute the value of C.

Nonlinear flows and optimality for functional inequalities – p.7/19

SLIDE 8

Caffarelli-Kohn-Nirenberg (CKN) inequalities

Rd

|v|p |x|b p dx 2/p ≤ Ca,b

Rd

|∇v|2 |x|2 a dx ∀ v ∈ Da,b with a ≤ b ≤ a + 1 if d ≥ 3 , a < b ≤ a + 1 if d = 2 , a = d−2

2

p = 2 d d − 2 + 2 (b − a) b − a → 0 ⇐ ⇒ p →

2d d−2

b − (a + 1) → 0 ⇐ ⇒ p → 2+ 1 Ca,b = inf

Da,b

Rd

|∇v|2 |x|2 a dx

Rd

|v|p |x|b p dx

2/p

Nonlinear flows and optimality for functional inequalities – p.8/19

SLIDE 9

The symmetry issue

Rd

|v|p |x|b p dx 2/p ≤ Ca,b

Rd

|∇v|2 |x|2 a dx ∀ v ∈ Da,b Ca,b = best constant for general functions v C∗

a,b = best constant for radially symmetric functions v

C∗

a,b ≤ Ca,b

Up to scalar multiplication and dilation, the optimal radial function is v∗

a,b(x) =

1 + |x|− 2a (1+a−b)

b−a

−

b−a 1+a−b

Questions: is optimality (equality) achieved ? do we have va,b = v∗

a,b ?

Nonlinear flows and optimality for functional inequalities – p.9/19

SLIDE 10

Symmetry and symmetry breaking (d ≥ 3)

Case a > 0, Existence and symmetry : Th. Aubin, G. Talenti, E. Lieb, Chou-Chu, P .L. Lions, Horiuchi,... Case a < 0, symmetry breaking: Catrina-Wang, Felli-Schneider. Case a < 0 : Lin, Wang; Dolbeault, E., Tarantello (d=2); Betta-Brock-Mercaldo-Posteraro (b > 0) Case a < 0 : Dolbeault, E., Loss, Tarantello bF S(a) = d(d − 2 − 2a) 2

(d − 2 − 2a)2 + 4(d − 1)

− 1 2 (d − 2 − 2a)

Nonlinear flows and optimality for functional inequalities – p.10/19

SLIDE 11

Generalized Caffarelli-Kohn-Nirenberg inequalities (CKN)

Let d ≥ 3 . For any p ∈ [2, p(θ, d) :=

2d d−2θ ], there exists a positive constant C(θ, p, a) such that

Rd

|v|p |x|b p dx 2

p

≤ C(θ, p, a)

Rd

|∇v|2 |x|2 a dx θ

Rd

|v|2 |x|2 (a+1) dx 1−θ In the radial case, with Λ = (a − ac)2, the best constant when the inequality is restricted to radial functions is C∗

CKN(θ, p, a) and (see [Del Pino, Dolbeault, Filippas, Tertikas]):

CCKN(θ, p, a) ≥ C∗

CKN(θ, p, a) = C∗ CKN(θ, p) Λ

p−2 2p −θ

C∗

CKN(θ, p) =

2 πd/2

Γ(d/2)

2 p−1

p

(p−2)2

2+(2 θ−1) p

p−2

2 p

2+(2 θ−1) p 2 p θ

θ

4 p+2

6−p

2 p

Γ
2

p−2 + 1 2

√π Γ
2

p−2

p−2

p

and for θ small, we have proved that there is symmetry breaking for certain values of (Λ, p) such that u∗

Λ,p is stable! In principle in all cases where we have observed this phenomenum, θ ≤ 0.7

approx. (Dolbeault, E., Tarantello, Tertikas (2011))

Nonlinear flows and optimality for functional inequalities – p.11/19

SLIDE 12

An optimal symmetry result

With the change of variables : r → rα, v(r, ω) = w(rα, ω), and with n = d − b p α = d − 2 a − 2 α + 2 , Dw =

α ∂w

∂r , 1 r ∇ωw

p =

2n n−2

and the CKN the inequality becomes α1− 2

p

Rd |w|p dµ

2

p

≤ Ca,b

Rd |Dw|2 dµ ,

dµ := rn−1 dr dω “ = dx (Rn)” This inequality scales like a “critical Sobolev inequality" in Rn, but n does not need to be an integer... The parameters α and n vary in the ranges 0 < α < ∞ and d < n < ∞ and the Felli-Schneider curve in the (α, n) variables is given by α =

d − 1

n − 1 =: α FS THEOREM [2015].- If α ≤ α FS and d ≥ 2, optimality is achieved by radial functions.

Nonlinear flows and optimality for functional inequalities – p.12/19

SLIDE 13

Notations

If ∇ω denotes the gradient with respect to the angular variables ω ∈ Sd−1 and ∆ω is the Laplace-Beltrami operator on Sd−1, we define Dw =

α ∂w

∂r , 1 r ∇ωw

,

we define the self-adjoint operator L by Lw := − D∗ D w = α2 w′′ + α2 n − 1 r w′ + ∆ω w r2 The fundamental property of L is the fact that

Rd w1 Lw2 dµ = −
Rd Dw1 · Dw2 dµ

∀ w1, w2 ∈ D(Rd) ✄ Heuristics: we look for a monotonicity formula along a well chosen nonlinear flow, based on the analogy with the decay of the Fisher information along the fast diffusion flow in Rd

Nonlinear flows and optimality for functional inequalities – p.13/19

SLIDE 14

Fisher information decay and a fast diffusion equation

Let u = |w|p, p =

2 n n−2.

Up to multiplicative constants,

Rd |w|p dµ =
Rd u dµ,

and

Rd |Dw|2 dµ = I[u] ,

with I[u] :=

Rd u |Dp|2 dµ ,

p = m 1 − m um−1 and m = 1 − 1 n Here I is the Fisher information and p is the pressure function. Next, define the fast diffusion equation (flow) ∂u ∂t = Lum , m = 1 − 1 n ✄ STRATEGY: Assume that α ≤ αFS, 1) prove that for all t ≥ 0,

d dt

Rd u(t) dµ = 0 and

d dt I[u(t, ·)] ≤ 0,

2) prove that

d dt I[u(t, ·)] = 0 means, in particular, that u is radially symmetric.

Nonlinear flows and optimality for functional inequalities – p.14/19

SLIDE 15

Mass conservation and Fisher information decay along the fast diffusion flow

Easy to see: the mass

Rd u dµ is conserved along the flow.

With p = m 1 − m um−1 , I[u] :=

Rd u |Dp|2 dµ ,

Proposition 1.- Let u0 ≥ 0. Up to estimates near the origin and near infinity, d dtI[u(t, ·)] = − 2 (n − 1)n−1 K[p] , with K[p] :=

Rd

1 2 L |Dp|2 − Dp · DLp − 1 n (Lp)2

p1−n dµ

Proposition 2.- If α ≤ α FS, K[p] ≥ 0. Proposition 3.- If u0 ≥ 0 is a critical point, then 0 = I′[u0] · Lum

0 = d dt I[u(t, ·)]

t=0 = C K[p0].

Proposition 4.- If α ≤ α FS, K[p0] = 0 implies that p0 is independent of ω and p0 = a + b r2.

Nonlinear flows and optimality for functional inequalities – p.15/19

SLIDE 16

Proving decay (1/2)

K[p] :=

Rd k[p] p1−n dµ ,

k[p] := Q(p) − 1 n (L p)2 = 1 2 L |Dp|2 − Dp · D L p − 1 n (L p)2 Lemma.- Let n = 1 be any real number, d ∈ N, d ≥ 2, and consider a function p ∈ C3((0, ∞) × M), where (M, g) is a smooth, compact Riemannian manifold. Then we have k[p] = α4

1 − 1

n p′′ − p′ r − ∆ω p α2 (n − 1) r2 2 + 2 α2 1 r2

∇ωp′ − ∇ωp

r

2

+ 1 r4 kM[p] , with kM[p] := 1 2 ∆ω |∇ωp|2 − ∇ωp · ∇ω∆ω p −

1 n−1 (∆ω p)2 − (n − 2) α2 |∇ωp|2

Nonlinear flows and optimality for functional inequalities – p.16/19

SLIDE 17

Proving decay (2/2)

Proposition.- Assume that d ≥ 3, n > d and M = Sd−1. There is a positive constant ζ⋆ such that

Sd kM[p] p1−n dω ≥ (n − 2)
α2

FS − α2 Sd |∇ωp|2 p1−n dω

+ ζ⋆ (n − d)

Sd |∇ωp|4 p1−n dω

Proof based on the Bochner-Lichnerowicz-Weitzenböck formula. So, we have shown that K[p] ≥ sum of squares + (n − 2)

α2

FS − α2 Sd |∇ωp|2 p1−n dω

and therefore, if α ≤ αFS, K[p] ≥ 0.

Nonlinear flows and optimality for functional inequalities – p.17/19

SLIDE 18

End of the proof

If α ≤ αFS (resp. Λ ≤ ΛFS), and if p0 is a critical point of the E-L equations for CKN, written in the good variables, then 0 = K[p0] ≥

Rd α4
1 − 1

n p′′

0 − p′

r − ∆ω p0 α2 (n − 1) r2 2 p1−n dµ +

Rd (n − 2)
α2

FS − α2

|∇ωp0|2 p1−n dµ +

Rd ζ⋆ (n − d) |∇ωp0|4 p1−n

dµ , where ζ⋆ > 0 and n > d. So, ∇ωp0 ≡ 0, that is, p0 does not depend on ω, which means radial symmetry. Moreover, p′′

0 − p′ r − ∆ω p0 α2 (n−1) r2 ≡ 0, which implies that for some a, b > 0,

p0 = a + b r2.

Nonlinear flows and optimality for functional inequalities – p.18/19

SLIDE 19

Extensions : The robustness of the method

With the definitions wLq,γ(Rd) :=

Rd |w|q |x|−γ dx

1/q , wLq(Rd) := wLq,0(Rd) , consider the family of (subcritical) Caffarelli-Kohn-Nirenberg interpolation inequalities given by wL2p,γ(Rd) ≤ Cβ,γ,p ∇wϑ

L2,β(Rd) w1−ϑ Lp+1,γ (Rd)

∀ w ∈ Hp

β,γ(Rd) .

(1)

Here the parameters β, γ and p are subject to the restrictions d ≥ 2 , γ − 2 < β < d − 2 d γ , γ ∈ (−∞, d) , p ∈ (1, p⋆] with p⋆ := d − γ d − β − 2

(2)

and the exponent ϑ is determined by the scaling invariance, i.e., ϑ = (d − γ) (p − 1) p

d + β + 2 − 2 γ − p (d − β − 2)

.

REMARK. The “critical" case p = p⋆ corresponds to the kind of inequalities discussed above.
REMARK. In a recent paper with J. Dolbeault, M. Loss and M. Muratori we have extended our

methodology, but several important changes have to be made in order to fit the sub-criticality of the inequalities: Renyi entropies instead of Fisher information, other regularity results, etc. )

Nonlinear flows and optimality for functional inequalities – p.19/19