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

Nonlinear flows and optimality for functional inequalities Maria J. Esteban CEREMADE CNRS & Universit´ e Paris-Dauphine IN COLLABORATION WITH J. D OLBEAULT , M. L OSS Dedicated to Jean-Michel Coron, on his 60th birthday Nonlinear flows and optimality for functional inequalities – p.1/19

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

Sobolev-like inequalities on the sphere On the d -dimensional sphere, let us consider the interpolation inequality d d �∇ u � 2 p − 2 � u � 2 p − 2 � u � 2 ∀ u ∈ H 1 ( S d , dµ ) , L 2 ( S d ) + L 2 ( S d ) ≥ (1) L p ( S d ) where the measure dµ is the uniform probability measure on S d ⊂ R d +1 corresponding to the measure induced by the Lebesgue measure on R d +1 , and the exposant p ≥ 1 , p � = 2 , is such that p ≤ 2 ∗ := 2 d d − 2 if d ≥ 3 . 2 d The case p = d − 2 corresponds to the Sobolev inequality (equivalent via the stereographic projection). � d − 2 �� � 2 d d R d |∇ v | 2 dx ≥ S ∀ u ∈ H 1 ( R d ) , d − 2 dx R d | v | 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 d 2 +1 ( d − 1) 2 < 2 ∗ ). Nonlinear flows and optimality for functional inequalities – p.3/19

Linear flow method Let us define ρ = | u | p . The two inequalities below are equivalent d d �∇ u � 2 p − 2 � u � 2 p − 2 � u � 2 L 2 ( S d ) + L 2 ( S d ) ≥ L p ( S d ) . � 2 ��� � � d � 1 2 p p | 2 dω ≥ p dω S d |∇ ρ S d ρ dω − S d ρ . p − 2 If we define the functionals E p and I p respectively by � 2 ��� � � 1 � 1 p 2 p | 2 dω , p dω I p [ ρ ] := S d |∇ ρ E p [ ρ ] := − p � = 2 , S d ρ dω S d ρ if p − 2 then the above inequalities amount to I p [ ρ ] ≥ d E p [ ρ ] . To establish such inequalities, one can use the heat flow ∂ρ ∂t = ∆ ρ d where ∆ denotes the Laplace-Beltrami operator on S d . We have �� � S d ρ dω = 0 dt d d If p ≤ 2 # , dt E p [ ρ ] = − I p [ ρ ] and dt I p [ ρ ] ≤ − d I p [ ρ ] . Nonlinear flows and optimality for functional inequalities – p.4/19 Details of the computation based on the carré du champ will be given below. However, there is a

Nonlinear versus linear flow We want to prove I p [ ρ ] − d E p [ ρ ] ≥ 0 . For p ≤ 2 # , d � � I p [ ρ ] − d E p [ ρ ] ≤ ( − d + d ) I p [ ρ ] = 0 . dt Not difficult to prove that ρ converges to a constant as t → + ∞ and � � lim I p [ ρ ] − d E p [ ρ ] = 0 . t → + ∞ What if 2 # < p < 2 ∗ ? LEMMA [Dolbeault, E., Loss]. When 2 # < p < 2 ∗ , we can find a function ρ 0 such that ρ solution ∂ρ of ∂t = ∆ ρ , ρ ( t = 0) = ρ 0 , and d � �� I p [ ρ ] − d E p [ ρ ] t =0 > 0 . � dt � dt = ∆ ρ m , for a well-chosen m � = 1 . d ρ Then, we can get the same result by considering the flow 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

Prove rigidity directly, “without the flow": heuristics On R d , show that minimizers of E [ v ] does not depend on the angles ω , only on r . 1) AIM: 2) Define flow (linear, nonlinear), d dt v = H [ v ] . show that it is well defined for all times. dt E [ v ( t )] = −| A [ v ( t )] | − | C | |∇ ω v ( t ) | 2 ≤ 0 . d 3) 4) If E bounded below, for any initial value v 0 , when t → + ∞ , v ( t ) → w , minimizer. And |∇ ω w | = 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 v 0 = w . d dt E [ v ( t )] | t =0 = −| A [ w ] | − | C | |∇ ω w | 2 = E ′ [ w ] · H [ w ] = 0 . So, ∇ ω w ≡ 0 . Nonlinear flows and optimality for functional inequalities – p.6/19

Attainability and value of best constants in functional inequalities F ( Dv, v, x ) ≤ C G ( D 2 v, 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

Caffarelli-Kohn-Nirenberg (CKN) inequalities � 2 /p | v | p |∇ v | 2 �� � | x | b p dx ≤ C a,b | x | 2 a dx ∀ v ∈ D a,b R d R d a � = d − 2 with a ≤ b ≤ a + 1 if d ≥ 3 , a < b ≤ a + 1 if d = 2 , 2 2 d p = d − 2 + 2 ( b − a ) 2 d b − a → 0 ⇐ ⇒ p → d − 2 b − ( a + 1) → 0 ⇐ ⇒ p → 2 + |∇ v | 2 � | x | 2 a dx 1 R d = inf � 2 /p C a,b D a,b | v | p �� | x | b p dx R d Nonlinear flows and optimality for functional inequalities – p.8/19

The symmetry issue � 2 /p | v | p |∇ v | 2 �� � ≤ C a,b ∀ v ∈ D a,b | x | b p dx | x | 2 a dx R d R d C a,b = best constant for general functions v C ∗ a,b = best constant for radially symmetric functions v C ∗ a,b ≤ C a,b Up to scalar multiplication and dilation, the optimal radial function is b − a 1 + | x | − 2 a (1+ a − b ) � − � 1+ a − b v ∗ a,b ( x ) = b − a Questions: is optimality (equality) achieved ? do we have v a,b = v ∗ a,b ? Nonlinear flows and optimality for functional inequalities – p.9/19

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 d ( d − 2 − 2 a ) − 1 b F S ( a ) = 2 ( d − 2 − 2 a ) ( d − 2 − 2 a ) 2 + 4( d − 1) � 2 Nonlinear flows and optimality for functional inequalities – p.10/19

Generalized Caffarelli-Kohn-Nirenberg inequalities (CKN) 2 d Let d ≥ 3 . For any p ∈ [2 , p ( θ, d ) := d − 2 θ ] , there exists a positive constant C ( θ, p, a ) such that � 2 � θ �� � 1 − θ | v | p |∇ v | 2 | v | 2 �� �� p ≤ C ( θ, p, a ) | x | b p dx | x | 2 a dx | x | 2 ( a +1) dx R d R d R d In the radial case, with Λ = ( a − a c ) 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]): p − 2 2 p − θ C CKN ( θ, p, a ) ≥ C ∗ CKN ( θ, p, a ) = C ∗ CKN ( θ, p ) Λ � p − 2 � p − 2 � � 2 p − 1 � 6 − p � � p − 2 + 1 2 p � θ � Γ ( p − 2) 2 2 π d/ 2 � � 2 p � 2+(2 θ − 1) p p 4 2 p 2 C ∗ CKN ( θ, p ) = √ π Γ Γ( d/ 2) 2+(2 θ − 1) p 2 p θ p +2 � � 2 p − 2 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

An optimal symmetry result r �→ r α , v ( r, ω ) = w ( r α , ω ) , With the change of variables : and with n = d − b p = d − 2 a − 2 � ∂r , 1 � α ∂w + 2 , D w = r ∇ ω w α α 2 n p = and the CKN the inequality becomes n − 2 � 2 �� � α 1 − 2 p R d | w | p dµ R d | D w | 2 dµ , dµ := r n − 1 dr dω “ = dx ( R n )” ≤ C a,b p This inequality scales like a “critical Sobolev inequality" in R n , 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 � d − 1 α = 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

Notations If ∇ ω denotes the gradient with respect to the angular variables ω ∈ S d − 1 and ∆ ω is the Laplace-Beltrami operator on S d − 1 , we define � α ∂w ∂r , 1 � D w = r ∇ ω w , we define the self-adjoint operator L by L w := − D ∗ D w = α 2 w ′′ + α 2 n − 1 w ′ + ∆ ω w r 2 r The fundamental property of L is the fact that � � ∀ w 1 , w 2 ∈ D ( R d ) R d w 1 L w 2 dµ = − R d D w 1 · D w 2 dµ ✄ 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 R d Nonlinear flows and optimality for functional inequalities – p.13/19