The sharp constants in critical Sobolev embedding theorems via - PDF document

The sharp constants in critical Sobolev embedding theorems via optimal transport Alexander Nazarov (St.-Petersburg State University) St. Petersburg, 2010

photo by V. Zelenkov This commercial reads: Permanently sober loaders; . . . We select optimal transport

n � 2; 1 < p < n . W 1 p ( R n ) stands for the completion of ˙ C ∞ 0 ( R n ) with respect to the norm �∇ v � p . 1. Critical Sobolev embedding: �∇ v � p, R n K ( n, p ) = inf � v � p ∗ , R n > 0 . v ∈ ˙ W 1 p ( R n ) \{ 0 } 2. Critical trace embedding: �∇ v � p, R n + K 1 ( n, p ) = inf > 0 . p ( R n � v � p ∗∗ ,∂ R n v ∈ ˙ W 1 + ) \{ 0 } + Here p ∗ = p ∗∗ = ( n − 1) p np n − p ; n − p . NB : Without loss of generality, one can assume v � 0.

How to find the sharp con- stants in 1 and 2 ? Critical Sobolev embedding. 1. The classical approach (T. Aubin, 1976; G. Talenti, 1976): Step 1: Symmetrization, i.e. the rear- rangement mapping any level set to the ball of the same volume centered in ori- gin. It is well known (G. P´ olya, G. Szeg¨ o, 1940s) that this rearrangement dimin- ishes our functional. Step 2: Thus, the problem is reduced to a one-dimensional inequality which was considered by G.A. Bliss (1930).

Alternative approach based on the op- timal transport (D. Cordero-Erausquin, B. Nazaret, C. Villani, 2004). Consider two probability measures in R n with smooth densities F and G and bounded supports. Then there exists the Brenier map T = ∇ ϕ such that for all measurable functions ψ � � ψ ( x ) G ( x ) dx = ψ ( T ( x )) F ( x ) dx. (1) R n R n Moreover, the function ϕ is convex and satisfies the Monge–Amp` ere equation F ( x ) = G ( ∇ ϕ ( x )) · det( D 2 ϕ ( x )) . (2) almost everywhere w.r.t. the measure Here D 2 ϕ is a.e.-Hessian matrix Fdx . of ϕ which exists by A.D. Aleksandrov’s theorem.

By (1), G 1 − 1 G − 1 � � n ( x ) dx = n ( ∇ ϕ ( x )) F ( x ) dx. R n R n Using (2) and the Hadamard inequality, we obtain G 1 − 1 � n ( x ) dx = R n 1 n ( D 2 ϕ ( x )) F 1 − 1 � = det n ( x ) dx � R n � 1 ∆ ϕ ( x ) F 1 − 1 � n ( x ) dx. (3) n R n Since ϕ is convex, we can change ∆ ϕ , understood as a.e.-Laplacian in the right- hand side of (3), to the full distributional Laplacian.

Integrating by parts, we get G 1 − 1 � n ( x ) dx � R n � − 1 �∇ ϕ ( x ) , ∇ ( F 1 − 1 � n )( x ) � dx. (4) n R n Put F = v p ∗ , G = u p ∗ . Then � v � p ∗ , Ω = � u � p ∗ , Ω = 1, and (4) becomes u p ∗ (1 − 1 � n ) ( x ) dx � R n n ( p − 1) � − p ( n − 1) � n − p ( x ) �∇ ϕ ( x ) , ∇ v ( x ) � dx. v n ( n − p ) R n (Note that the exponent in the last integral equals p ∗ /p ′ ). Now we apply the H¨ older inequality and arrive at

u p ∗ (1 − 1 n ) ( x ) dx � p ( n − 1) � n ( n − p ) �∇ v � p, R n · R n � 1 /p ′ v p ∗ ( x ) |∇ ϕ ( x ) | p ′ dx � � · . R n By (1), v p ∗ ( x ) |∇ ϕ ( x ) | p ′ dx = u p ∗ ( y ) | y | p ′ dy. � � R n R n This gives R n u p ∗ (1 − 1 n ) ( x ) dx � � 1 /p ′ � p ( n − 1) n ( n − p ) �∇ v � p, R n . R n u p ∗ ( x ) | x | p ′ dx � � Since the Brenier map ϕ is not contained in the last inequality, it remains valid for all u and v normalized in L p ∗ ( R n ).

Now we observe that the equality in 1 n ( D 2 ϕ ) � 1 n ∆ ϕ implies D 2 ϕ = C I , det and thus, we can assume ∇ ϕ ( x ) = Cx . Further, the equality in the H¨ older in- equality means v p ∗ /p ′ ∇ ϕ = C ∇ v . This implies v = v ( | x | ) and provides a 1st or- der ODE for v . Solving it, we obtain the Bliss function h ( x ) = ( a + b | x | p ′ ) 1 − n p . Direct calculation shows that we really have the equality for u = v = Ch . In particular, this means �∇ v � p, R n � v � p ∗ , R n � �∇ h � p, R n � h � p ∗ , R n , � 1 �� 1 1 � p ′ � � n . n − p n p , n and K ( n, p ) = n p ω n − 1 · B p ′ + 1 p − 1

2. Critical trace embedding. Escobar (1988) conjectured that the minimizer in the half-space is w ( x ) = | x − ε e | − n − p p − 1 , (5) with e = (0 , . . . , 0 , − 1), and proved it for p = 2 using the conformal invariance � of the quotient �∇ v � 2 , R n + . B. � v � 2 ∗∗ ,∂ R n + Nazaret (2006) proved this conjecture by the optimal transport approach. Now we consider two probability mea- sures in R n + with smooth densities F and G and bounded supports. Then the identity (1) becomes � � ψ ( x ) G ( x ) dx = ψ ( T ( x )) F ( x ) dx. (6) R n R n + +

Just as earlier, we obtain n ( x ) dx � 1 G 1 − 1 ∆ ϕ ( x ) F 1 − 1 � � n ( x ) dx. n R n R n + + Integrating by parts, we get G 1 − 1 � n n ( x ) dx � R n + F 1 − 1 � n ( x ) �∇ ϕ ( x ) , n � d Σ − � ∂ R n + �∇ ϕ ( x ) , ∇ ( F 1 − 1 � − n )( x ) � dx. R n + By definition of the Brenier map, for all x ∈ R n + one has ∇ ϕ ( x ) ∈ R n + . Therefore, �∇ ϕ ( x ) , n � � 0 on ∂ R n + , and G 1 − 1 � n n ( x ) dx � R n + �∇ ϕ ( x ) , ∇ ( F 1 − 1 � � − n )( x ) � dx. (7) R n +

Adding to both parts of (7) the integral � e , ∇ ( F 1 − 1 � n )( x ) � dx = R n + F 1 − 1 � = n ( x ) � e , n � d Σ = ∂ R n + F 1 − 1 � = n ( x ) d Σ , ∂ R n + we arrive at F 1 − 1 G 1 − 1 � � n ( x ) d Σ + n n ( x ) dx � ∂ R n R n + + � e − ∇ ϕ ( x ) , ∇ ( F 1 − 1 � n )( x ) � dx. (8) � R n + Put F = v p ∗ , G = u p ∗ . Then � v � p ∗ , R n + = + = 1, and (8) becomes � u � p ∗ , R n

n ( p − 1) � v � p ∗∗ + � ( n − 1) p � n − p ( x ) · v p ∗∗ ,∂ R n n − p R n + · � e − ∇ ϕ ( x ) , ∇ v ( x ) � dx − n � u � p ∗∗ + . p ∗∗ , R n By the H¨ older inequality, � v � p ∗∗ + � ( n − 1) p �∇ v � p, R n + · p ∗∗ ,∂ R n n − p � 1 v p ∗ ( x ) | e − ∇ ϕ ( x ) | p ′ dx p ′ � � · − R n + − n � u � p ∗∗ + . p ∗∗ , R n By (6), v p ∗ ( x ) | e − ∇ ϕ ( x ) | p ′ dx = � R n + u p ∗ ( y ) | e − y | p ′ dy. � = R n + This gives

� v � p ∗∗ + � ( n − 1) p �∇ v � p, R n + · p ∗∗ ,∂ R n n − p � 1 u p ∗ ( x ) | e − x | p ′ dx p ′ � � · − R n + − n � u � p ∗∗ + . (9) p ∗∗ , R n Note that both sides of (9) do not con- tain the Brenier map. Hence, by approx- imation, this inequality remains valid for all u and v normalized in L p ∗ ( R n + ). Now we specify (7) by setting u = Cw , with w defined in (5) and C = � w � − 1 + . p ∗ , R n W 1 p ( R n ˙ Then for any v ∈ + ) such that + = 1, we have � v � p ∗ , R n � v � p ∗∗ + � A �∇ v � p, R n + − B, (10) p ∗∗ ,∂ R n

where 1 n ( p − 1) B = n C p ∗∗ · I ; A = ( n − 1) p p ′ , · C n − p · I n − p dx I = � w � p ∗∗ � + = | x − e | ( n − 1) p ′ . p ∗∗ , R n R n + W 1 ˙ For arbitrary v ∈ p (Ω), without nor- malization, (9) can be rewritten as fol- lows: � p ∗∗ � K ( v ) � AK ( v ) − B, J ( v ) i.e. K p ∗∗ ( v ) J p ∗∗ ( v ) � F ( K ( v )) ≡ AK ( v ) − B, where J ( v ) = �∇ v � p, Ω K ( v ) = �∇ v � p, Ω , . � v � p ∗∗ ,∂ Ω � v � p ∗ , Ω

By elementary calculus, the function F achieves its minimum at the point p ( n − 1) B n ( p − 1) A = n − p 1 p = K ( w ) , p − 1 C I and therefore, K p ∗∗ ( w ) � n ( p − 1) p − 1 J p ∗∗ ( v ) � n − p I � n − p AK ( w ) − B = n − p . p − 1 If v = u = Cw , then the Brenier map is the identity. Direct calculation shows that all the inequalities become equali- ties. This means �∇ v � p, R n �∇ w � p, R n + + , � � v � p ∗∗ ,∂ R n � w � p ∗∗ ,∂ R n + + and � 1 1 � p ′ � � �� ( n − 1) p ′ . ω n − 2 n − p n − 1 n − 1 K 1 ( n, p ) = · B 2 , p − 1 2 2( p − 1)

The following observation is by A. Nazarov (to appear in Algebra and Analysis, 2010, N5). Theorem . Let Ω be a convex circular cone with aperture 2 θ . Then the mini- mum for the critical trace embedding in Ω is provided by the function (5). The proof runs almost without changes. In particular, this implies � n ( p − 1) p − 1 1 � n − p ( n − 1) p sin ( n − 1) p I p ∗∗ ( θ ) , K 1 ( n, p ; Ω) = p − 1 with dx I = � w � p ∗∗ � p ∗∗ , Ω = | x − e | ( n − 1) p ′ . Ω

Remark . The value of I for circular cones can be calcu- lated explicitly: 2 − 1 2 a − 3 n a − n − 1 � n 2 , a − n � � � I = π 2 Γ B · 2 2 1 2 − a 2 ( θ ) P · sin n − a − 1 � � cos( θ ) , n − a − 3 2 where a = ( n − 1) p 2( p − 1) while P µ ν ( x ) is the Legendre function. Theorem remains valid for any convex cone Ω, if its supporting hyperplanes at almost every point have a constant an- gle θ with the axis x n . The simplest example of such cone is a dyhedral an- gle less than half-space. Another inter- esting example is a cone supported by arbitrary simplex in S n − 1 . It is worth to note that for nonconvex cone of such type ( θ > π 2 ), the function (5) does not provide minimum in the critical trace embed- ding, though it is a stationary point.