A remark on the concentration compactness principle in critical dimension
aa r X i v : . [ m a t h . C A ] S e p A REMARK ON THE CONCENTRATION COMPACTNESSPRINCIPLE IN CRITICAL DIMENSION
FENGBO HANG
Abstract.
We prove some refinements of concentration compactness princi-ple for Sobolev space W ,n on a smooth compact Riemannian manifold ofdimension n . As an application, we extend Aubin’s theorem for functions on S n with zero first order moments of the area element to higher order momentscase. Our arguments are very flexible and can be easily modified for functionssatisfying various boundary conditions or belonging to higher order Sobolevspaces. Introduction
Let n ≥
2, Ω ⊂ R n be a bounded open subset with smooth boundary. In [M], itis showed that for any u ∈ W ,n (Ω) \ { } , Z Ω exp a n | u | nn − k∇ u k nn − L n (Ω) dx ≤ c ( n ) | Ω | . (1.1)Here | Ω | is the volume of Ω and a n = n (cid:12)(cid:12) S n − (cid:12)(cid:12) n − . (1.2) (cid:12)(cid:12) S n − (cid:12)(cid:12) is the volume of S n − under the standard metric. For convenience, we willuse this notation a n throughout the paper. (1.1) can be viewed as a limit case ofthe Sobolev embedding theorem.To study extremal problems related to (1.1), a concentration compactness theo-rem [Ln, Theorem 1.6 on p196, Remark 1.18 on p199] was proved. For n = 2, theargument is elegant. The approach is recently applied in [CH] on smooth Riemannsurfaces to deduce refinements of the concentration compactness principle (see [CH,Section 2]). These refinements are crucial in extending Aubin’s classical theorem on S for functions with zero first order moments of the area element (see [A, Corollary2 on p159]) to higher order moments cases, motivated from similar inequalities on S (see [CH, GrS, OPS, W]).For n ≥
3, due to the subtle analytical difference between weak convergence in L and L p , p = 2, [Ln, p197] has to use special symmetrization process to gain thepointwise convergence of the gradient of functions considered. Unfortunately, aspointed out in [CCH], this argument is not sufficient to derive [Ln, Remark 1.18].More accurate argument is presented in [CCH]. More recently, [LLZ] deduce similarresults on domains in Heisenberg group without using symmetrization process.The main aim of this note is to extend the analysis in [CH] from dimension 2to dimensions at least 3. In particular we will prove refinements of concentrationcompactness principle (Proposition 2.1 and Theorem 2.1). Our approach also doesnot use symmetrization process and is more close to [CH, Ln]. It can be easily modified for functions satisfying various boundary conditions or belonging to higherorder Sobolev spaces.Throughout the paper, we will assume ( M n , g ) is a smooth compact Riemannianmanifold with dimension n ≥
2. For an integrable function u on M , we denote u = 1 µ ( M ) Z M udµ. (1.3)Here µ is the measure associated with the Riemannian metric g . The Moser-Trudinger inequality (see [F]) tells us that for every u ∈ W ,n ( M ) \ { } with u = 0,we have Z M exp a n | u | nn − k∇ u k nn − L n ( M ) dµ ≤ c ( M, g ) . (1.4)Here a n is given in (1.2).It follows from (1.4) and Young’s inequality that for any u ∈ W ,n ( M ) with u = 0, we have the Moser-Trudinger-Onofri inequalitylog Z M e nu dµ ≤ α n k∇ u k nL n + c ( M, g ) . (1.5)Here α n = (cid:18) n − n (cid:19) n − | S n − | . (1.6)We will use this notation α n throughout the paper.In Section 2 we will derive some refinements of concentration compactness prin-ciple in dimension n . These refinements will be used in Section 3 to extend Aubin’stheorem on S n to vanishing higher order moments case. In Section 4, we discussmodifications of our approach applied to higher order Sobolev spaces and Sobolevspaces on surfaces with nonempty boundary.Last but not least, we would like to thank the referee for careful reading of themanuscript and many valuable suggestions.2. Concentration compactness principle in critical dimension
As usual, for u ∈ W ,n ( M ), we denote k u k W ,n ( M ) = (cid:16) k u k nL n ( M ) + k∇ u k nL n ( M ) (cid:17) n . We start with a basic consequence of Moser-Trudinger inequality (1.4). It shouldbe compared with [CH, Lemma 2.1].
Lemma 2.1.
For any u ∈ W ,n ( M ) and a > , we have Z M e a | u | nn − dµ < ∞ . (2.1) Proof.
We could use the same argument as in the proof of [CH, Lemma 2.1]. Insteadwe modify the approach a little bit so that it also works for higher order Sobolevspaces. Without losing of generality, we can assume u is unbounded. Let ε > v ∈ C ∞ ( M ) such that k u − v k W ,n ( M ) < ε. ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 3
We denote w = u − v , then u = v + w = v + w + w − w. It follows that | u | ≤ k v k L ∞ + | w | + | w − w | . Hence | u | nn − ≤ n − ( k v k L ∞ + | w | ) nn − + 2 n − | w − w | nn − . We get e a | u | nn − ≤ e n − a ( k v k L ∞ + | w | ) nn − e n − a | w − w | nn − ≤ e n − a ( k v k L ∞ + | w | ) nn − e a n | w − w | nn − k∇ w k nn − Ln when ε is small enough. It follows from (1.4) that Z M e a | u | nn − dµ ≤ c ( M, g ) e n − a ( k v k L ∞ + | w | ) nn − < ∞ . Now we are ready to prove a localized version of [Ln, Theorem 1.6]. The readershould compare it with [CH, Lemma 2.2].
Proposition 2.1.
Assume u i ∈ W ,n ( M n ) such that u i = 0 , u i ⇀ u weakly in W ,n ( M ) and |∇ u i | n dµ → |∇ u | n dµ + σ (2.2) as measure. If x ∈ M and p ∈ R such that < p < σ ( { x } ) − n − , then for some r > , sup i Z B r ( x ) e a n p | u i | nn − dµ < ∞ . (2.3) Here a n = n (cid:12)(cid:12) S n − (cid:12)(cid:12) n − . (2.4) Proof.
Fix p ∈ (cid:16) p, σ ( { x } ) − n − (cid:17) , then σ ( { x } ) < p n − . (2.5)We can find a ε > ε ) σ ( { x } ) < p n − (2.6)and (1 + ε ) p < p . (2.7)Let v i = u i − u , then v i ⇀ W ,n ( M ), v i → L n ( M ). For any ϕ ∈ C ∞ ( M ), we have k∇ ( ϕv i ) k nL n ≤ ( k ϕ ∇ v i k L n + k v i ∇ ϕ k L n ) n ≤ ( k ϕ ∇ u i k L n + k ϕ ∇ u k L n + k v i ∇ ϕ k L n ) n ≤ (1 + ε ) k ϕ ∇ u i k nL n + c ( n, ε ) k ϕ ∇ u k nL n + c ( n, ε ) k v i ∇ ϕ k nL n . FENGBO HANG
It follows thatlim sup i →∞ k∇ ( ϕv i ) k nL n ≤ (1 + ε ) (cid:18)Z M | ϕ | n dσ + Z M | ϕ | n |∇ u | n dµ (cid:19) + c ( n, ε ) k ϕ ∇ u k nL n = (1 + ε ) Z M | ϕ | n dσ + c ( n, ε ) Z M | ϕ | n |∇ u | n dµ. Note that we can find a ϕ ∈ C ∞ ( M ) such that ϕ | B r ( x ) = 1 for some r > ε ) Z M | ϕ | n dσ + c ( n, ε ) Z M | ϕ | n |∇ u | n dµ < p n − . Hence for i large enough, we have k∇ ( ϕv i ) k nL n < p n − . In other words, k∇ ( ϕv i ) k nn − L n < p . We have Z B r ( x ) e a n p | v i − ϕv i | nn − dµ ≤ Z M e a n p | ϕv i − ϕv i | nn − dµ ≤ Z M e a n | ϕvi − ϕvi | nn − k ∇ ( ϕvi ) k nn − Ln dµ ≤ c ( M, g ) . Next we observe that | u i | nn − = | ( v i − ϕv i ) + u + ϕv i | nn − ≤ (1 + ε ) | v i − ϕv i | nn − + c ( n, ε ) | u | nn − + c ( n, ε ) | ϕv i | nn − , hence e a n | u i | nn − ≤ e (1+ ε ) a n | v i − ϕv i | nn − e c ( n,ε ) | u | nn − e c ( n,ε ) | ϕv i | nn − . Since e (1+ ε ) a n | v i − ϕv i | nn − is bounded in L p ε ( B r ( x )), e c ( n,ε ) | u | nn − ∈ L q ( B r ( x )) forany 0 < q < ∞ (by Lemma 2.1), e c ( n,ε ) | ϕv i | nn − → i → ∞ and p ε > p , itfollows from Holder inequality that e a n | u i | nn − is bounded in L p ( B r ( x )). Corollary 2.1.
Assume u i ∈ W ,n ( M ) such that u i = 0 and k∇ u i k L n ≤ . Wealso assume u i ⇀ u weakly in W ,n ( M ) and |∇ u i | n dµ → |∇ u | n dµ + σ (2.8) as measure. Let K be a compact subset of M and κ = max x ∈ K σ ( { x } ) . (2.9)(1) If κ < , then for any ≤ p < κ − n − , sup i Z K e a n p | u i | nn − dµ < ∞ . (2.10) ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 5 (2) If κ = 1 , then σ = δ x for some x ∈ K , u = 0 and after passing to asubsequence, e a n | u i | nn − → c δ x (2.11) as measure for some c ≥ .Proof. First assume κ <
1. For any x ∈ K , we have1 ≤ p < κ − n − ≤ σ ( { x } ) − n − . (2.12)By the Proposition 2.1 we can find r x > i Z B rx ( x ) e a n p | u i | nn − dµ < ∞ . (2.13)We have an open covering K ⊂ [ x ∈ K B r x ( x ) , hence there exists x , · · · , x N ∈ K such that K ⊂ N [ k =1 B r k ( x k ) . Here r k = r x k . For any i , Z K e a n p | u i | nn − dµ ≤ N X k =1 Z B rk ( xk ) e a n p | u i | nn − dµ ≤ N X k =1 sup j Z B rk ( xk ) e a n p | u j | nn − dµ< ∞ . Next assume κ = 1, then for some x ∈ K , σ ( { x } ) = 1. Since Z M |∇ u | n dµ + σ ( M ) ≤ , we see u must be a constant function and σ = δ x . Using u = 0, we see u = 0.After passing to a subsequence, we can assume u i → r >
0, it followsfrom the first case that e a n | u i | nn − is bounded in L q ( M \ B r ( x )) for any q < ∞ ,hence e a n | u i | nn − → L ( M \ B r ( x )). This together with the fact Z M e a n | u i | nn − dµ ≤ c ( M, g )implies that after passing to a subsequence, e a n | u i | nn − → c δ x as measure forsome c ≥ Theorem 2.1.
Assume α > , m i > , m i → ∞ , u i ∈ W ,n ( M n ) such that u i = 0 and log Z M e nm i u i dµ ≥ αm ni . (2.14) We also assume u i ⇀ u weakly in W ,n ( M ) , |∇ u i | n dµ → |∇ u | n dµ + σ as measureand e nm i u i R M e nm i u i dµ → ν (2.15) FENGBO HANG as measure. Let (cid:8) x ∈ M : σ ( { x } ) ≥ α − n α (cid:9) = { x , · · · , x N } , (2.16) here α n = (cid:18) n − n (cid:19) n − | S n − | , (2.17) then ν = N X i =1 ν i δ x i , (2.18) here ν i ≥ and P Ni =1 ν i = 1 .Proof. Assume x ∈ M such that σ ( { x } ) < α − n α , then we claim that for some r > ν ( B r ( x )) = 0. Indeed we fix p such that α n − n α − n − < p < σ ( { x } ) − n − , (2.19)it follows from Proposition 2.1 that for some r > Z B r ( x ) e a n p | u i | nn − dµ ≤ c, (2.20)here c is a positive constant independent of i . By Young’s inequality we have nm i u i ≤ a n p | u i | nn − + α n m ni p n − , (2.21)hence Z B r ( x ) e nm i u i dµ ≤ ce αnmnipn − . It follows that R B r ( x ) e nm i u i dµ R M e nm i u i dµ ≤ ce (cid:16) αnpn − − α (cid:17) m ni . In particular, ν ( B r ( x )) ≤ lim inf i →∞ R B r ( x ) e nm i u i dµ R M e nm i u i dµ = 0 . We get ν ( B r ( x )) = 0. The claim is proved.Clearly the claim implies ν ( M \ { x , · · · , x N } ) = 0 . (2.22)Hence ν = P Ni =1 ν i δ x i , with ν i ≥ P Ni =1 ν i = 1.It is worth pointing out that the arguments for Proposition 2.1 and Theorem 2.1can be easily modified to work for functions satisfying various boundary conditionsor belonging to higher order Sobolev spaces. We will discuss these examples inSection 4. ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 7 A generalization of Aubin inequality to higher order momentscase on S n Here we will extend Aubin’s inequality for functions on S n with zero first ordermoments of the area element (see [A, Corollary 2, p159]) to higher order momentscases. For n = 2, this is done in [CH].First we introduce some notations. For a nonnegative integer k , we denote P k = P k (cid:0) R n +1 (cid:1) = (cid:8) all polynomials on R n +1 with degree at most k (cid:9) ;(3.1) ◦ P k = ◦ P k (cid:0) R n +1 (cid:1) = (cid:26) p ∈ P k : Z S n pdµ = 0 (cid:27) . (3.2)Here µ is the standard measure on S n . Definition 3.1.
For m ∈ N , let N m ( S n )= { N ∈ N : ∃ x , · · · , x N ∈ S n and ν , · · · , ν N ∈ [0 , ∞ ) s.t. ν + · · · + ν N = 1 and for any p ∈ ◦ P m , ν p ( x ) + · · · + ν N p ( x N ) = 0 . (cid:27) = { N ∈ N : ∃ x , · · · , x N ∈ S n and ν , · · · , ν N ∈ [0 , ∞ ) s.t. for any p ∈ P m , ν p ( x ) + · · · + ν N p ( x N ) = 1 | S n | Z S n pdµ. (cid:27) ,and N m ( S n ) = min N m ( S n ) . (3.3)As in [CH], every choice of ν , · · · , ν N and x , · · · , x N in Definition 3.1 corre-sponds to an algorithm for numerical integration of functions on S n (see [Co, HSW]for further discussion). Theorem 3.1.
Assume u ∈ W ,n ( S n ) such that R S n udµ = 0 (here µ is the standardmeasure on S n ) and for every p ∈ ◦ P m , R S n pe nu dµ = 0 , then for any ε > , we have log Z S n e nu dµ ≤ (cid:18) α n N m ( S n ) + ε (cid:19) k∇ u k nL n + c ( m, n, ε ) . (3.4) Here N m ( S n ) is defined in Definition 3.1 and α n = (cid:18) n − n (cid:19) n − | S n − | . Proof.
Let α = α n N m ( S n ) + ε . If the inequality is not true, then there exists v i ∈ W ,n ( S n ) such that v i = 0, R S n pe nv i dµ = 0 for all p ∈ ◦ P m andlog Z S n e nv i dµ − α k∇ v i k nL n → ∞ as i → ∞ . In particular R S n e nv i dµ → ∞ . Sincelog Z S n e nv i dµ ≤ α n k∇ v i k nL n + c ( n ) , FENGBO HANG we see k∇ v i k L n → ∞ . Let m i = k∇ v i k L n and u i = v i m i , then m i → ∞ , k∇ u i k L n =1 and u i = 0. After passing to a subsequence, we have u i ⇀ u weakly in W ,n ( S n ) ;log Z S n e nm i u i dµ − αm ni → ∞ ; |∇ u i | n dµ → |∇ u | n dµ + σ as measure; e nm i u i R S n e nm i u i dµ → ν as measure.Let (cid:8) x ∈ S n : σ ( { x } ) ≥ α − n α (cid:9) = { x , · · · , x N } , (3.5)then it follows from Theorem 2.1 that ν = N X i =1 ν i δ x i . (3.6)Moreover ν i ≥ P Ni =1 ν i = 1 and N X i =1 ν i p ( x i ) = 0for any p ∈ ◦ P m . Hence N ∈ N m ( S n ). It follows from σ ( S n ) ≤ α − n αN ≤ , We get α ≤ α n N ≤ α n N m ( S n ) . This contradicts with the choice of α .Indeed, what we get from the above argument is the following Theorem 3.2. If u ∈ W ,n ( S n ) such that R S n udµ = 0 (here µ is the standardmeasure on S n ) and for every p ∈ ◦ P m , (cid:12)(cid:12)(cid:12)(cid:12)Z S n pe nu dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ b ( p ) , (3.7) here b ( p ) is a nonnegative number depending only on p , then for any ε > , log Z S n e nu dµ ≤ (cid:18) α n N m ( S n ) + ε (cid:19) k∇ u k nL n + c ( m, n, b, ε ) . (3.8) Here N m ( S n ) is defined in Definition 3.1 and α n = (cid:18) n − n (cid:19) n − | S n − | . We remark that the constant α n N m ( S n ) + ε is almost optimal in the following sense:If a ≥ c ∈ R such that for any u ∈ W ,n ( S n ) with u = 0 and R S n pe nu dµ = 0for every p ∈ ◦ P m , we havelog Z S n e nu dµ ≤ a k∇ u k nL n + c, (3.9) ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 9 then a ≥ α n N m ( S n ) . This claim can be proved almost the same way as the argumentin [CH, Lemma 3.1]. For reader’s convenience we sketch the proof here.First we note that we can rewrite the assumption as for any u ∈ W ,n ( S n ) with R S n pe nu dµ = 0 for every p ∈ ◦ P m , we havelog Z S n e nu dµ ≤ a k∇ u k nL n + nu + c. (3.10)Assume N ∈ N , x , · · · , x N ∈ S n and ν , · · · , ν N ∈ [0 , ∞ ) s.t. ν + · · · + ν N = 1and for any p ∈ ◦ P m , ν p ( x ) + · · · + ν N p ( x N ) = 0. We will prove a ≥ α n N .The above remark follows. Without losing of generality we can assume ν i > ≤ i ≤ N and x i = x j for 1 ≤ i < j ≤ N .For x, y ∈ S n , we denote xy as the geodesic distance between x and y on S n . For r > x ∈ S n , we denote B r ( x ) as the geodesic ball with radius r and center x i.e. B r ( x ) = { y ∈ S n : xy < r } .Let δ > ≤ i < j ≤ N , B δ ( x i ) ∩ B δ ( x j ) = ∅ .For 0 < ε < δ , we let φ ε ( t ) = nn − log δε , < t < ε ; nn − log δt , ε < t < δ ;0 , t > δ. (3.11)If b ∈ R , then we write φ ε,b ( t ) = φ ε ( t ) + b, < t < δ ; b (cid:0) − tδ (cid:1) , δ < t < δ ;0 , t > δ. (3.12)Let v ( x ) = N X i =1 φ ε, n log ν i ( xx i ) , (3.13)then calculation shows Z S n e nv dµ = (cid:12)(cid:12) S n − (cid:12)(cid:12) δ n n − ε − nn − + O (cid:18) log 1 ε (cid:19) (3.14)as ε → + .For p ∈ ◦ P m , using N X i =1 ν i p ( x i ) = 0 , (3.15)we can show Z S n e nv pdµ = O (cid:18) log 1 ε (cid:19) (3.16)as ε → + .To get a test function satisfying orthogornality condition, we need to do somecorrections. Let us fix a base of ◦ P m (cid:12)(cid:12)(cid:12)(cid:12) S n , namely p | S n , · · · , p l | S n , here p , · · · , p l ∈ ◦ P m . We first claim that there exists ψ , · · · , ψ l ∈ C ∞ c S n \ N [ i =1 B δ ( x i ) ! such that the determinant det (cid:20)Z S n ψ j p k dµ (cid:21) ≤ j,k ≤ l = 0 . (3.17)Indeed, fix a nonzero smooth function η ∈ C ∞ c S n \ N [ i =1 B δ ( x i ) ! , then ηp , · · · , ηp l are linearly independent. It follows that the matrix (cid:20)Z S n η p j p k dµ (cid:21) ≤ j,k ≤ l is positive definite. Then ψ j = η p j satisfies the claim.It follows from (3.17) that we can find β , · · · , β l ∈ R such that Z S n e nv + l X j =1 β j ψ j p k dµ = 0 (3.18)for k = 1 , · · · , l . Moreover β j = O (cid:18) log 1 ε (cid:19) (3.19)as ε → + . As a consequence we can find a constant c > l X j =1 β j ψ j + c log 1 ε ≥ log 1 ε . (3.20)We define u as e nu = e nv + l X j =1 β j ψ j + c log 1 ε . (3.21)Note this u will be the test function we use to prove our remark.It follows from (3.18) that R S n e nu pdµ = 0 for all p ∈ ◦ P m . Moreover using (3.14)and (3.19) we see Z S n e nu dµ = (cid:12)(cid:12) S n − (cid:12)(cid:12) δ n n − ε − nn − + O (cid:18) log 1 ε (cid:19) (3.22)= (cid:12)(cid:12) S n − (cid:12)(cid:12) δ n n − ε − nn − (1 + o (1)) , hence log Z S n e nu dµ = nn − ε + O (1) (3.23)as ε → + . Calculation shows u = o (cid:18) log 1 ε (cid:19) (3.24)and Z S n |∇ u | n dµ = (cid:18) nn − (cid:19) n (cid:12)(cid:12) S n − (cid:12)(cid:12) N log 1 ε + o (cid:18) log 1 ε (cid:19) . (3.25)We plug u into (3.10) and get nn − ε ≤ (cid:18) nn − (cid:19) n (cid:12)(cid:12) S n − (cid:12)(cid:12) N a log 1 ε + o (cid:18) log 1 ε (cid:19) . Divide log ε on both sides and let ε → + , we see a ≥ α n N . ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 11
It is clear that N ( S n ) = 2. Hence Aubin’s theorem [A, Corollary 2 on p159]follows from Theorem 3.1. Lemma 3.1. N ( S n ) = n + 2 . Corollary 3.1.
Assume u ∈ W ,n ( S n ) such that R S n udµ = 0 (here µ is the stan-dard measure on S n ) and for every p ∈ ◦ P , (cid:12)(cid:12)(cid:12)(cid:12)Z S n pe nu dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ b ( p ) , here b ( p ) is a nonnegative number depending only on p , then for any ε > , wehave log Z S n e nu dµ ≤ (cid:18) α n n + 2 + ε (cid:19) k∇ u k nL n + c ( n, b, ε ) . (3.26)We will prove Lemma 3.1 after some preparations. In R N , we have the hyper-plane H = (cid:8) x ∈ R N : x + · · · + x N = 1 (cid:9) . (3.27)Here x , · · · , x N are the coordinates of x . Let e , · · · , e N be the standard base of R N and y = 1 N ( e + · · · + e N ) . (3.28)We denote Σ = ( x ∈ H : | x − y | = r N − N ) . (3.29)Note that Σ is N − q N − N . Lemma 3.2.
For any p ∈ P (cid:0) R N (cid:1) , we have | Σ | Z Σ pdS = 1 N N X i =1 p ( e i ) . (3.30) Here dS is the standard measure on Σ , and | Σ | is the measure of Σ under dS .Proof. For 1 ≤ i ≤ N , we have1 | Σ | Z Σ x i dS = 1 | Σ | Z Σ x dS. Moreover 1 | Σ | Z Σ ( x + · · · + x N ) dS = 1 , hence | Σ | R Σ x dS = N . It follows that for 1 ≤ i ≤ N ,1 | Σ | Z Σ x i dS = 1 N . (3.31)To continue we observe that for 1 ≤ i ≤ N ,1 | Σ | Z Σ x i dS = 1 | Σ | Z Σ x dS. On the other hand, 1 | Σ | Z Σ N X i =1 (cid:18) x i − N (cid:19) dS = N − N , developing the identity out we get for 1 ≤ i ≤ N ,1 | Σ | Z Σ x i dS = 1 N . (3.32)At last we claim for 1 ≤ i < j ≤ N ,1 | Σ | Z Σ x i x j dS = 0 . (3.33)Indeed this follows from 1 | Σ | Z Σ x i x j dS = 1 | Σ | Z Σ x x dS ;1 | Σ | Z Σ ( x + · · · + x N ) dS = 1 , and (3.32). Lemma 3.2 follows from (3.31), (3.32) and (3.33).Now we are ready to prove Lemma 3.1. Proof of Lemma 3.1.
Assume N ∈ N ( S n ), we claim N ≥ n + 2. If this is not thecase, then N < n + 2. We can find x , · · · , x N ∈ S n , ν , · · · , ν N ≥ ν p ( x ) + · · · + ν N p ( x N ) = 1 | S n | Z S n pdµ for every p ∈ P (cid:0) R n +1 (cid:1) . In particular ν + · · · + ν N = 1; ν x + · · · + ν N x N = 0 . Let V = span { x , · · · , x N } , then dim V ≤ N − ≤ n . Hence we can find a nonzero vector ξ ∈ V ⊥ . Let p ( x ) = ( ξ · x ) , then0 = ν p ( x ) + · · · + ν N p ( x N ) = 1 | S n | Z S n pdµ > . A contradiction. Hence N ( S n ) ≥ n + 2.On the other hand, it follows from Lemma 3.2 and dilation, translation andorthogornal transformation that the vertex of a regular ( n + 1)-simplex embeddedin the unit ball, namely x , · · · , x n +2 , satisfies1 n + 2 p ( x ) + · · · + 1 n + 2 p ( x n +2 ) = 1 | S n | Z S n pdµ for every p ∈ P (cid:0) R n +1 (cid:1) . Hence N ( S n ) ≤ n + 2. ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 13 Further discussions
In this section we show that our approach above can be modified without mucheffort for higher order Sobolev spaces and Sobolev spaces on surfaces with nonemptyboundary. For reader’s convenience we carefully write down theorems in everycase considered, although they seem a little repetitive. Another reason is thesestatements are of interest themselves.4.1. W s, ns ( M n ) for even s . Recall ( M n , g ) is a smooth compact Riemannianmanifold with dimension n . Let s ∈ N be an even number strictly less than n , wehave the usual Sobolev space W s, ns ( M ) with norm k u k W s, ns ( M ) = s X k =0 Z M (cid:12)(cid:12) D k u (cid:12)(cid:12) ns dµ ! sn (4.1)for u ∈ W s, ns ( M ). Here D k is the differentiation associated with the naturalconnection of g . Standard elliptic theory tells us k u k W s, ns ≤ c ( M, g ) (cid:0)(cid:13)(cid:13) ∆ s u (cid:13)(cid:13) L ns + k u k L ns (cid:1) . (4.2)We also have the Poincare inequality k u − u k L ns ≤ c ( M, g ) (cid:13)(cid:13) ∆ s u (cid:13)(cid:13) L ns (4.3)for u ∈ W s, ns ( M ).Let a s,n = n | S n − | π n s Γ (cid:0) s (cid:1) Γ (cid:0) n − s (cid:1) ! nn − s . (4.4)Here Γ is the usual Gamma function i.e.Γ ( α ) = Z ∞ t α − e − t dt (4.5)for α >
0. The Moser-Trudinger inequality (see [BCY, F]) tells us that for every u ∈ W s, ns ( M ) \ { } with u = 0, we have Z M exp a s,n | u | nn − s (cid:13)(cid:13) ∆ s u (cid:13)(cid:13) nn − s L ns dµ ≤ c ( M, g ) . (4.6)It follows from (4.6) and Young’s inequality that for any u ∈ W s, ns ( M ) with u = 0,we have the Moser-Trudinger-Onofri inequalitylog Z M e nu dµ ≤ α s,n (cid:13)(cid:13) ∆ s u (cid:13)(cid:13) ns L ns + c ( M, g ) . (4.7)Here α s,n = s (cid:18) n − sn (cid:12)(cid:12) S n − (cid:12)(cid:12)(cid:19) n − ss Γ (cid:0) n − s (cid:1) π n s Γ (cid:0) s (cid:1) ! ns . (4.8)Again we start from a basic qualitative property of functions in W s, ns ( M ). Lemma 4.1.
For any u ∈ W s, ns ( M ) and a > , we have Z M e a | u | nn − s dµ < ∞ . (4.9) This follows from (4.6) exactly in the same way as the proof of Lemma 2.1.Indeed the argument there is written in a way working for higher order Sobolevspaces as well.Next we will derive an analogy of Proposition 2.1 for W s, ns ( M ). Proposition 4.1.
Assume s ∈ N is an even number strictly less than n , u i ∈ W s, ns ( M n ) such that u i = 0 , u i ⇀ u weakly in W s, ns ( M ) and (cid:12)(cid:12) ∆ s u i (cid:12)(cid:12) ns dµ → (cid:12)(cid:12) ∆ s u (cid:12)(cid:12) ns dµ + σ (4.10) as measure. If x ∈ M and p ∈ R such that < p < σ ( { x } ) − sn − s , then for some r > , sup i Z B r ( x ) e a s,n p | u i | nn − s dµ < ∞ . (4.11) Here a s,n = n | S n − | π n s Γ (cid:0) s (cid:1) Γ (cid:0) n − s (cid:1) ! nn − s . (4.12) Proof.
Fix p ∈ (cid:16) p, σ ( { x } ) − sn − s (cid:17) , then σ ( { x } ) < p n − ss . (4.13)We can find a ε > ε ) σ ( { x } ) < p n − ss (4.14)and (1 + ε ) p < p . (4.15)Let v i = u i − u , then v i ⇀ W s, ns ( M ), v i → W s − , ns ( M ). For any ϕ ∈ C ∞ ( M ), we have (cid:12)(cid:12) ∆ s ( ϕv i ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ϕ ∆ s v i (cid:12)(cid:12) + c s − X k =0 (cid:12)(cid:12) D s − k ϕ (cid:12)(cid:12) (cid:12)(cid:12) D k v i (cid:12)(cid:12) ≤ (cid:12)(cid:12) ϕ ∆ s u i (cid:12)(cid:12) + (cid:12)(cid:12) ϕ ∆ s u (cid:12)(cid:12) + c s − X k =0 (cid:12)(cid:12) D s − k ϕ (cid:12)(cid:12) (cid:12)(cid:12) D k v i (cid:12)(cid:12) . Hence (cid:13)(cid:13) ∆ s ( ϕv i ) (cid:13)(cid:13) ns L ns ≤ (cid:13)(cid:13) ϕ ∆ s u i (cid:13)(cid:13) L ns + (cid:13)(cid:13) ϕ ∆ s u (cid:13)(cid:13) L ns + c s − X k =0 (cid:13)(cid:13)(cid:12)(cid:12) D s − k ϕ (cid:12)(cid:12) (cid:12)(cid:12) D k v i (cid:12)(cid:12)(cid:13)(cid:13) L ns ! ns ≤ (1 + ε ) (cid:13)(cid:13) ϕ ∆ s u i (cid:13)(cid:13) ns L ns + c ( ε ) (cid:13)(cid:13) ϕ ∆ s u (cid:13)(cid:13) ns L ns + c ( ε ) s − X k =0 (cid:13)(cid:13)(cid:12)(cid:12) D s − k ϕ (cid:12)(cid:12) (cid:12)(cid:12) D k v i (cid:12)(cid:12)(cid:13)(cid:13) ns L ns . ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 15
Letting i → ∞ ,lim sup i →∞ (cid:13)(cid:13) ∆ s ( ϕv i ) (cid:13)(cid:13) ns L ns ≤ (1 + ε ) (cid:18)Z M | ϕ | ns dσ + Z M | ϕ | ns (cid:12)(cid:12) ∆ s u (cid:12)(cid:12) ns dµ (cid:19) + c ( ε ) (cid:13)(cid:13) ϕ ∆ s u (cid:13)(cid:13) ns L ns = (1 + ε ) Z M | ϕ | ns dσ + c ( ε ) Z M | ϕ | ns (cid:12)(cid:12) ∆ s u (cid:12)(cid:12) ns dµ. We can find a ϕ ∈ C ∞ ( M ) such that ϕ | B r ( x ) = 1 for some r > ε ) Z M | ϕ | ns dσ + c ( ε ) Z M | ϕ | ns (cid:12)(cid:12) ∆ s u (cid:12)(cid:12) ns dµ < p n − ss . Hence for i large enough, we have (cid:13)(cid:13) ∆ s ( ϕv i ) (cid:13)(cid:13) ns L ns < p n − ss . In other words, (cid:13)(cid:13) ∆ s ( ϕv i ) (cid:13)(cid:13) nn − s L ns < p . We have Z B r ( x ) e a s,n p | v i − ϕv i | nn − s dµ ≤ Z M e a s,n p | ϕv i − ϕv i | nn − s dµ ≤ Z M e a s,n | ϕvi − ϕvi | nn − s (cid:13)(cid:13)(cid:13)(cid:13) ∆ s ( ϕvi ) (cid:13)(cid:13)(cid:13)(cid:13) nn − sL ns dµ ≤ c ( M, g ) . Next we observe that | u i | nn − s = | ( v i − ϕv i ) + u + ϕv i | nn − s ≤ (1 + ε ) | v i − ϕv i | nn − s + c ( ε ) | u | nn − s + c ( ε ) | ϕv i | nn − s , hence e a s,n | u i | nn − s ≤ e (1+ ε ) a s,n | v i − ϕv i | nn − s e c ( ε ) | u | nn − s e c ( ε ) | ϕv i | nn − s . Since e (1+ ε ) a s,n | v i − ϕv i | nn − s is bounded in L p ε ( B r ( x )), e c ( ε ) | u | nn − s ∈ L q ( B r ( x )) forany 0 < q < ∞ (by Lemma 4.1), e c ( ε ) | ϕv i | nn − s → i → ∞ and p ε > p , it followsfrom Holder inequality that e a s,n | u i | nn − s is bounded in L p ( B r ( x )).Proposition 4.1 together with a covering argument implies the following corollary,just like the proof of Corollary 2.1. Corollary 4.1.
Assume u i ∈ W s, ns ( M ) such that u i = 0 and (cid:13)(cid:13) ∆ s u i (cid:13)(cid:13) L ns ≤ . Wealso assume u i ⇀ u weakly in W s, ns ( M ) and (cid:12)(cid:12) ∆ s u i (cid:12)(cid:12) ns dµ → (cid:12)(cid:12) ∆ s u (cid:12)(cid:12) ns dµ + σ (4.16) as measure. Let K be a compact subset of M and κ = max x ∈ K σ ( { x } ) . (4.17) (1) If κ < , then for any ≤ p < κ − sn − s , sup i Z K e a s,n p | u i | nn − s dµ < ∞ . (4.18)(2) If κ = 1 , then σ = δ x for some x ∈ K , u = 0 and after passing to asubsequence, e a s,n | u i | nn − s → c δ x (4.19) as measure for some c ≥ . We are ready to derive an analogy of Theorem 2.1 for W s, ns ( M ). Theorem 4.1.
Let s ∈ N be an even number strictly less than n . Assume α > , m i > , m i → ∞ , u i ∈ W s, ns ( M n ) such that u i = 0 and log Z M e nm i u i dµ ≥ αm ns i . (4.20) We also assume u i ⇀ u weakly in W s, ns ( M ) , (cid:12)(cid:12) ∆ s u i (cid:12)(cid:12) ns dµ → (cid:12)(cid:12) ∆ s u (cid:12)(cid:12) ns dµ + σ asmeasure and e nm i u i R M e nm i u i dµ → ν (4.21) as measure. Let (cid:8) x ∈ M : σ ( { x } ) ≥ α − s,n α (cid:9) = { x , · · · , x N } , (4.22) here α s,n = s (cid:18) n − sn (cid:12)(cid:12) S n − (cid:12)(cid:12)(cid:19) n − ss Γ (cid:0) n − s (cid:1) π n s Γ (cid:0) s (cid:1) ! ns , (4.23) then ν = N X i =1 ν i δ x i , (4.24) here ν i ≥ and P Ni =1 ν i = 1 .Proof. Assume x ∈ M such that σ ( { x } ) < α − s,n α , then we claim for some r > ν ( B r ( x )) = 0. Indeed we fix p such that α sn − s s,n α − sn − s < p < σ ( { x } ) − sn − s , (4.25)it follows from Proposition 4.1 that for some r > Z B r ( x ) e a s,n p | u i | nn − s dµ ≤ c, (4.26)here c is a positive constant independent of i . By Young’s inequality we have nm i u i ≤ a s,n p | u i | nn − s + α s,n m ns i p n − ss , (4.27)hence Z B r ( x ) e nm i u i dµ ≤ ce αs,nm nsip n − ss . ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 17
It follows that R B r ( x ) e nm i u i dµ R M e nm i u i dµ ≤ ce αs,np n − ss − α ! m nsi . In particular, ν ( B r ( x )) ≤ lim inf i →∞ R B r ( x ) e nm i u i dµ R M e nm i u i dµ = 0 . We get ν ( B r ( x )) = 0. The claim is proved.Clearly the claim implies ν ( M \ { x , · · · , x N } ) = 0 . (4.28)Hence ν = P Ni =1 ν i δ x i , with ν i ≥ P Ni =1 ν i = 1.The analogy of Theorem 3.2 is the following Theorem 4.2.
Let s ∈ N be an even number strictly less than n . If u ∈ W s, ns ( S n ) such that R S n udµ = 0 (here µ is the standard measure on S n ) and for every p ∈ ◦ P m , (cid:12)(cid:12)(cid:12)(cid:12)Z S n pe nu dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ b ( p ) , (4.29) here b ( p ) is a nonnegative number depending only on p , then for any ε > , log Z S n e nu dµ ≤ (cid:18) α s,n N m ( S n ) + ε (cid:19) (cid:13)(cid:13) ∆ s u (cid:13)(cid:13) ns L ns + c ( m, n, b, ε ) . (4.30) Here N m ( S n ) is defined in Definition 3.1 and α s,n = s (cid:18) n − sn (cid:12)(cid:12) S n − (cid:12)(cid:12)(cid:19) n − ss Γ (cid:0) n − s (cid:1) π n s Γ (cid:0) s (cid:1) ! ns . (4.31) Proof.
Let α = α s,n N m ( S n ) + ε . If the inequality (4.30) is not true, then there exists v i ∈ W s, ns ( S n ) such that v i = 0, (cid:12)(cid:12)(cid:12)(cid:12)Z S n pe nv i dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ b ( p ) , for all p ∈ ◦ P m and log Z S n e nv i dµ − α (cid:13)(cid:13) ∆ s v i (cid:13)(cid:13) ns L ns → ∞ as i → ∞ . In particular R S n e nv i dµ → ∞ . By (4.7),log Z S n e nv i dµ ≤ α s,n (cid:13)(cid:13) ∆ s v i (cid:13)(cid:13) ns L ns + c ( n ) , hence (cid:13)(cid:13) ∆ s v i (cid:13)(cid:13) L ns → ∞ . Let m i = (cid:13)(cid:13) ∆ s v i (cid:13)(cid:13) L ns and u i = v i m i , then m i → ∞ , (cid:13)(cid:13) ∆ s u i (cid:13)(cid:13) L ns = 1 and u i = 0. After passing to a subsequence, we have u i ⇀ u weakly in W s, ns ( S n ) ;log Z S n e nm i u i dµ − αm ns i → ∞ ; (cid:12)(cid:12) ∆ s u i (cid:12)(cid:12) ns dµ → (cid:12)(cid:12) ∆ s u (cid:12)(cid:12) ns dµ + σ as measure; e nm i u i R S n e nm i u i dµ → ν as measure.Let (cid:8) x ∈ S n : σ ( { x } ) ≥ α − s,n α (cid:9) = { x , · · · , x N } , (4.32)then it follows from Theorem 4.1 that ν = N X i =1 ν i δ x i . (4.33)Moreover ν i ≥ P Ni =1 ν i = 1 and N X i =1 ν i p ( x i ) = 0for any p ∈ ◦ P m . It follows that α − s,n αN ≤ , and N ∈ N m ( S n ). We get α ≤ α s,n N ≤ α s,n N m ( S n ) . This contradicts with the choice of α .Similar as before, the constant α s,n N m ( S n ) + ε is almost optimal. It is also interestingto compare Theorem 4.2 with [BCY, lemma 4.3] and [CY2, lemma 4.6].4.2. Paneitz operator in dimension . Let (cid:0) M , g (cid:1) be a smooth compact Rie-mannian manifold with dimension 4. The Paneitz operator is given by (see [CY2,HY]) P u = ∆ u + 2 div ( Rc ( ∇ u, e i ) e i ) −
23 div ( R ∇ u ) . (4.34)Here e , e , e , e is a local orthonormal frame with respect to g . The associated Q curvature is Q = −
16 ∆ R − | Rc | + 16 R . (4.35)In 4-dimensional conformal geometry, P and Q play the same roles as − ∆ andGauss curvature in 2-dimensional conformal geometry.For u ∈ C ∞ ( M ), let E ( u ) = Z M P u · udµ (4.36)= Z M (cid:18) (∆ u ) − Rc ( ∇ u, ∇ u ) + 23 R |∇ u | (cid:19) dµ. ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 19
By this formula, we know E ( u ) still makes sense for u ∈ H ( M ) = W , ( M ).In [CY2], it is shown that if P ≥ P = { constant functions } , then forany u ∈ H ( M ) \ { } with u = 0, Z M e π u E ( u ) dµ ≤ c ( M, g ) . (4.37)In particular log Z M e u dµ ≤ π E ( u ) + c ( M, g ) . (4.38)Note that 32 π = a , and π = α , .We want to remark that if P ≥ P = { constant functions } , then forany u ∈ H ( M ) with u = 0, k u k L ≤ c ( M, g ) E ( u ) . (4.39)On the other hand, it follows from standard elliptic theory (or the Bochner’s iden-tity) that k u k H ≤ c ( M, g ) (cid:16) k ∆ u k L + k u k L (cid:17) ≤ c ( M, g ) (cid:16) E ( u ) + k u k H (cid:17) ≤ c ( M, g ) E ( u ) + 12 k u k H + c ( M, g ) k u k L ≤ k u k H + c ( M, g ) E ( u ) . We have used the interpolation inequality in between. It follows that k u k H ≤ c ( M, g ) E ( u ) . (4.40)It is also worth pointing out that on the standard S , P ≥ P = { constant functions } . Moreover in [Gur, GurV], some general criterion for suchpositivity condition to be valid were derived. Proposition 4.2.
Let (cid:0) M , g (cid:1) be a smooth compact Riemannian manifold with P ≥ and ker P = { constant functions } . Assume u i ∈ H ( M ) such that u i = 0 and E ( u i ) ≤ . We also assume u i ⇀ u weakly in H ( M ) and (∆ u i ) dµ → (∆ u ) dµ + σ (4.41) as measure. Let K be a compact subset of M and κ = max x ∈ K σ ( { x } ) . (4.42)(1) If κ < , then for any ≤ p < κ , sup i Z K e π pu i dµ < ∞ . (4.43)(2) If κ = 1 , then σ = δ x for some x ∈ K , u = 0 and after passing to asubsequence, e π u i → c δ x (4.44) as measure for some c ≥ . Proof.
Since u i ⇀ u weakly in H ( M ), we see u i → u in H ( M ). It follows that u = 0 and E ( u i ) → E ( u ) + σ ( M ) . Since E ( u i ) ≤
1, we get E ( u ) + σ ( M ) ≤ . By the assumption on P we know E ( u ) ≥ E ( u ) = 0 if and only if u = 0, hence σ ( M ) ≤
1. With these facts at hand, Proposition 4.2 follows from Proposition 4.1by the same argument as in the proof of Corollary 2.1.On the standard S , we have P = ∆ −
2∆ and E ( u ) = Z S (cid:16) (∆ u ) + 2 |∇ u | (cid:17) dµ ≥ k ∆ u k L for u ∈ H (cid:0) S (cid:1) . It follows from this inequality and Theorem 4.2 that Proposition 4.3. If u ∈ H (cid:0) S (cid:1) such that R S udµ = 0 (here µ is the standardmeasure on S ) and for every p ∈ ◦ P m , (cid:12)(cid:12)(cid:12)(cid:12)Z S pe nu dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ b ( p ) , (4.45) here b ( p ) is a nonnegative number depending only on p , then for any ε > , log Z S e u dµ ≤ (cid:18) π N m ( S ) + ε (cid:19) Z S (cid:16) (∆ u ) + 2 |∇ u | (cid:17) dµ + c ( m, b, ε ) . (4.46) Here N m (cid:0) S (cid:1) is defined in Definition 3.1. In view of Proposition 4.3 and the recent proof of sharp version of Aubin’sMoser-Trudinger inequality on S in [GuM], it is tempting to conjecture that for u ∈ H (cid:0) S (cid:1) with R S udµ = 0 and Z S x i e u dµ ( x ) = 0 for 1 ≤ i ≤
5, (4.47)we have log (cid:18) | S | Z S e u dµ (cid:19) ≤ π Z S (cid:16) (∆ u ) + 2 |∇ u | (cid:17) dµ. (4.48)In a recent work [Gu], some progress has been made toward proving (4.48) foraxially symmetric functions.4.3. W s, ns ( M n ) for odd s . Let s ∈ N be an odd number strictly less than n .Denote a s,n = n | S n − | π n s Γ (cid:0) s +12 (cid:1) Γ (cid:0) n − s +12 (cid:1) ! nn − s . (4.49)The Moser-Trudinger inequality (see [F]) tells us that for every u ∈ W s, ns ( M ) \ { } with u = 0, Z M exp a s,n | u | nn − s (cid:13)(cid:13)(cid:13) ∇ ∆ s − u (cid:13)(cid:13)(cid:13) nn − s L ns dµ ≤ c ( M, g ) . (4.50) ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 21
This implies the Moser-Trudinger-Onofri inequalitylog Z M e nu dµ ≤ α s,n (cid:13)(cid:13)(cid:13) ∇ ∆ s − u (cid:13)(cid:13)(cid:13) ns L ns + c ( M, g ) . (4.51)Here α s,n = s (cid:18) n − sn (cid:12)(cid:12) S n − (cid:12)(cid:12)(cid:19) n − ss Γ (cid:0) n − s +12 (cid:1) π n s Γ (cid:0) s +12 (cid:1) ! ns . (4.52) Proposition 4.4.
Assume s ∈ N is an odd number strictly less than n , u i ∈ W s, ns ( M n ) such that u i = 0 , u i ⇀ u weakly in W s, ns ( M ) and (cid:12)(cid:12)(cid:12) ∇ ∆ s − u i (cid:12)(cid:12)(cid:12) ns dµ → (cid:12)(cid:12)(cid:12) ∇ ∆ s − u (cid:12)(cid:12)(cid:12) ns dµ + σ (4.53) as measure. If x ∈ M and p ∈ R such that < p < σ ( { x } ) − sn − s , then for some r > , sup i Z B r ( x ) e a s,n p | u i | nn − s dµ < ∞ . (4.54) Here a s,n = n | S n − | π n s Γ (cid:0) s +12 (cid:1) Γ (cid:0) n − s +12 (cid:1) ! nn − s . (4.55)Since the proof of Proposition 4.4 is almost identical to the proof of Proposition4.1, we omit it here. The same will happen to Corollary 4.2 and Theorems 4.3, 4.4below. Corollary 4.2.
Assume u i ∈ W s, ns ( M ) such that u i = 0 and (cid:13)(cid:13)(cid:13) ∇ ∆ s − u i (cid:13)(cid:13)(cid:13) L ns ≤ .We also assume u i ⇀ u weakly in W s, ns ( M ) and (cid:12)(cid:12)(cid:12) ∇ ∆ s − u i (cid:12)(cid:12)(cid:12) ns dµ → (cid:12)(cid:12)(cid:12) ∇ ∆ s − u (cid:12)(cid:12)(cid:12) ns dµ + σ (4.56) as measure. Let K be a compact subset of M and κ = max x ∈ K σ ( { x } ) . (4.57)(1) If κ < , then for any ≤ p < κ − sn − s , sup i Z K e a s,n p | u i | nn − s dµ < ∞ . (4.58)(2) If κ = 1 , then σ = δ x for some x ∈ K , u = 0 and after passing to asubsequence, e a s,n | u i | nn − s → c δ x (4.59) as measure for some c ≥ . Theorem 4.3.
Let s ∈ N be an odd number strictly less than n . Assume α > , m i > , m i → ∞ , u i ∈ W s, ns ( M n ) such that u i = 0 and log Z M e nm i u i dµ ≥ αm ns i . (4.60) We also assume u i ⇀ u weakly in W s, ns ( M ) , (cid:12)(cid:12)(cid:12) ∇ ∆ s − u i (cid:12)(cid:12)(cid:12) ns dµ → (cid:12)(cid:12)(cid:12) ∇ ∆ s − u (cid:12)(cid:12)(cid:12) ns dµ + σ as measure and e nm i u i R M e nm i u i dµ → ν (4.61) as measure. Let (cid:8) x ∈ M : σ ( { x } ) ≥ α − s,n α (cid:9) = { x , · · · , x N } , (4.62) here α s,n = s (cid:18) n − sn (cid:12)(cid:12) S n − (cid:12)(cid:12)(cid:19) n − ss Γ (cid:0) n − s +12 (cid:1) π n s Γ (cid:0) s +12 (cid:1) ! ns , (4.63) then ν = N X i =1 ν i δ x i , (4.64) here ν i ≥ and P Ni =1 ν i = 1 . Theorem 4.4.
Let s ∈ N be an odd number strictly less than n . If u ∈ W s, ns ( S n ) such that R S n udµ = 0 (here µ is the standard measure on S n ) and for every p ∈ ◦ P m , (cid:12)(cid:12)(cid:12)(cid:12)Z S n pe nu dµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ b ( p ) , (4.65) here b ( p ) is a nonnegative number depending only on p , then for any ε > , log Z S n e nu dµ ≤ (cid:18) α s,n N m ( S n ) + ε (cid:19) (cid:13)(cid:13)(cid:13) ∇ ∆ s − u (cid:13)(cid:13)(cid:13) ns L ns + c ( m, n, b, ε ) . (4.66) Here N m ( S n ) is defined in Definition 3.1 and α s,n = s (cid:18) n − sn (cid:12)(cid:12) S n − (cid:12)(cid:12)(cid:19) n − ss Γ (cid:0) n − s +12 (cid:1) π n s Γ (cid:0) s +12 (cid:1) ! ns . (4.67)4.4. Functions on compact surfaces with boundary.
In this subsection, weassume (cid:0) M , g (cid:1) is a smooth compact surface with nonempty boundary and asmooth Riemannian metric g .4.4.1. Functions with zero boundary value.
We denote H ( M ) = W , ( M ). Itfollows from [CY1] that for every u ∈ H ( M ) \ { } , Z M e π u k∇ u k L dµ ≤ c ( M, g ) . (4.68)As a consequence, log Z M e u dµ ≤ π k∇ u k L + c ( M, g ) . (4.69) Lemma 4.2.
For u ∈ H ( M ) and a > , we have Z M e au dµ < ∞ . (4.70) Proof.
Indeed, fix ε > v ∈ C ∞ c ( M ) such that k∇ u − ∇ v k L < ε .With this v and (4.68) at hands, we can proceed exactly the same way as in theproof of Lemma 2.1. ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 23
Proposition 4.5.
Assume u i ∈ H ( M ) such that u i ⇀ u weakly in H ( M ) , and |∇ u i | dµ → |∇ u | dµ + σ (4.71) as measure. If x ∈ M , p ∈ R satisfies < p < σ ( { x } ) , then for some r > , sup i Z B r ( x ) e πpu i dµ < ∞ . (4.72) Proof.
Fix p ∈ (cid:16) p, σ ( { x } ) (cid:17) , then σ ( { x } ) < p . (4.73)We can find a ε > ε ) σ ( { x } ) < p (4.74)and (1 + ε ) p < p . (4.75)Let v i = u i − u , then v i ⇀ H ( M ), v i → L ( M ). For any ϕ ∈ C ∞ ( M ), we have k∇ ( ϕv i ) k L ≤ ( k ϕ ∇ v i k L + k v i ∇ ϕ k L ) ≤ ( k ϕ ∇ u i k L + k ϕ ∇ u k L + k v i ∇ ϕ k L ) ≤ (1 + ε ) k ϕ ∇ u i k L + c ( ε ) k ϕ ∇ u k nL n + c ( ε ) k v i ∇ ϕ k nL n . It follows that lim sup i →∞ k∇ ( ϕv i ) k L ≤ (1 + ε ) (cid:18)Z M ϕ dσ + Z M ϕ |∇ u | dµ (cid:19) + c ( ε ) k ϕ ∇ u k L = (1 + ε ) Z M ϕ dσ + c ( ε ) Z M ϕ |∇ u | dµ. We can find a ϕ ∈ C ∞ ( M ) such that ϕ | B r ( x ) = 1 for some r > ε ) Z M ϕ dσ + c ( ε ) Z M ϕ |∇ u | dµ < p . Hence for i large enough, k∇ ( ϕv i ) k L < p . Note that ϕv i ∈ H ( M ). We have Z B r ( x ) e πp v i dµ ≤ Z M e πp ( ϕv i ) dµ ≤ Z M e π ( ϕvi ) k ∇ ( ϕvi ) k L dµ ≤ c ( M, g ) . Next we observe that u i = ( v i + u ) ≤ (1 + ε ) v i + c ( ε ) u , hence e πu i ≤ e π (1+ ε ) v i e c ( ε ) u . Since e π (1+ ε ) v i is bounded in L p ε ( B r ( x )), e c ( ε ) u ∈ L q ( B r ( x )) for any 0 < q < ∞ (by Lemma 4.2), and p ε > p , it follows from Holder inequality that e πu i isbounded in L p ( B r ( x )). Corollary 4.3.
Assume u i ∈ H ( M ) such that k∇ u i k L ≤ . We also assume u i ⇀ u weakly in H ( M ) and |∇ u i | dµ → |∇ u | dµ + σ (4.76) as measure. Let K be a compact subset of M and κ = max x ∈ K σ ( { x } ) . (4.77)(1) If κ < , then for any ≤ p < κ , sup i Z K e πp | u i | nn − dµ < ∞ . (4.78)(2) If κ = 1 , then σ = δ x for some x ∈ K , u = 0 and after passing to asubsequence, e πu i → c δ x (4.79) as measure for some c ≥ . With Proposition 4.5, we can derive Corollary 4.3 exactly in the same way asthe proof of Corollary 2.1.
Theorem 4.5.
Let (cid:0) M , g (cid:1) be a smooth compact Riemann surface with nonemptyboundary. Assume α > , m i > , m i → ∞ , u i ∈ H ( M ) and log Z M e m i u i dµ ≥ αm i . (4.80) We also assume u i ⇀ u weakly in H ( M ) , |∇ u i | dµ → |∇ u | dµ + σ as measureand e m i u i R M e m i u i dµ → ν (4.81) as measure. Let { x ∈ M : σ ( { x } ) ≥ πα } = { x , · · · , x N } , (4.82) then ν = N X i =1 ν i δ x i , (4.83) here ν i ≥ and P Ni =1 ν i = 1 . Theorem 4.5 follows from Proposition 4.5 by the same arguments as in the proofof Theorem 2.1.
ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 25
Functions with no boundary conditions.
Let H ( M ) = W , ( M ). It followsfrom [CY1] that for every u ∈ H ( M ) \ { } with u = 0, we have Z M e π u k∇ u k L dµ ≤ c ( M, g ) . (4.84)As a consequence, log Z M e u dµ ≤ π k∇ u k L + c ( M, g ) . (4.85) Lemma 4.3.
For any u ∈ H ( M ) and a > , we have Z M e au dµ < ∞ . (4.86)Using (4.84), the same argument as the proof of Lemma 2.1 implies Lemma 4.3. Proposition 4.6.
Let (cid:0) M , g (cid:1) be a smooth compact Riemann surface with nonemptyboundary. Assume u i ∈ H ( M ) such that u i = 0 , u i ⇀ u weakly in H ( M ) and |∇ u i | dµ → |∇ u | dµ + σ (4.87) as measure. Given x ∈ M and p ∈ R such that < p < σ ( { x } ) . (1) If x ∈ M \ ∂M , then for some r > , sup i Z B r ( x ) e πpu i dµ < ∞ . (4.88)(2) If x ∈ ∂M , then for some r > , sup i Z B r ( x ) e πpu i dµ < ∞ . (4.89) Proof.
Fix p ∈ (cid:16) p, σ ( { x } ) (cid:17) , then σ ( { x } ) < p . (4.90)We can find a ε > ε ) σ ( { x } ) < p (4.91)and (1 + ε ) p < p . (4.92)Let v i = u i − u , then v i ⇀ H ( M ), v i → L ( M ). For any ϕ ∈ C ∞ ( M ), we have k∇ ( ϕv i ) k L ≤ ( k ϕ ∇ v i k L + k v i ∇ ϕ k L ) ≤ ( k ϕ ∇ u i k L + k ϕ ∇ u k L + k v i ∇ ϕ k L ) ≤ (1 + ε ) k ϕ ∇ u i k L + c ( ε ) k ϕ ∇ u k nL n + c ( ε ) k v i ∇ ϕ k nL n . It follows that lim sup i →∞ k∇ ( ϕv i ) k L ≤ (1 + ε ) (cid:18)Z M ϕ dσ + Z M ϕ |∇ u | dµ (cid:19) + c ( ε ) k ϕ ∇ u k L = (1 + ε ) Z M ϕ dσ + c ( ε ) Z M ϕ |∇ u | dµ. If x ∈ M \ ∂M , then we can find a ϕ ∈ C ∞ c ( M ) such that ϕ | B r ( x ) = 1 for some r > ε ) Z M ϕ dσ + c ( ε ) Z M ϕ |∇ u | dµ < p . Hence for i large enough, we have k∇ ( ϕv i ) k L < p . Note that ϕv i ∈ H ( M ). We have Z B r ( x ) e πp v i dµ ≤ Z M e πp ( ϕv i ) dµ ≤ Z M e π ( ϕvi ) k ∇ ( ϕvi ) k L dµ ≤ c ( M, g ) . Next we observe that u i = ( v i + u ) ≤ (1 + ε ) v i + c ( ε ) u , hence e πu i ≤ e π (1+ ε ) v i e c ( ε ) u . Since e π (1+ ε ) v i is bounded in L p ε ( B r ( x )), e c ( ε ) u ∈ L q ( B r ( x )) for any 0 < q < ∞ (by Lemma 4.2), and p ε > p , it follows from Holder inequality that e πu i isbounded in L p ( B r ( x )).If x ∈ ∂M , then (4.84) and similar arguments as in the proof of Proposition 2.1tells us sup i Z B r ( x ) e πpu i dµ < ∞ . Corollary 4.4.
Assume u i ∈ H ( M ) such that u i = 0 and k∇ u i k L ≤ . We alsoassume u i ⇀ u weakly in H ( M ) and |∇ u i | dµ → |∇ u | dµ + σ (4.93) as measure. Let K be a compact subset of M and κ = max x ∈ K \ ∂M σ ( { x } ) ; (4.94) κ = max x ∈ K ∩ ∂M σ ( { x } ) ; (4.95) ONCENTRATION COMPACTNESS PRINCIPLE IN CRITICAL DIMENSION 27 (1) If κ < , then for any ≤ p < min n κ , κ o , sup i Z K e πpu i dµ < ∞ . (4.96)(2) If κ = 1 , then σ = δ x for some x ∈ ∂M , u = 0 and after passing to asubsequence, e πu i → c δ x (4.97) as measure for some c ≥ .Proof. First assume κ <
1. We claim for any x ∈ K , there exists a r x > i Z B rx ( x ) e πpu i dµ < ∞ . Once this claim is proved, we can deducesup i Z K e πpu i dµ < ∞ by the covering argument in the proof of Corollary 2.1.If x ∈ K \ ∂M , then p < κ ≤ σ ( { x } ) . Hence p < σ ( { x } ) . It follows from Proposition 4.6 that for some r > i Z B r ( x ) e πpu i dµ = sup i Z B r ( x ) e π · p u i dµ < ∞ . If x ∈ K ∩ ∂M , then p < κ ≤ σ ( { x } ) . It follows from Proposition 4.6 again that for some r > i Z B r ( x ) e πpu i dµ < ∞ . This proves the claim.The case when κ = 1 can be handled the same way as in the proof of Corollary2.1. Theorem 4.6.
Let (cid:0) M , g (cid:1) be a smooth compact Riemann surface with nonemptyboundary. Assume α > , m i > , m i → ∞ , u i ∈ H ( M ) such that u i = 0 and log Z M e m i u i dµ ≥ αm i . (4.98) We also assume u i ⇀ u weakly in H ( M ) , |∇ u i | dµ → |∇ u | dµ + σ as measureand e m i u i R M e m i u i dµ → ν (4.99) as measure. Let { x ∈ M \ ∂M : σ ( { x } ) ≥ πα } ∪ { x ∈ ∂M : σ ( { x } ) ≥ πα } (4.100)= { x , · · · , x N } , then ν = N X i =1 ν i δ x i , (4.101) here ν i ≥ and P Ni =1 ν i = 1 .Proof. Assume x ∈ M \ ∂M with σ ( { x } ) < πα , then we claim that for some r > ν ( B r ( x )) = 0. Indeed we fix p such that14 πα < p < σ ( { x } ) , (4.102)it follows from Proposition 4.6 that for some r > Z B r ( x ) e πpu i dµ ≤ c, (4.103)here c is a positive constant independent of i . We have2 m i u i ≤ πpu i + m i πp , (4.104)hence Z B r ( x ) e m i u i dµ ≤ ce m i πp . It follows that R B r ( x ) e m i u i dµ R M e m i u i dµ ≤ ce ( πp − α ) m i . In particular, ν ( B r ( x )) ≤ lim inf i →∞ R B r ( x ) e m i u i dµ R M e m i u i dµ = 0 . We get ν ( B r ( x )) = 0.If x ∈ ∂M with σ ( { x } ) < πα , then similar argument shows for some r > ν ( B r ( x )) = 0.Clearly these imply ν ( M \ { x , · · · , x N } ) = 0 . (4.105)Hence ν = P Ni =1 ν i δ x i , with ν i ≥ P Ni =1 ν i = 1. References [A] Aubin.
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