Energy identity for a class of approximate biharmonic maps into sphere in dimension four

We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in $L^p$ for some $p>1$. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on $\mathbb R^4$.

In this paper we will discuss the limiting behavior of weakly convergent sequences of approximate (extrinsic) biharmonic maps {u m } ⊂ W 2,2 (Ω, S k ) in dimension n = 4, especially an energy identity and oscillation convergence. First we recall the notion of approximate (extrinsic) biharmonic maps. Definition 1.1 A map u ∈ W 2,2 (Ω, S k ) is called an approximate biharmonic map if there exists a bi-tension field h ∈ L 1 loc (Ω, R k+1 ) such that ∆ 2 u + |∆u| 2 + ∆|∇u| 2 + 2 ∇u, ∇∆u u = h (1.1) in the distribution sense. In particular, if h = 0 then u is called a biharmonic map to S k .
It is an important observation that biharmonic maps are invariant under dilations in R n for n = 4. Such a property leads to non-compactness of biharmonic maps in dimension 4, which prompts recent studies by Wang [20] and Hornung-Moser [7] concerning the failure of strong convergence for weakly convergent biharmonic maps. Roughly speaking, the results in [20] and [7] assert that the failure of strong convergence occurs at finitely many concentration points of Hessian energy, where finitely many bubbles (i.e. nontrivial biharmonic maps on R 4 ) are generated, and the total Hessian energies from these bubbles account for the total loss of Hessian energies during the process of convergence.
Our first result is to extend the results from [20] and [7] to the context of suitable approximate biharmonic maps to S k . More precisely, we have Assume u m ⇀ u in W 2,2 and h m ⇀ h in L p . Then (i) u is an approximate biharmonic map to S k with h as its bi-tension field.
(ii) There exist a nonnegative integer L depending only on M and {x 1 , · · · , x L } ⊂ Ω such that u m → u strongly in W 2,2 loc ∩ C 0 loc (Ω\{x 1 , · · · , x L }, S k ). (iii) For 1 ≤ i ≤ L, there exist a positive integer L i depending only on M and nontrivial smooth biharmonic map ω ij from R 4 to S k with finite Hessian energy, The idea to prove Theorem 1.2 is based on the duality between the Lorentz spaces L 2,1 and L 2,∞ (see §2 below for the definitions and basic properties). More precisely, we can bound the L 2,1 -norm of ∇ 2 u m in the neck region, while showing the L 2,∞ -norm of ∇ 2 u m can be arbitrarily small in the neck region. Our argument to estimate ∇ 2 u m L 2,1 relies heavily on the symmetry of S k -a property that was earlier utilized by [2], [20], [18], and [9] in the study of biharmonic maps. However, the argument to establish the estimation of ∇ 2 u m L 2,∞ does not utilize the symmetry of S k and hence holds for any target manifold N .
We conjecture that Theorem 1.1 remains to be true for any target manifold N . For a general target manifold N , it remains to be a difficult question on how to obtain L 2,1 -estimate for ∇ 2 u similar to (2.3) in Theorem 2.3. In a forthcoming paper [24], we will employ a different approach similar to [7] to prove Theorem 1.1 under the stronger assumption that the bi-tension fields are bounded in L p for some p > 4 3 . We would like to remark that the corresponding L 2,1 -estimate of ∇u plays a very important role in the study of harmonic maps by Hélein [6]. Later, Lin-Rivierè [13] (and Lin-Wang [14] respectively) utilized the duality between L 2,1 and L 2,∞ to study the energy quantization effect of harmonic maps (and approximate harmonic maps respectively) to S k in higher dimensions. See also Laurain-Rivierè [12] for some most recent related works.
From the view of point for applications, Theorem 1.2 can be a useful extension of the results by [20] and [7]. A typical application of Theorem 1.2 is to study asymptotic behavior at time infinity for the heat flow of biharmonic maps in dimension 4.
Let's review some recent studies on the heat flow of biharmonic maps undertaken by Lamm [10], Gastel [4], Wang [23], and Moser [15]. For a given compact Riemannian manifold N ⊂ R k+1 without boundary, the equation of heat flow of (extrinsic) biharmonic maps into N is to seek u : Ω × [0, +∞) → N that solves (see Lamm [10]): where u 0 ∈ W 2,2 (Ω, N ) is a given map, P(y) : R k+1 → T y N is the orthogonal projection from R k+1 to the tangent space of N at y ∈ N , and B(y)(X, Y ) = −∇ X P(y)(Y ), ∀X, Y ∈ T y N , is the second fundamental form of N ⊂ R k+1 . Note that any t-independent solution u : Ω → N of (1.5) is a biharmonic map to N . In dimension n = 4, Lamm [10] established the existence of global smooth solutions to (1.5)-(1.7) for u 0 ∈ W 2,2 (Ω, N ) with small W 2,2 -norm, and Gastel [4] and Wang [23] independently showed that there exists a unique global weak solution to (1.5))-(1.7) for any initial data u 0 ∈ W 2,2 (Ω, N ) that has at most finitely many singular times. Moreover, such a solution enjoys the energy inequality: Recently, Moser [15] was able to show the existence of a global weak solution to (1.5)-(1.7) for any target manifold N in dimensions n ≤ 8. It follows from (1.8) that there exists a sequence t m ↑ ∞ such that u m := u(·, t m ) ∈ W 2,2 (Ω, N ) satisfies (i) τ 2 (u m ) := u t (t m ) L 2 → 0; and (ii) u m satisfies in the distribution sense − ∆ 2 u m + ∆(B(u m )(∇u m , ∇u m )) + 2∇ · ∆u m , ∇(P(u m )) − ∆(P(u m )), ∆u m = τ 2 (u m ). (1.9) In particular, when N = S k , by Definition 1.1 {u m } is a sequence of approximate biharmonic maps to S k , which are bounded in W 2,2 and have their bi-tension fields bounded in L 2 .
(iii) For 1 ≤ i ≤ L, there exist a positive integer L i and nontrivial biharmonic maps {ω ij } L i j=1 on R 4 with finite Hessian energies such that (1.10) and In a forthcoming article [24], we will show Theorem 1.3 remains to be true for a general target manifold N .
The paper is organized as follows. In §2, we establish the Hölder continuity for any approximate biharmonic map with its bi-tension field in L p for some p > 1, and L 2,1 -estimate of its Hessian ∇ 2 u. In §3, we show the strong convergence under the smallness condition of Hessian energy and set up the bubbling process. In §4, we show no concentration of ∇ 2 u L 2,∞ (·) in the neck region. In §5, we apply the duality between L 2,1 and L 2,∞ to show neither Hessian energy nor oscillation can concentrate in the neck region, which proves Theorem 1.2.

A priori estimates of approximate biharmonic maps
This section is devoted to the estimate of L 2,1 norm of ∇ 2 u for an approximate biharmonic map in terms of its Hessian energy and L p -norm of its bi-tension field for some p > 1, and the Hölder continuity estimate under the smallness condition on its Hessian energy.
First we recall the definition and some basic properties of Lorentz spaces L 2,1 and L 2,∞ on R n (see [6] for more detail).

3)
and  [20]. Let × denote the wedge product in R k+1 . First observe that the equation (1.1) is equivalent to: Since by scaling the case δ = 1 can be reduced to the case δ = 1, we assume δ = 1 for simplicity. Let h ∈ L p (R 4 ) be an extension of h such that Now we consider the Hodge decomposition of the 1-form It is well-known [8] that there exist a function F ∈ W 1,4 (R 4 ) and a 2- and It is easy to see that H satisfies By Proposition 2.2 and Hölder inequality, d u × d u ∈ L 2,1 (R 4 ). Hence, by the Calderon-Zgymund's L p,q -theory, we conclude that ∇ 2 H ∈ L 2,1 (R 4 ), and and Then it is readily seen that and Now we want to estimate F 1 , F 2 , F 3 as follows. For F 1 , since and by Hölder inequality . (2.20) Hence, by Proposition 2.2 we have that ∇ 2 F 1 ∈ L 2,1 (R 4 ) and . (2.21) For F 2 , we have where I β (f ) is the Riesz potential of order β (0 < β ≤ 4) defined by It follows from Adams [1] (see also [20]) that ∇ 2 F 2 ∈ L 2p 2−p (R 4 ) and . .

Blow up analysis and energy inequality
This section is devoted to ǫ 0 -compactness lemma and preliminary steps on the blow up analysis of approximate biharmonic maps with bi-tension fields bounded in L p for some p > 1. In particular, we will indicate that (1.10) holds with "= " replaced by "≥". First we have ≤ C.
Hence we may assume that lim m,l→∞ ) be a cut-off function of B 1 2 , multiplying the equations of u m and u l by (u m − u l )φ 2 and integrating over B 1 , we obtain It is easy to see For IV , observe that for 1 < r < 4 with 1 4 + 1 q + 1 r = 1, we have is an approximate biharmonic map with bi-tension field h.
Proof. Let ǫ 0 > 0 be given by Corollary 2.4, and define Then by a simple covering argument we have that Σ is a finite set. In fact For any x 0 ∈ Ω \ Σ, there exists r 0 > 0 such that lim inf Hence Corollary 2.4 and Lemma 3.1 imply that there exists α ∈ (0, 1) such that ). This proves that u m → u in W 2,2 loc ∩ C 0 loc (Ω \ Σ). It is clear that u ∈ W 2,2 (Ω) is an approximate biharmonic map with bi-tension field h ∈ L p (Ω). Applying Corollary 2.4 again, we conclude that u ∈ C 0 (Ω, S k ). ✷

Proof of Theorem1.2:
The proof of (1.10) with "≥" is similar to [20] Then v m is an approximate biharmonic map, with bi-tension field h m (x) = r 4 m h(x m + r m x), that satisfies Thus Corollary 2.4 and Lemma 3.1 imply that, after taking possible subsequences, there exists a nontrivial biharmonic map ω : Performing such a blow-up argument near any x i ∈ Σ, 1 ≤ i ≤ L, we can find all possible nontrivial biharmonic maps {ω ij } ∈ W 2,2 (R 4 ) for 1 ≤ j ≤ L i , with L i ≤ CM ǫ −2 0 . It is not hard to see (1.10) holds with "=" replaced by "≥". In order prove "≤" of (1.10), we need to show that the L 2,∞ -norm of u m over any neck region is arbitrarily small. This will be done in the next section. We will return to the proof of Theorem 1.2 in §5.

L 2,∞ -estimate in the neck region
In this section, we first show that there is no concentration of ∇ 2 u L 2,∞ in the neck region. Then use the duality between L 2,1 and L 2,∞ to prove Theorem 1.2 by showing that there is no Hessian energy concentration in the neck region. More precisely, we have Lemma 4.1 For any ǫ > 0, suppose that u ∈ W 2,2 (B 1 , S k ) is an approximate biharmonic map, whose bi-tension field h ∈ L p (B 1 ) for some p > 1, satisfying that for 0 < δ < 1 2 , R > 1, and 0 < r < 2δ R , Proof. First recall that It suffices to estimate λ 2 x ∈ B δ \B 2Rr : |∇ 2 u| > λ for λ > 1.
We may assume that δ = 2 K ǫ λ for some positive integer K ≥ 1. There are two cases to consider: (i) ǫ λ ≥ 2Rr. Then we have (ii) ǫ λ < 2Rr. Then we may assume that there exists 1 ≤ i 0 ≤ K such that 2Rr = 2 i 0 ǫ λ so that It is not hard to see that the case (ii) can be done by the same way as the case (i). Thus we only need to prove (i). To simplify the presentation, introduce where we have used |∇u| 2 = |∆u · u| ≤ |∆u| on B i and (4.1). Let h i : R 4 → R k+1 be an extension of h from B i to R 4 such that where G(·) is the fundamental solution of ∆ 2 on R 4 , and w i : Then it is easy to see that Now we have Claim. For 0 ≤ i ≤ K − 1, .
For III, by integration by parts we have Therefore, we have .
Similar to III, we can estimate IV by .
Putting the estimates of I, II, III, IV together, we have .
Since w i is a biharmonic function, the standard estimate and (4.12) imply that w i ∈ C ∞ (B i ), and sup x∈A i . (4.13) This yields (4.10). It follows from the above Claim that sup λ≥1 λ 2 = I + II.

Since Hölder inequality implies
It is easy to see and .
Putting these estimates together we can obtain .
This clearly implies (4.2). The proof is complete. ✷

Proof of Theorem 1.2 and 1.3
This section is devoted to the proof of "=" of (1.10). The argument is based on the duality between L 2,1 and L 2,∞ .
Competition of Proof of Theorem 1.2: For simplicity, we may assume Σ = {0} ⊂ Ω is a single point. In particular, u m → u in W 2,2 loc (B r 1 \ {0}) for some r 1 > 0. By an induction argument similar to that of [3] in the context of harmonic maps, we may assume that there is only one bubble in B r 1 , i.e. L 1 = 1. Then for any ǫ > 0, there exist r m ↓ 0, R ≥ 1 sufficiently large, and 0 < δ ≤ ǫ p 4(p−1) such that for m sufficiently large, it holds Therefore by the duality between L 2,1 and L 2,∞ , we have Since ǫ > 0 is arbitrary, this yields that (1.10) holds. It is easy to see that (1.11) follows from (1.10) and the pointwise inequality |∇u m | 2 ≤ |∆u m |. The proof of Theorem 1.2 is now complete. Let v m : B 2Rrm → S k be a minimizing biharmonic map extension of u m such that (v m , ∇v m ) = (u m , ∇u m ) on ∂B 2Rrm (x 0 ). Then we would have (cf. [17]) Now we define w m : B δ (x 0 ) → S k by letting Then w m is an approximate biharmonic map to S k with bi-tension field h m given by Therefore by Theorem 2.3 we have Since there exists a harmonic function w m on B δ (x 0 ), with ∇ 2 w m L 2 (B δ (x 0 )) ≤ Cǫ

Thus (5.5) follows. ✷
Proof of Theorem 1.3: It follows from the energy inequality (1.8) that there exists t m ↑ +∞ such that u m (·) = u(·, t m ) is an approximate biharmonic map into S k with bi-tension fields h m = u t (·, t m ) ∈ L 2 (Ω) satisfying .
Therefore we may assume that after taking another subsequence, u m ⇀ u ∞ in W 2,2 (Ω, S k ). It is easy to see that u ∞ is a biharmonic map so that u ∞ ∈ C ∞ (Ω, S k ) (see [20]). All other conclusions follow directly from Theorem 1.2. ✷