Energy identity of approximate biharmonic maps to Riemannian manifolds and its application

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1 Energy identity of approximate biharmonic maps to Riemannian manifolds and its application Changyou Wang Shenzhou Zheng December 9, 011 Abstract We consider in dimension four wealy convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in L p for p > 3. 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 R. As a corollary, we obtain an energy identity for the heat flow of biharmonic maps at time infinity. 1 Introduction This is a continuation of our previous wor [6] on the blow-up analysis of approximate biharmonic maps in dimension. In [6], we obtained in dimension four an energy identity for approximate biharmonic maps into sphere with bounded L p bi-tension field for p > 1, and an energy identity of the heat flow of biharmonic map into sphere at time infinity. The aim of this paper is to extend the main theorems of [6] to any compact Riemannian manifold without boundary, under the additional assumption that the bi-tension fields are bounded in L p for p > 3. The main results of this paper was announced in [6]. Let Ω R be a bounded smooth domain, and (N, h be a compact n-dimensional Riemannian manifold without boundary, embedded into K-dimensional Euclidean space R K. Recall the Sobolev space W l,p (Ω, N, 1 l < + and 1 p < +, is defined by W l,p (Ω, N = { v W l,p (Ω, R K : v(x N } a.e. x Ω. In this paper we will discuss the limiting behavior of wealy convergent sequences of approximate (extrinsic biharmonic maps {u } W, (Ω, N in dimension n =, especially an energy identity during the process of convergence. First we recall the notion of approximate (extrinsic biharmonic maps. Definition 1.1 A map u W, (Ω, N is called an approximate biharmonic map if there exists a bi-tension field h L 1 loc (Ω, RK such that u = (B(u( u, u + u, (P(u (P(u, u + h (1.1 in the distribution sense, where P(y : R K T y N is the orthogonal projection from R K 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. In particular, if h = 0 then u is called a biharmonic map to N. Department of Mathematics, University of Kentucy, Lexington, KY 0506, cywang@ms.uy.edu. Department of Mathematics, Beijing Jiaotong University, Beijing 1000, P. R. China, shzhzheng@bjtu.edu.cn. 1

2 Note that biharmonic maps to a Riemannian manifold N are critical points of the hessian energy functional E (u = u dx Ω over W, (Ω, N. Biharmonic maps are higher order extensions of harmonic maps. The study of regularity of biharmonic maps has generated considerable interests after the initial wor by Chang- Wang-Yang [3], the readers can refer to Wang [, 3, ], Strzeleci [19], Lamm-Rivière [1], Struwe [1], Scheven [17, 18] (see also Ku [10] and Gong-Lamm-Wang [6] for the boundary regularity. In particular, the interior regularity theorem asserts that the smoothness of W, -biharmonic maps holds in dimension n =, and the partial regularity of stationary W, -biharmonic maps holds in dimensions n 5. It is an important observation that biharmonic maps are invariant under dilations in R n for n =. Such a property leads to non-compactness of biharmonic maps in dimension, which prompts recent studies by Wang [] and Hornung-Moser [8] concerning the failure of strong convergence for wealy convergent biharmonic maps. Roughly speaing, the results in [] and [8] 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 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 [] and [8] to the context of suitable approximate biharmonic maps to a compact Riemannian manifold N. More precisely, we have Theorem 1. For n =, suppose {u } W, (Ω, N is a sequence of approximate biharmonic maps, which are bounded in W, (Ω, N and have their bi-tension fields h bounded in L p for p > 3, i.e. ( M := sup u W, + h L p < +. (1. Assume u u in W, and h h in L p. Then (i u is an approximate biharmonic map to N with h as its bi-tension field. (ii there exist a nonnegative integer L depending on M and L points {x 1,, x L } Ω such that u u strongly in W, loc C0 loc (Ω\{x 1,, x L }, N. (iii For 1 i L, there exist a positive integer L i depending on M and L i nontrivial smooth biharmonic map ω ij from R to N with finite hessian energy, 1 j L i, such that lim u = B ri (x i B ri (x i L i u + ω ij, R (1.3 j=1 and lim u = B ri (x i B ri (x i L i u + ω ij, R (1. j=1 where r i = 1 min { x i x j, dist(x i, Ω}. 1jL, j i As an application of Theorem 1., we study asymptotic behavior at time infinity for the heat flow of biharmonic maps in dimension. Let s review the studies on the heat flow of biharmonic maps undertaen by Lamm [11], Gastel [5], Wang [5], and Moser [16]. The equation of heat flow of (extrinsic biharmonic maps into N is

3 to see u : Ω [0, + N that solves: u t + u = (B(u( u, u + u, (P(u (P(u, u, Ω (0, + (1.5 u = u 0, Ω {0} (1.6 (u, u ν = (u 0, u 0, Ω (0, +, ν (1.7 where u 0 W, (Ω, N is a given map. Note that any time independent solution u : Ω N of (1.5 is a biharmonic map to N. In dimension n =, Lamm [11] established the existence of global smooth solutions to (1.5-(1.7 for u 0 W, (Ω, N with small W, -norm, and Gastel [5] and Wang [5] independently showed that there exists a unique global wea solution to (1.5-(1.7 for any initial data u 0 W, (Ω, N that has at most finitely many singular times. Moreover, such a solution enjoys the energy inequality: T u t + u (T u 0, 0 < T < +. (1.8 0 Ω Ω Ω Recently, Moser [16] showed the existence of a global wea 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 such that u := u(, t W, (Ω, N satisfies (i τ (u := u t (t satisfies τ (u L 0; and (ii u satisfies in the distribution sense u + (B(u ( u, u + u, (P(u (P(u, u = τ (u. (1.9 By Definition 1.1 {u } is a sequence of approximate biharmonic maps to N, which are bounded in W, and have their bi-tension fields bounded in L. Hence, as an immediate corollary, we obtain Theorem 1.3 For n = and u 0 W, (Ω, N, let u : Ω R + N, with u L (R +, W, (Ω and u t L (R +, L (Ω, be a global wea solution of (1.5-(1.7 that satisfies the energy inequality (1.8. Then there exist t +, a biharmonic map u C W, (Ω, N with u = u 0 on Ω, and a nonnegative integer L and L points {x 1,, x L } Ω such that (i u := u(, t u in W, (Ω, N. (ii u u in Cloc 0 W, loc (Ω \ {x 1,, x L }, N. (iii for 1 i L, there exist a positive integer L i and L i nontrivial biharmonic maps {ω ij } L i j=1 on R with finite hessian energies such that lim u = B ri (x i B ri (x i L i u + ω ij, R (1.10 j=1 and lim u = B ri (x i B ri (x i L i u + ω ij, R (1.11 j=1 where r i = 1 min { x i x j, dist(x i, Ω}. 1jL, j i The main ideas to prove Theorem 1. can be outlined as follows. First, we adapt the arguments from [3, ] to establish an ε 0 regularity theorem for any approximate biharmonic map u with 3

4 bi-tension field h L p for p > 1, which asserts that u C α for any α (0, min{1, (1 1 p } and u L p. Second, we prove that for p > 3 there is no concentration of angular hessian energy in the nec region by comparing the approximate biharmonic maps with radial biharmonic functions over annulus. This is a well nown technique in harmonic maps in dimension two (see []. For biharmonic maps in dimension, it was derived by Hornung-Moser [8]. Third, we use a Pohozaev type argument to control the radial hessian energy by the angular hessian energy and L p -norm of bi-tension fields in the nec region. The assumption p 3 seems to be necessary to validate the Pohozaev type argument, since we need u (x u L 1 and h (x u L 1. It remains to be an open question whether Theorem 1. holds for 1 < p 3. The paper is organized as follows. In, we establish the Hölder continuity and W,p -regularity for any approximate biharmonic map with its bi-tension field in L p for p > 1. In 3, we show the strong convergence under the smallness condition of hessian energy and set up the bubbling process. In, we prove Theorem 1. by establishing (i there is no concentration of angular hessian energy in the nec region; and (ii control the radial hessian energy in the nec region by angular hessian energy and L p -norm of bi-tension field through a Pohozaev type argument. Acnowledgement. The first author is partially supported by NSF The second author is partially supported by NSFC grant This wor is conducted while the second author is visiting the University of Kentucy, he would lie to than Department of Mathematics for its support and hospitality. Added Note. After we completed this paper, we noticed from a closely related preprint posted in arxiv.org at December 3, 011 by Laurain and Rivière (arxiv: v1, in which they claimed Theorem 1. holds for all p > 1. Since the main ideas of our proof are different from theirs, we believe that our wor here shall have its own interest. A priori estimates of approximate biharmonic maps In this section, we will establish both Hölder continuity and W,p -regularity for any approximate biharmonic map with bi-tension field h L p for p > 1, under the smallness condition of hessian energy. The proof of Hölder continuity is based on suitable modifications and extensions of that by Wang [3] Theorem A on the regularity of biharmonic maps to the context of approximate biharmonic maps in dimension. The proof of a bootstrap type argument, which may have its own interest. It is needed for the ε 0 -compactness lemma and Pohozaev argument for approximate biharmonic maps. Denote by B r (x R the ball with center x and radius r, and B r = B r (0. First, we have Lemma.1 For any α (0, min{1, (1 1 p }, there exist ε 0 > 0 such that if u W, (B, N is an approximate biharmonic map with its bi-tension field h L p (B 1 for p > 1, and satisfies ( u + u ε 0. (.1 B then u C α (B 1, N and [ ( u ]Cα(B C ε 0 + h L 1 p (B 1. (.

5 Proof. We follow [3], 3 and closely and only setch the main steps of proof here. First, by choosing ε 0 sufficiently small, Proposition 3. and Theorem 3.3 of [3] imply that there exists an adopted frame {e α } m α=1 (m = dimn along with u T N on B 1 such that its connection form A = ( De α, e β satisfies: d A = 0 in B 1 ; x A = 0 on B 1 A L (B 1 + A L (B 1 C u L (B 1 (.3 A L,1 (B 1 + A L,1 (B 1 C( u L (B 1 + u L (B 1. Here L r,s denotes Lorentz spaces for 1 r < +, 1 s, see [3] or [7] for its definition and basic properties. Utilizing {e α } m α=1, we can rewrite (1.1 into the following form (see [3] Lemma.1: for 1 α m, u, e α = G(u, e α + h, e α, (. where u, e α + u, e α G(u, e α = + ( β ( u, e β e α, e β ( u T, e β e α, e β β u, e β B(u(e α, u, e β + (B(u( u, u, e α. As in [3] Lemma.3, we let ũ W, (R, R K, {ẽ α } n α=1, Ã and h be the extensions of u, {e α } n α=1, A and h from B 1 to R such that ẽ α 1, Ãαβ = ẽ α, ẽ β, and ũ L (R C u L (B 1, ũ L (R C u L (B 1 ; ũ L, (R C u L, (B 1, ũ L, (R C u L, (B 1 ; (.6 Ã L,1 (R + Ã L,1 (R C( u L (B 1 + u L (B 1 ; h L q (R C h L q (B 1, 1 q p. Let Γ(x y = c ln x y be the fundamental solution of in R. Set W α (x = Γ(x yg(ũ, ẽ α (ydy + R Γ(x y h, ẽ α (ydy R Then we have For J α, observe that (.5 = J α 1 (x + J α (x x R. (.7 W α = G(ũ, ẽ α + h, ẽ α, in R. (.8 J α (x = yγ(x y h, ẽ α (y dy. R Hence by Calderon-Zygmund s W,p -theorey, we have J α W,p (R and J α C h h C L. (.9 L p (R L p (R p (B 1 By Sobolev embedding theorem, we have that J1 α L p (B 1, with p = p p for 1 < p < or p is any finite number for p, and J α h C L. (.10 L p (B 1 p (B 1 5

6 By Hölder inequality, (.9 and (.10 imply that for any θ (0, 1, it holds J α + J α L, 1 (B θ L, (B θ Cθ(1 p J α 1 L p (B θ Cθ(1 p L h. (.11 p (B 1 For J1 α, we follow exactly [3] Lemma., Lemma.5, and Lemma.6 to obtain that J1 α L, (R and ( J1 α u Cε L, (R 0 L, (B + u. (.1 1 L, (B 1 By Sobolev embedding theorem in Lorentz spaces, (.1 yields J1 α C J α L, (R 1. (.13 L, (R Combining (.1 with (.13 yields J1 α + J L, (R 1 α L, (R Cε 0 ( u L, (B 1 + u L, (B 1. (.1 Now, as in [3] Lemma.6, we consider the Hodge decomposition of the 1-form dũ, ẽ α. It is well-nown [9] that there exist F α W 1, (R and H α W 1, (R, R such that dũ, ẽ α = df α + d H α, dh α = 0 in R, (.15 L F α + Hα L ũ C L u C L. (.16 (R (R (R (B 1 It is easy to see that H α satisfies and H α = dũ dẽ α in R, (.17 F α = u, e α = W α in B 1. (.18 By Calderon-Zygmund s L r,s -theory and Sobolev embedding theorem, we have that H α L, (R and L H α + L dũ H α C L dẽα Cε, (R, (R, (R 0 u. (.19 L, (B 1 By (.18, we have that F α W α is a biharmonic function on B 1. By the standard estimate of biharmonic functions, we have (see [3] Lemma.7 that for any θ (0, 1, it holds (F α W α + (F α W α L, (B θ L, (B θ (Fα Cθ( W α + (F α W α. (.0 L, (B 1 L, (B 1 Putting (.11, (.1, (.19, and (.0 together, we can argue, similar to [3] page 8, to reach that for any θ (0, 1, it holds u + u L u C(ε + θ( + u L, (B θ, (B 1 L, (B θ L, (B 1 +Cθ (1 1 p h L p (B 1. (.1 6

7 It is readily seen that for any α (0, min{1, (1 1 p }, we can choose both θ (0, 1 and ε 0 (0, 1 sufficiently small so that u + u θα( L u + h u + L. (. L, (B θ L, (B θ, (B 1 L, (B 1 p (B 1 In fact, by iterating (. finitely many times on B 1 (x for x B 1, we would have that for 0 < r 1, u + u Crα( L u + h u + L. (.3 L, (B r(x L, (B r(x, (B 1 L, (B 1 p (B 1 Since L q (B r (x L s, (B r (x for any 1 q < s, (.3 implies that for any 1 < q <, r q ( u q + u q Cr α( u + u h + p. (. L (B 1 L (B 1 L p (B 1 B r(x This, with the help of Morrey s decay lemma, immediately implies that u C α (B 1 and (. holds. This completes the proof of Lemma.1. In order to show ε 0 -compactness and Pohozaev argument for approximate biharmonic maps, we will establish the higher order Sobolev type regularity for approximate biharmonic maps. The proof utilizes Adams Reisz potential estimate between Morry spaces, we briefly recall Morrey spaces and Adams estimates (see [1] and [] for more details. For an open set U R, 1 p < +, 0 < λ, the Morrey space M p,λ (U is defined by M p,λ (U = { } f L p (U : f p = sup r λ f M p,λ B r U B p < +. (.5 r The wea Morrey space M p,λ (U is the set of functions f L p, (U satisfying { } f p sup ρ λ n f p M p,λ (U L B p, (B r <. (.6 r U For 0 < β <, let I β (f be the Riesz potential of order defined by I β (f(x Recall Adams estimate [1] in dimension : R f(y x y β dy, x R. (.7 Lemma. (1 For any β > 0, 0 < λ, 1 < p < λ β, if f M p,λ (R, then I β (f M p,λ (R, where p = λp λ pβ. Moreover, I β (f M p,λ (R C f M p,λ (R. (.8 ( For 0 < β < λ, if f M 1,λ (R, then I β (f M Now we are ready to prove I β (f λ λ β M,λ (R λ λ β,λ (R. Moreover, C f M 1,λ (R. (.9 7

8 Lemma.3 There exists ε 0 > 0 such that if u W, (B 1, N is an approximate biharmonic map with its bi-tension field h L p (B 1 for p > 1, and satisfies B 1 u + u ε 0. (.30 Then u W,p (B 1, N and 8 u L p (B 1 8 ( u C L + h u + L (B 1 L (B 1 p (B 1 Proof. By (. of Lemma.1, we have that for any α (0, min{1, (1 1 p }, We now divide the proof into three steps. Step1. There exists α 0 (0, α] such that u M, α 0 (B 1 u M, α 0 (B 1 + u. (.31 osc Br(xu Cr α, B r (x B 1. (.3 M, α 0 (B 1, u M, α 0 (B 1, and ( u C L + h u + L (B 1 L (B 1 p (B 1. (.33 Observe that (.33 is a refined version of (.3 and (.. It is obtained by the hole filling argument as follows. For any B r (x B 1, let φ C0 (B r(x such that 0 φ 1, φ 1 on B r (x. Multiplying (1.1 by φ(u u x,r, where u x,r is the average of u on B r (x, and integrating over B r (x, we would have (φ(u u x,r B r(x B r(x + C B r(x ((1 φ(u u x,r (φ(u u x,r + C B r(x u (φ(u u x,r ( u u (φ(u u x,r + C u + h osc Br(xu. (.3 B r(x It is not hard to see that by Hölder inequality, Sobolev s inequality and (.3, (.3 implies ( u + u θ ( u + u + Cr α, (.35 C B r (x where θ = C+1 < 1. Now we can iterate (.35 finitely many times and achieve that there exists α 0 (0, α such that ( sup r α 0 ( u + u C u L (B 1 + u L (B 1 + h p L p (B 1. (.36 B r(x B 1 B r(x B r(x This yields (.33. Step. Set p > by p = { p p if 1 < p < any < q < + if p. 8

9 Then u W, p (B 1 and u L p (B 1 + u L p (B 1 ( u C L + h u + L (B 1 L (B 1 p (B 1 To show (.37, let ũ, h : R R K be extensions of u and h on B 1 and such that. (.37 ũ M, α 0 (R C u M, α 0 (B 1, ũ M, α 0 (R C u M, α 0 (B 1, (.38 h L p (R C h L p (B 1. (.39 Define w : R R K by w(x = Γ(x y h(y dy + y Γ(x y(b(ũ( ũ, ũ(y dy R R y Γ(x y ũ, (P(ũ (y dy Γ(x y (P(ũ, ũ (y dy R R = w 1 (x + w (x + w 3 (x + w (x, x R. (.0 Then it is readily seen that (u w = 0 on B 1, (.1 or u w is a biharmonic function on B 1. Now we estimate w i, 1 i, as follows. For w 1, by Calderon-Zygmund s W,p -theory we have that w 1 W,p (R so that w 1 L p (B 1 and w 1 L p (B 1 C h L p (B 1. (. For w 3, since ũ ũ M 3, α 0 (R, w 3 CI ( ũ ũ and w 3 CI 1 ( ũ ũ, Lemma. implies that w 3 M ( α 0, α 3α 0 0 (R, w 3 M ( α 0, α 3α 0 0 (R, and w 3 ( α 0 + w M 3α, α 3 ( α (R M 3α, α 0 0 (R C ũ ũ ũ C M 3, α 0 ũ (R M, α 0 (R M, α 0 (R C u u M, α 0 (B 1 M, α 0 (B 1 u L C( + h u + L (B 1 L (B 1 p (B 1. (.3 For w, it is easy to see that w CI 3 ( ũ + ũ and w CI ( ũ + ũ. Since ( ũ + ũ M 1, α 0 (R, Lemma. implies that w M w M α 0, α 1 α 0 0 (R and w α α, α M 0 0 w α 0 (R M u L C( + h u + L (B 1 L (B 1 p (B 1 1 α, α 0 0 (R α 0, α 1 α 0 0 (R and. (. 9

10 For w, since w CI 1 ( ũ, ũ M α 0, α 1 α 0 0 (R and w α 0 M 1 α, α 0 0 (R M, α 0 (R, Lemma. implies that w C ũ M, α 0 (R u L C( + h u + L. (.5 (B 1 L (B 1 p (B 1 It is not hard to see from (.1 and the standard estimate on biharmonic { function, and } the estimates (., (.3, (., and (.5 that there exist 1 < q < min p 3p, α 0 (1 α 0, α 0 3α 0 and { } 0 < α 1 min α 0, ( 3pq p such that u M q, α 1 (B 3 and u M q, α 1 (B 3, and 8 8 ( u + u u C L + h u + L. (.6 (B 1 L (B 1 p (B 1 M q, α 1 (B 3 8 M q, α 1 (B 3 8 With (.6, we can repeat the same argument to bootstrap the integrablity of u and finally get that u L p (B 1 and (.37 holds. Step 3. u L p (B 1 and 8 u L p (B 1 8 ( u C L + h u + L (B 1 L (B 1 p (B 1 To prove (.7, first observe that the equation (1.1 can be written as. (.7 u = div(e(u + G(u + h, in B 1, (.8 where E(u = (B(u( u, u + u, (P(u and G(u = (P(u, u so that E(u C( u 3 + u u, G(u C( u + u. By (.37 and Sobolev s inequality, we have u L p E(u L p 8 5p (B 1 3p (B 1. Note that we can write u = u 1 + u + u 3 + u in B 1 so that G(u L p, where p (B 1 and u 1 = G(u in B 1 ; (u 1, u 1 = (0, 0 on B 1, (.9 u = h in B 1 ; (u, u = (0, 0 on B 1, (.50 u 3 = div(e(u B 1 ; (u 3, u 3 = (0, 0 on B 1, (.51 and u = 0 B 1 ; (u 3, u 3 = (u, u on B 1. (.5 By Calderon-Zygmund s L q -theory, we have ( W u 1 p + u W u C( L + h, u + L, (.53 p (B 1,p (B 1 (B 1 L (B 1 p (B 1 and u 3 W 3, p p (B 1 + u W,p (B 1 5 ( u C( L + h u + L (B 1 L (B 1 p (B 1, (.5 10

11 p 3, In particular, we can conclude that u W u 3, p W p (B 15 By Hölder inequality, (.55 then implies p (B 1 5 and u L C(( + h u + L (B 1 L (B 1 p (B 1 div(e(u C( 3 u u + u + u 3 + u L p Hence applying W,q -estimate of (.51 yields that u 3 W, p p W u 3, ( u p p C( L + u (B 18 (B 1 p (B 1 8 and p (B 1 5. (.55 h + L. (.56 L (B 1 p (B 1 Since p > p, by combining (.56 with (.53 and (.5, we finally obtain that u W,p (B 1 8 and (.7 holds. This completes the proof of Lemma.3. 3 Blow up analysis and energy inequality This section is devoted to ɛ 0 -compactness lemma and preliminary steps on the blow up analysis of approximate biharmonic maps with bi-tension fields bounded in L p for p > 1. First we have Lemma 3.1 For n =, there exists an ɛ 0 > 0 such that if {u } W, (B 1, N is a sequence of approximate biharmonic maps satisfying ( sup u L (B 1 + u L (B 1 ɛ 0, (3.1 and u u in W, (B 1 and h h in L p (B 1 for some p > 1. Then u C 0 W,p (Ω, N is an approximate biharmonic map with bi-tension field h, and lim u u = 0. (3. W, (B 1 Proof. The first assertion follows easily from (1.1 and (3.. To show (3., it suffices to show that {u } is a Cauchy sequence in W, (B 1. By (3.1 and Lemma.1, there exist α (0, 1 and q > such that [ ] u sup Cα(B + L u C. 3 q (B 3 Hence we may assume that lim,l L u u l = 0. (B 3 For η C0 (B 3 be a cut-off function of B 1, multiplying the equations of u and u l by (u u l φ. 11

12 and integrating over B 1, we obtain (u u l φ B 1 (u u l ( (u u l φ + u u l φ + B 1 +3 ( u l + u u u l φ B 1 + u u (u (u u l φ B 1 + u l u l (u l (u u l φ B 1 = I + II + III + IV + V. B 1 h h l u u l φ It is easy to see I C( (u u l L (B 3 + u u l L (B 3 0, II C h h l L 1 (B 3 u u l L (B 3 0, III C( u L (B 3 + u l L (B 3 u u l L (B 3 0. For IV, observe that for 1 < r < with q + 1 r = 1, we have ( IV C u L (B 3 u L (B 3 u u l L (B 3 + u L q (B 3 u L (B 3 (u u l L r (B 3 0, since (u u l L r (B 3 0. Similarly, we can show V 0. Hence {u } is a Cauchy sequence in W, (B 1. This completes the proof. Lemma 3. Under the same assumptions as Theorem 1., there exists a finite subset Σ Ω such that u u in W, loc C0 loc (Ω\Σ, N. Moreover, u W,p C 0 (Ω, N is an approximate biharmonic map with bi-tension field h. Proof. Let ɛ 0 > 0 be given by Lemma.1, and define Σ := { } x Ω : lim inf ( u + u > ɛ 0. (3.3 r>0 B r(x Then by a simple covering argument we have that Σ is a finite set and H 0 (Σ 1 ɛ sup ( u + u < +. 0 Ω For any x 0 Ω \ Σ, there exists r 0 > 0 such that lim inf ( u + u ɛ 0. B r0 (x 0 1

13 Hence Lemma.1 and Lemma 3.1 imply that there exists α (0, 1 such that C u C, α (B r 0 (x 0 so that u u in C 0 W, (B r 0 (x 0. This proves that u u in W, loc C0 loc (Ω \ Σ. It is clear that u W, (Ω is an approximate biharmonic map with bi-tension field h L p (Ω. Applying Lemma.1 and Lemma.3 again, we conclude that u C 0 (Ω, N W,p (Ω, N. Proof of Theorem 1.: The proof of (1.10 with = replaced by is similar to [] Lemma 3.3. Here we setch it. For any x 0 Σ, there exist r 0 > 0, x x 0 and r 0 such that { } max ( u + u = ɛ 0 x B r0 (x 0 B r (x = ( u + u. B r (x ( Define v (x = u (x + r x : r 1 B r0 (x 0 \ {x } N. Then v is an approximate biharmonic map, with bi-tension field h (x = r h(x + r x, that satisfies and B 1 (x ( v + v ɛ 0 h (, x r 1 B r0 (x 0 \ {x }, and L p (r 1 (Br 0 (x 0\{x } B 1 (0 (1 1 r p L h 0. p (Ω ( v + v = ɛ 0, Thus Lemma.3 and Lemma 3.1 imply that, after taing possible subsequences, there exists a nontrivial biharmonic map ω : R N with ɛ 0 ( ω + ω < + R such that v ω in W, loc C0 loc (R. Performing such a blow-up argument at all x i Σ, 1 i L, we can find all possible nontrivial biharmonic maps {ω ij } W, (R for 1 j L i, with L i CMɛ 0. Moreover, by the lower semicontinuity, we have that the part of (1.10 holds. The other half,, of (1.10 will be proved in the next section. No hessian energy concentration in the nec region In order to show the part of (1.10, we need to show that there is no concentration of hessian energy in the nec region. This will be done in two steps. The first step is to show that there is no angular hessian energy concentration in the nec region by comparing with radial biharmonic functions over dydaic annulus. The second step is to use the type of almost hessian energy monotonicity inequality, which is obtained by the Pohozaev type argument, to control the radial component of hessian energy by the angular component of hessian energy. We require p > 3 in both steps. Suppose that {u } W, (B 1, N is a sequence of approximate biharmonic maps satisfying for some p > 3, ( u + u + h p C, 1. (.1 B 1 13

14 Without loss of generality, we assume that u u in W,, h h in L p, and u u in W, loc (B 1\{0}, N but not in W, (B 1, N. Furthermore, as in Ding-Tian [], we may assume that the total number of bubbles generated at 0 is L = 1. Then for any ε > 0 there exist r 0, R 1 p sufficiently large, and 0 < δ ε (p 1 so that for sufficiently large, the following property holds ( u + u ɛ, 1 B ρ \B ρ 16 Rr ρ 16δ. (. Step 1. Angular hessian energy estimate in the nec region: Since p > 3, it follows from (., Lemma.1, Lemma.3, and Sobolev embedding theorem that for any α (0, (1 1 p, u C α W,p (B ρ \ B ρ, u C 0 (B ρ \ B ρ, and [ ( u ]Cα(Bρ\Bρ + u L (Bρ\Bρ C ε + ρ (1 1 p Cε, 1 8 Rr ρ 8δ. (.3 It follows from Lemma.3 that 3 L u Cε, 1 3 (Bρ \B ρ 8 Rr ρ 8δ. (. To handle the contributions of various boundary terms during the argument, by Fubini s theorem we assume that R > 0 and δ > 0 are chosen so that for sufficiently large, the following property r ( u + u + 3 u 3 8 sup ( u + u + 3 u 3 Cε, (.5 B r B r \B r holds for r = 1 Rr, Rr, δ, δ, δ. For simplicity, we assume (.5 holds for all 1. Here we indicate (.5 for r = Rr : set ũ = u (r x : B δr 1 N. Then by Fatou s lemma we have R lim inf ( ũ + ũ + 3 ũ 3 lim inf ( ũ + ũ + 3 ũ 3. 1 R B r B R \B 1 R By Fubini s theorem and scalings, this easily imply (.5 for r Rr (for simplicity, we can assume r = Rr. Let N N be such that N = [ δ Rr ]. Set A i := B i+1 Rr \ B i Rr and B i := B i+ Rr \ B i 1 Rr, 1 i N 1. (.6 Now we define a radial biharmonic function v on the annulus B δ \B Rr as follows. For simplicity, N 1 we may assume δ Rr is a positive integer so that B δ \B Rr = A i. For 0 i N 1, v (x = v ( x satisfies v = 0 in A i, v (r = B u i+1, v Rr (r = u B i+1 Rr r, if r = i+1 Rr, v (r = B u i, v Rr (r = u B i Rr r, if r = i Rr. i=0 (.7 1

15 Here denotes the average integral. By the standard estimate of biharmonic functions (see, e.g., [8] Lemma 5.1 and (.3, we have that v W,p (A i for 0 i N 1 and [v ]Cα(Ai C ([u ]Cα(Ai + [ u ]L (Ai Cε. In particular, we have sup osc A i (u v Cε. (.8 0iN 1 We now perform the estimate, similar to the arguments by [0] or [] on harmonic maps and [8] on biharmonic maps. First, since u v W,p (A i, we can apply the Green s identity to get that for 0 i N 1, (u v (u v = (u v ( (u v + (u v A i A i A i ν (u v (u v. (.9 ν Summing over 0 i N 1, we obtain B δ \B Rr (u v = = N 1 i=0 ( + ( N 1 i=0 ( A i A i B δ B δ A i B δ (u v (u v (u v (u v B Rr ν ( (u v (u v B Rr ν ( (u v u (u v + u B δ B Rr ν u ν (u v. (.10 B Rr Here we haved use that v = 0 in A i, and (u v v v = B ρ ν B ρ ν (u v = 0 for ρ = δ and Rr due to the radial form of v and the choices of boundary conditions of v. We can chec that the last two terms in the right hand side of (.10 converge to zero as. In fact, by (.5, Hölder inequality, and (. we have that (u v u u u + ( u u B δ ν B δ B δ B δ ( 1 ( 1 C δ u δ u Cε 3. (.11 B δ B δ Similarly, (u v B Rr ν u u u + ( B Rr C (Rr u B Rr 15 u u B Rr B Rr 1 ( 1 Rr u BRr Cε 3. (.1

16 For the last term, by Fubini s theorem and (.8 we have u ( B δ ν (u v C δ 3 u 3 B δ ( Cɛ 3 u 3 B δ \B δ and, similarly, B Rr u ν (u v C (Rr 3 u 3 B Rr 3 max B δ u v 3 3 ( Cε 3 u 3 B Rr \B 1 Rr Cε. (.13 max u v B Rr 3 Cε. (.1 Therefore we conclude that the last two terms in the right hand side of (.10 converge to zero as. For the first term in the right hand side of (.10, we proceed as follows. First, we can rewrite the equation for u as u = div(e(u + G(u + h, (.15 where Hence A i ( E(u C u u + u 3 (, G(u C u + u. (.16 u (u v = div(e(u (u v + G(u (u v + h (u v A i A i A i = I i + IIi + IIIi. (.17 By (.8 we have ( II i C u + u osc A i u Cε A i and III i Cosc A i u For I i, by integration by parts we obtain I i = E(u (u v + so that after summing over 0 i N 1 we have N 1 i=0 I i = N 1 A i i=0 A i N 1 C i=0 ( + C A i A i u + u (.18 h Cε h. (.19 A i A i E(u (u v ν, (.0 ( E(u (u v + E(u (u v ν B δ B Rr A i B δ + ( u u + u 3 (u v B Rr ( u u + u 3 u v IV + V. (.1 16

17 By (.5 and Hölder inequality, we see that V Cε. For IV, we proceed the estimate as follows. By Nirenberg s interpolation inequality and (.8, we have ( u 1 ] 1 ( C [u ( u + u 1 A i BMO(B i B i ( i Rr ( Cε 1 ( u + u 1 ( i Rr. (. Since v is a biharmonic function on A i, we also have ( A i (u v 1 B i ( C u 1 A i C Cε 1 [u ] 1 BMO(B i ( ( B i B i u + u ( i Rr 1 ( u + u 1 ( i Rr. (.3 Therefore we have IV = N 1 i=0 A i N 1 C Cε ( i=0 A i ( N 1 i=0 u u (u v A i u 1 ( u 1 ( (u v 1 A i A i u 1 ( N 1 ( Cε u B δ \B Rr i=0 B i 1 ( Bδ\B 1 u + u 1 ( i Rr Rr u + u 1 x. (. Applying Lemma 5. in [8], we have the following Hardy inequality: Bδ\B 1 Rr u x ( u + B δ \B 1 Rr By (.5 and Hölder inequality, we have that 1 ( 1 u C δ u δ B δ B δ and, similarly, 1 u C (Rr u Rr B 1 B Rr 1 Rr B δ B 1 Rr ( 1 C u B 8δ \B δ 1 ( 1 C u B Rr \B 1 Rr ( 1 x u. (.5 Cε, (.6 Cε. (.7 17

18 Substituting (.5, (.6 and (.7 into (., we obtain ( IV Cε u + Cε. (.8 B δ \B 1 Rr Using the same argument to estimate the second term of IV, we have ( IV Cε u + Cε. (.9 B δ \B Rr Combining the estimates together yields ( (u v Cε u + u + h + Cε. (.30 B δ \B Rr B δ \B 1 Rr This, combined with Calderon-Zygmund s W, estimate, yields ( (u v Cε u + u + h + Cε. (.31 B δ \B Rr B δ \B 1 Rr Step. Control of radial component of hessian energy in the nec region: Since v is radial, it is easy to see that (.31 yields ( T u Cε u + u + h + Cε B δ \B Rr B δ \B 1 Rr Cε. (.3 Here T u = u r ( u denotes the tangential component of u. Next, we want to apply the Pohozaev type argument to W,p -approximate biharmonic maps u with L p bi-tension field h for p 3 to control u by B δ \B Rr r T u and B δ \B Rr h L p (B δ. This type of argument is well-nown in the blow up analysis of harmonic or approximate harmonic maps on Riemann surfaces (see [0], [1], [15], and [13]. In the context of biharmonic maps, it was first derived by Hornung-Moser [8]. By (. and Lemma.3, we see that u W,p (B δ \ B 1 Rr. While in B 1 Rr, since ũ (x = u (r x : B R N is an approximate biharmonic map that converges to the bubble ω 1, we conclude that ũ W,p (B R < + and hence u W,p (B 1 Rr by scaling. From this, we then see u W,p (B δ. Since x u L (B δ and p 3, we see that u (x u L 1 (B δ and h (x u L 1 (B δ. Since the equation (1.1 implies that ( u h (x T u (xn a.e. x B δ. Note also that x u (x T u (xn for a.e. x B δ. Multiplying the equation (1.1 by x u and integrating over B r for any 0 < r δ, we have u (x u = h (x u. (.33 B r B r Applying Green s identity, we have = u (x u B r u (x u + r B r B r r ( u u r u ( u B r r + r u. (.3 r 18

19 Direct calculations yield B r u (x u = B r = ( u x B r ( u div B r + u u x = r Br. (.35 Putting (.35, (.3, and (.33 together yields u r Br + r r ( u u r u ( u B r r + r u r = h (x u. (.36 B r Applying integration by parts multi-times to (.36 in the same way as [3] or Angelsberg [], we can obtain that for a.e. 0 < r δ, ( u u,α + x β u,αβ = B r B r r + x u r + d ( xα u,β u,αβ + x u dr B r r r 3 u r + 1 h (x u. (.37 r B r Recall that in the spherical coordinate, we have where S 3 u = u,rr + 3 r u,r + 1 r S 3u, denotes the Laplace operator on the standard three sphere S 3. Hence we have u = [ u,rr + 9 B r B r r u,r + 6 ] r u,ru,rr [ 1 + r S 3u + (u,rr + 3 r u,r ( r S ] 3u. (.38 On the other hand, we have = B r B r B r ( u,α + x β u,αβ r + x u r [ 1 r u,r + u,rr + r S 3u,r + 8 ] r u,ru,rr. (.39 Substituting (.38 and (.39 into (.37 and integrating over r [Rr, δ], we obtain [3 u,rr + 3 r u,r + ] r u,ru,rr ( + B δ \B Rr B δ \B Rr B δ [ 1 r S 3u + (u,rr + 3 r u,r ( δ r S ] 3u + B Rr ( x α u,β u,αβ r Rr x u r 3 + u r B r h u,r = I + II + III. (.0 19

20 By Hölder inequality, (.3, (.5, (.6, and (.7, we have ( 1 ( I T u + u T u 1 + Cε Cε. B δ \B Rr B δ \B Rr Bδ\BRr For II we have II δ h L p (B δ u L p p 1 (B δ Cδ u L (B δ Cδ, where we have used the fact p p 3 so that p 1. We use (.5 to estimate III as follows. First we have ( x α u,β u,αβ x u B δ r r 3 + u r [ ] C u u + δ 1 u B δ B δ ( 1 ( 1 ( 1 C δ u δ u + C δ u B δ B δ B [ δ ] C u L (B δ \B δ u L (B δ \B δ + u L (B δ \B δ Cε. Similarly, by (.5 we have ( x α u,β u,αβ x u B Rr r r 3 + u r [ ] C u u + (Rr 1 u B Rr B Rr 1 C (Rr ( 1 u Rr B Rr BRr u 1 + C (Rr u B Rr [ C u L (B Rr \B 1 u Rr L (B Rr \B 1 + u Rr L (B Rr \B 1 Rr ] Cε. Therefore, by putting these estimates together, we have [3 u,rr + 3 r u,r + ] r u,ru,rr C(ε + δ. (.1 B δ \B Rr Competition of Proof of Theorem 1.: Since r u,ru,rr ( u,rr + 1 r u,r, (.1 implies this, combined with (.3, implies B δ \B Rr u,rr C(ε + δ, B δ \B Rr u C(ε + δ. 0

21 Thus there is no concentration of hessian energy in the nec region. It is well nown that this yields the energy identity (1.10. To show (1.11, observe that Nirenberg s interpolation inequality and (1.10 imply ( u L (B δ \B Rr C u L (B δ u L (B δ + u L (B δ \B 1 Rr C(ɛ + δ + o(1 where we have used that u L (B δ = u L (B δ + o(1 = o(1. Thus (1.11 also holds. Proof of Corollary 1.3: It follows from the energy inequality (1.8 that there exists t + such that u ( = u(, t is an approximate biharmonic map into N with bi-tension field h = u t (, t L (Ω satisfying L h = ut (, t 0. (Ω L (Ω Moreover, u W, (Ω C u0 W, (Ω. Therefore we may assume that after taing another subsequence, u u in W, (Ω, N. It is easy to see that u is a biharmonic map so that u C (Ω, N (see [3]. All other conclusions follow directly from Theorem 1.. References [1] Adams, D. R.: A note on Riesz potentials. Due Math. J. (, (1975. [] G. Angelsberg, A monotonicity formula for stationary biharmonic maps. Math. Z., 5 (006, [3] Chang, A., Wang, L., Yang, P.: A regularity theory of biharmonic maps. Comm. Pure Appl. Math. 5, (1999. [] Ding, W. Y., Tian, G.: Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom. 3, (1995. [5] Gastel, A.: The extrinsic polyharmonic map heat flow in the critical dimension. Adv. Geom. 6 no., (006. [6] Gong, H. J., Lamm, T., Wang, C. Y.: Boundary regularity for a class of biharmonic maps. Calc. Var. Partial Differential Equations, to appear. [7] Hélein, F.: Harmonic Maps, Conservation Laws, and Moving Frames. Cambridge Tracts in Mathematics, 150, Cambridge: Cambridge University Press (00. [8] Hornung, P., Moser, R.: Energy identity for instrinsically biharmonic maps in four dimensions. Anal. PDE, to appear. [9] Iwaniec, T., Martin, G.: Quasiregular mappings in even dimensions. Acta Math. 170, 9-81 (

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