The qualitative behavior at the free boundary for approximate harmonic maps from surfaces

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{u_n\}$$\end{document}{un} be a sequence of maps from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold N with free boundary on a smooth submanifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\subset N$$\end{document}K⊂N satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _n \ \left( \Vert \nabla u_n\Vert _{L^2(M)}+\Vert \tau (u_n)\Vert _{L^2(M)}\right) \le \Lambda , \end{aligned}$$\end{document}supn‖∇un‖L2(M)+‖τ(un)‖L2(M)≤Λ,where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (u_n)$$\end{document}τ(un) is the tension field of the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_n$$\end{document}un. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time.


Introduction
Let (M, g) be a compact Riemannian manifold with smooth boundary and (N , h) be a compact Riemannian manifold of dimension n. Let K ⊂ N be a k−dimensional closed submanifold where 1 ≤ k ≤ n. For a mapping u ∈ C 2 (M, N ), the energy density of u is defined by where u * h is the pull-back of the metric tensor h. The energy of the mapping u is defined as A critical point of the energy E over C(K ) is a harmonic map with free boundary u(∂ M) on K . The problem of the existence, uniqueness and regularity of such harmonic maps with a free boundary was first systematically investigated in [8]. By Nash's embedding theorem, (N , h) can be isometrically embedded into some Euclidean space R N . Then we can get the Euler-Lagrange equation where A is the second fundamental form of N ⊂ R N and g is the Laplace-Beltrami operator on M which is defined by The tension field τ (u) is defined by τ (u) = − g u + A(u)(∇u, ∇u). (

1.3)
Thus, u is a harmonic map if and only if τ (u) = 0. When we consider a limit of a sequence of maps with uniformly L 2 -bounded tension fields, the domain may decompose into several pieces (a phenomenon called bubbling or blow-up), and the limit map satisfies the equations or bounds on each piece. The question is whether the sum of the energies of the limit map on those pieces equals the limit of the energies of the approximating maps. Affirmative results are called energy identity and no neck property, and the approach is called blow-up theory; the precise definitions will be given below. Because the problem is conformally invariant only in dimension 2, the analysis usually needs to be restricted to that case, and this will also apply to this paper.
When M is a closed surface, the compactness problem and the blow-up theory (energy identity and no neck property) for a sequence of maps {u n } from M to N with uniformly L 2 -bounded tension fields τ (u n ) and uniformly bounded energy has been extensively studied (see e.g. [6,13,29,31,32,48]), since the fundamental work of Sacks-Uhlenbeck [38]. For sequences of general bounded tension fields, see [20,21,26,49]. For sequences of solutions of more general elliptic systems with an antisymmetric structure, we refer to [16,18]. For corresponding results about harmonic map flows, see e.g. [24,31,32,44,47]. For results of other types of approximate sequences for harmonic maps, see e.g. [4,11,13,15,23]. For the energy identity of harmonic maps from higher dimensional domains, see [25].
In this paper, we shall study the blow-up analysis for a sequence of maps {u n } from a compact Riemann surface M with smooth boundary ∂ M to a compact Riemannian manifold N with uniformly L 2 -bounded tension fields τ (u n ), uniformly bounded energy and with free boundary u n (∂ M) on K . Since the interior case is already well understood, we shall focus on the case where the energy concentration occurs at the free boundary and complete the blow-up theory at the free boundary for a bubbling sequence. When boundary blow-up occurs, the corresponding neck domains are in general not simply half annuli and hence a finer decomposition of the neck domains would be necessary in order to carry out the neck analysis (see Sect. 5).
In fact, we shall first address the regularity problem at the free boundary for weak solutions (see Sect. 3) of − g u + A(u)(∇u, ∇u) = F in M (1.4) for some F ∈ L p (M), p > 1 and under the free boundary constraint (1.1), as it provides some necessary elliptic estimates at the free boundary, which form the analytical foundation of the blow-up theory for the sequence {u n } (see Sect. 4). We would like to remark that the regularity at the free boundary for weak solutions of (1.4) can be proved by applying the classical reflection methods for the harmonic map case by Gulliver-Jost [8] and Scheven [39] or a modified reflection method in [3] and [43] which combines Hélein's moving frame method [10] and Scheven's reflection method [39] so that the technique of Rivière-Struwe in [35] (which holds true also in dimension 2) can be applied. The latter was developed for Dirac-harmonic maps which includes harmonic maps as a special case. In this paper, we shall present an alternative approach without using moving frames (see Sect. 3). Now, we state our first main result: Theorem 1.1 Let u n : M → N be a sequence of W 2,2 maps with free boundary u n (∂ M) on K (1 ≤ k ≤ n), satisfying where τ (u n ) is the tension field of u n . We define the blow-up set

5)
where D M r (x) = {y ∈ M| dist(x, y) ≤ r } denotes the geodesic ball in M and > 0 is a constant whose value will be given in (5.3). Then S is a finite set { p 1 , ..., p I }. By taking subsequences, {u n } converges in W 2,2 loc (M\S) to some limit map u 0 ∈ W 2,2 (M, N ) with free boundary on K and there are finitely many bubbles: a finite set of harmonic spheres w l i : S 2 → N , l = 1, ..., l i , and a finite set of harmonic disks w k i : D 1 (0) → N , k = 1, ..., k i with free boundaries on K , where l i , k i ≥ 0 and l i + k i ≥ 1, i = 1, ..., I , such that is a connected set. Here, harmonic spheres are minimal spheres and harmonic disks with free boundary on K are minimal disks with free boundary on K .
In contrast to the Dirichlet problem where, due to the pointwise boundary condition, no blow-up at the boundary is possible. Here, a blow-up may occur at the boundary and produce one or more harmonic disks with the same free boundary K as the original maps. We should also mention that the Plateau boundary condition for minimal surfaces can also be seen as a free boundary condition where the target set K is a Jordan curve. Here, the monotonicity condition and the three-point normalization that are usually imposed prevent a boundary blow-up, however, see [8] and the systematic discussion in [13].
Our results in the above theorem apply to some classical problems like minimal surfaces in Riemannian manifolds with free boundaries, harmonic functions with free boundary (c.f. [17]) as well as to pseudo holomorphic curves in sympletic manifolds with totally real boundary conditions and Lagrangian boundary conditions, c.f. [7,12,28,51,53] and to string theory where the free boundary represents a D-brane, c.f. [14].
The reason why we work with a sequence of maps with uniformly L 2 -bounded tension fields and with free boundary is that we want to apply our results in Theorem 1.1 to the following heat flow for harmonic maps with free boundary: The existence of a global weak solution of (1.7-1.10) with finitely many singularities was considered by Ma [27], following the pioneering works by Struwe [44,45]. For higher dimensional cases, we refer to [2,46]. For other work on the harmonic map flow with free boundary, see [19]. For the harmonic map flow with Dirichlet boundary condition, we refer to Chang [1].
Here, (1.12) is equivalent to say that the image of weak limit u ∞ and bubbles {ω i } m i=1 is a connected set as in Theorem 1.1.
For finite time blow-ups, we have (1.7-1.10) with T 0 as its singular time. Then there exist finite many bubbles To study the regularity or the qualitative behavior at the free boundary for approximate harmonic maps in this paper, we need some new observations. Firstly, we need to extend the solution across the free boundary as in the harmonic map case done by Scheven [39] and the main difficulty is to write the equation of the extended map into an elliptic system with an antisymmetric potential up to some transformation (see Proposition 3.3). Secondly, thanks to the free boundary condition, we can apply the Pohozaev's argument which was firstly introduced by Lin-Wang [24] for approximate harmonic maps, in the local region as D r (x) ∩ M with x ∈ ∂ M. See Lemma 4.3. This is crucial when we estimate the energy concentration in the neck domain. Thirdly, we have a finer observation of the neck domain. For the boundary blow-up point, the neck domains consist of some irregular half annulus. We will decompose these irregular neck domains into three parts as: interior parts, regular half annulus with the center points living on the boundary and the remaining parts. The first and third parts are easy to control due to the classical blow-up theory of (approximate) harmonic maps with interior blow-up points. In this paper, we focus on the energy concentration in the domains of the second parts.
Since the extended map satisfies an elliptic system with an antisymmetric potential up to some transformation and with some error term F (see Proposition 3.3), one can utilize the idea in [18] (with F = 0) with some modifications to get the energy identity. Here in the present paper, we shall adapt the methods in [5] developed for the interior bubbling case to get the energy identity and the no neck property in the free boundary case. To show the no neck property, namely, bubble tree convergence, we shall get the exponential decay of the energy by deriving a differential inequality on the neck region.
This paper is organized as follows. In Sect. 2, we recall some classical results which will be used in this paper. In Sect. 3, we derive a new form of the elliptic system for the extended map after involution across the boundary which will allow us to turn the boundary regularity problem into an interior regularity problem. As a corollary of this boundary regularity result, we prove a removability theorem for singularities at the free boundary. In Sect. 4, using the new equation of the involuted map, we obtain the small energy regularity in the free boundary case. The gap theorem and Pohozaev's identity in the free boundary case will also be established. In Sect. 5, we prove the energy identity and no neck property at the free boundary by decomposing the neck domain into several parts including a half annulus centered at the boundary and then using the involuted map's equation. Combining this with the interior blow-up theory, we complete the proof of Theorem 1.1. In Sect. 6, we apply Theorem 1.1 to the harmonic map flow with free boundary and prove Theorem 1.2 and Theorem 1.3.
Notation: D r (x 0 ) denotes the closed ball of radius r and center x 0 in R 2 . Denote Let a ≥ 0 be a constant, denote For convenience, we denote D r = D r (0), D = D 1 (0) and R 2 + = R 2 a when a = 0. Let T ⊂ ∂ M be a smooth boundary portion, denote In this paper, we use the notation g (or M ) to denote the Laplace-Beltrami operator on the Riemannian manifold (M, g) and use := ∂ 2 x + ∂ 2 y to denote the usual Laplace operator on R 2 .

Preliminary results
In this section, we will recall some well known results that are useful for our problem.
where τ (u) is the tension field of u. Moreover, by the Sobolev embedding W 2, p (R 2 ) ⊂ C 0 (R 2 ), we have Then u can be extended to a map in W 2,2 (D, N ).
Next, combining the regularity results for critical elliptical systems with an antisymmetric structure developed by Rivière [33] and Rivière-Struwe [35] with various extensions in e.g. [34,36,37,[40][41][42]54], we state the following It is well known that the harmonic map equation can be written as a critical elliptical system with an antisymmetric structure and hence we have the following (which can also be proved by using classical methods developed for the harmonic map case, see e.g.

Regularity at the free boundary
In this section, we will prove a regularity theorem for weak solutions of (1.4) and (1.
As an application, we derive the removability theorem for a local singularity at the free boundary.
We first need to define weak solutions of (1.4) and (1.1). For a weakly harmonic map with free boundary (i.e. F = 0), it is shown that the image of the map is contained in a small tubular neighborhood of K if the energy of the map is small, see Lemma 3.1 in [39]. The proof there requires the interior L ∞estimate for the gradient of the map. Here, we extend this localization property to the more general case of weak solutions of (1.4) with F ∈ L p (D + ) for some 1 < p ≤ 2 and derive certain oscillation estimate for the solution. In our case, there is in general no interior L ∞ -estimate for the gradient of the map.

Moreover, we have
Proof We shall follow the scheme of the proof of Lemma 3.1 in [39]. Take 2 = min{ 1 , } where 1 and are the corresponding constants in Lemma 2.1 and Theorem 2.5. By the interior regularity result Theorem 2.5, we know u ∈ W 2, p Let G x 0 be the fundamental solution of the Laplace operator with singularity at x 0 which satisfies and |∇η| ≤ C R , by Green's representation theorem and integrating by parts, we have for any 2 < s < 2 p 2− p . Thus, we obtain Combining the Poincaré inequality with the fact Integrating the above inequality, we get where the second inequality follows from the Poincaré inequality since d(u(x)) = 0 on ∂ 0 D + 5R (x 1 ) and the third inequality follows from the fact that Li p(d) = 1. Then, we have For any y 0 ∈ D + 1 4 Noting that D + , by a variant of the classical Poincaré inequality, we Therefore, Thus, the lemma follows immediately.
With the help of Lemma 3.2, we can extend the map to the whole disc D by involuting. Firstly, we consider 1 ≤ k ≤ n − 1. Without loss of generality, we may assume K ∩ ∂ N = ∅ in this paper. In fact, if K ∩ ∂ N = ∅, we extend the target manifold N smoothly across the boundary to another compact Riemannian manifold N , such that N ⊂ N and K ∩ ∂ N = ∅. Then we can consider N as a new target manifold.
Denote by K δ 0 the δ 0 -tubular neighborhood of K in N . Taking δ 0 > 0 small enough, then for any y ∈ K δ 0 , there exists a unique projection y ∈ K . Set y = ex p y {−ex p −1 y y}. So we may define an involution σ , i.e. σ 2 = I d as in [8,9,39] by Then it is easy to check that the linear operator Dσ : is the geodesic ball in N with the center point u(0, 1 2 ) and radius C 3 . Then we can define an extension of u to D 1 (0) that For k = n, we also use the above extension by replacing σ = I d. In the following part of this paper, we always state the argument for 1 ≤ k ≤ n − 1, since k = n is similar and easier. At this point, one can derive the regularity at the free boundary for weak solutions of (1.4) by applying classical methods in [8,39] for harmonic maps or the method in [3,43] which combines the method of moving frame and some modification of Rivière-Struwe's method in [35]. Now, we shall give our alternative approach which is also based on some extension of Rivière-Struwe's result.
In order to derive the equation of the involuted map u, we shall first define where is the projection map of R N → T y N . On one hand, Lemma 3.2 tells us that if we take 3 small enough (e.g. C 3 ≤ δ 1 ). Thus, (3.9) is well defined. On the other hand, noting that since (3.7) holds, if 3 is small enough (e.g. 4C 3 ≤ δ 1 ), then we know that and the notations P( u(x)), O( u(x)) in the sequel (see below) are well defined. It is easy to check that P(y) is invertible linear operator for any y ∈ B N δ 1 u 0, 1 2 , since the linear operator Dσ (y) is invertible. For simplicity, we still denote by P(y) the matrix corresponding to the linear operator P(y) under the standard orthonormal basis of R N . Moreover, the matrix P(y) and its inverse matrix P −1 (y) are smooth for y ∈ B N δ 1 u 0, 1 2 . So, there exists an orthogonal matrix O(y) which is also smooth, such that where P T is the transposed matrix and λ i (y), i = 1, ..., N is the eigenvalues of the positive symmetric matrix P T (y)P(y). It is easy to see that and the matrixes One can easily find that Q ∈ L ∞ ∩ W 1,2 (D, R N ) and is invertible. The involuted map satisfies the following proposition: There exists a positive where where ϒ u (·, ·) is a bounded bilinear form and F ∈ L p (D) which are defined by (3.21), satisfying

Proof
Step 1 Firstly, it is easy to see that u ∈ W 1,2 (D). Secondly, we prove that for any arbitrary test vector field V ∈ L ∞ ∩ W 1,2 We decompose V into the symmetric and antisymmetric part with respect to σ as in [39] Since σ 2 = I d, we have (x) (ρ(x)) = I d. Then, Noting Dσ : Since u is a weak solution of (1.4) in D + , we have (3.14) Moreover, there holds and Step 2 We claim: for any V ∈ L ∞ ∩ W 1,2 0 (D, R N ), there holds (3.17) In fact, on the one hand, by (3.12), we get On the other hand, we have Computing directly, we have Combining these equations, we obtain where the equality follows from that F(ρ (x)) ∈ T u(ρ (x)) N = T σ ( u) N . This is (3.17).
Step 3 In order to prove u is a weak solution of (3.10), take an arbitrary test vector (3.20) Noting that Q T = Q and Thus, (3.20) implies This is (3.19). We proved the first result of the lemma.
Step 4 If u ∈ W 2, p (D + ), according to the property of Dσ , it is easy to see u ∈ W 2, p (D) since u satisfies free boundary condition. Computing directly, we have By direct computing, we obtain Combining this with the fact that u satisfies Eq. (1.4) in D + , the Eq. (3.11) follows immediately by taking  To end this section, we derive the removability of a local singularity at the free boundary (see Theorem 2.3 for the interior case).

23)
then u can be extended to a map belonging to W 2, p (D + , N ).
Proof Applying a similar argument as in Lemma A.2 in [13], it is easy to see that u is a weak solution of (1.4) with F = g and with free boundary u(∂ 0 D + ) on K . By Theorem 3.4, we know u ∈ W 2, p (D + r ) for some small r > 0. Thus, u ∈ W 2, p (D + ).

Some basic analytic properties for the free boundary case
In this section, we will prove some basic lemmas for the free boundary case, such as small energy regularity (near the boundary), gap theorem, Pohozaev identity. Firstly, we prove a small energy regularity lemma near the boundary.
Taking 4 sufficiently small, we have So, we have proved the lemma in the case 1 < p < 2. Next, if p = 2, one can first derive the above estimate with p = 4 3 . Such an estimate gives a L 4 (D + 3/4 )− bound for ∇u. Then one can apply the W 2,2 −boundary estimate to the equation and get the conclusion of lemma with p = 2.
The gap theorem still holds for harmonic maps with free boundary.
where (u)(∇u, ∇u) = g αβ i jk (u) ∂u j ∂ x α ∂u k ∂ x β ∂ ∂ y i and i jk are the Christoffel symbol of N in local coordinates {y i } n i=1 .
Without loss of generality, we may assume M u i = 0, 1 ≤ i ≤ k. By standard elliptic estimates with Dirichlet boundary condition and Neumann boundary condition (see Lemma 2.6), we have If 5 is small, then u is a constant map.
Proof Since u(x) satisfies the equation with the free boundary u(∂ 0 D + ) on K , multiplying (x − x 0 )∇u to both sides of the above equation and integrating by parts, for any 0 < t < 1, we get where the last second equality follows from the fact that x−x 0 , − → n = 0 on ∂ 0 D + t (x 0 ). Then the conclusion of lemma follows immediately.

Corollary 4.4 Under the assumptions of Lemma 4.3, we have
Proof From Lemma 4.3, we have Integrating from t to 2t, we will get the conclusion of the corollary.

Energy identity and no neck property
In this section, we shall prove our main Theorem 1.1. We first consider the following simpler case of a single boundary blow-up point. N ) be a sequence of maps with tension fields τ (u n ) and with free boundaries u n (∂ 0 D + ) on K and satisfying Then there exist a subsequence of u n (still denoted by u n ) and a nonnegative integer L such that, for any i = 1, ..., L, there exist a point x i n , positive numbers λ i n and a nonconstant harmonic sphere w i or a nonnegative constant a i and a nonconstant harmonic disk w i (which we view as a map from R 2 a i ∪ {∞} → N ) with free boundary w i (∂R 2 a i ) on K such that:  After taking a subsequence, we may assume lim n→∞ d n r n = a ≥ 0. Then It is easy to see that v n (x) is defined in B n and satisfies where τ (v n (x)) = r 2 n τ (u n (x n + r n x)). Noting that for any x ∈ ∂ 0 B n := {x ∈ R 2 | x n + r n x ∈ ∂ 0 D + } on the boundary, 4 when n is big enough, by (5.4) and for any D 4R (0) ⊂ R 2 . Then there exist a subsequence of v n (also denoted by v n ) and a nontrivial harmonic map v 1 ∈ W 2,2 (R 2 a ) with free boundary v 1 (∂R 2 a ) on K such that for any R > 0, there hold In fact, by (5.7), we have when n is big enough. Then there exist a subsequence of v n (also denoted by v n ) and a harmonic map v ∈ W 2,2 (D + 3R (0)) such that 0, a)), Lastly, (5.9) follows from (5.7), (5.8), Sobolev embedding, Young's inequality and the fact that the measure of D 2R (0) ∩ B n \R 2 a goes to zero as n → ∞. 32 . By the conformal invariance of harmonic maps and the removable singularity Theorem 3.6, v 1 (x) can be extended to a nontrivial harmonic disk. In this case, we can see that v n (x) is defined in B n which tends to R 2 as n → ∞. Moreover, for any x ∈ R 2 , when n is sufficiently large, by (5.4), we have . (5.11) According to Lemma 2.1, there exist a subsequence of v n (we still denote it by v n ) and a harmonic map v 1 (x) ∈ W 1,2 (R 2 , N ) such that Besides, we know E(v 1 ; D 1 (0)) = 2 32 . By the standard theory of harmonic maps, v 1 (x) can be extended to a nontrivial harmonic sphere. We call the above harmonic sphere v 1 (x) or harmonic disk v 1 (x) the first bubble.
We will split the proof of Theorem 5.1 into two parts, energy identity and no neck result. Now, we begin to prove the energy identity.
Energy identity: By the standard induction argument in [6], we only need to prove the theorem in the case where there is only one bubble.
By Lemmas 2.1 and 4.1, there exist a subsequence of u n (still denoted by u n ) and a weak limit u ∈ W 2,2 (D + ) such that Step 1 We prove the energy identity for Case 1, i.e., lim n→∞ d n r n = a < ∞. Under the "one bubble" assumption, we first make the following: Claim: for any > 0, there exist δ > 0 and R > 0 such that |∇u n | 2 dx ≤ 2 for any t ∈ 1 2 r n R, 2δ (5.14) when n is large enough. In fact, if (5.14) is not true, then we can find t n → 0, such that lim n→∞ t n r n = ∞ and Then we have lim n→∞ d n t n = 0.
We set w n (x) := u n (x n + t n x) and B n := {x ∈ R 2 |x n + t n x ∈ D + }.
Then w n (x) lives in B n which tends to R 2 + as n → ∞. It is easy to see that 0 is an energy concentration point for w n . We have to consider the following two cases: (a) w n has no other energy concentration points except 0.
By Lemmas 2.1, 4.1 and the process of constructing the first bubble, passing to a subsequence, we may assume that w n converges to a harmonic map w(x) : R 2 + → N with free boundary w(∂R 2 + ) on K satisfying, for any R > 0, Noting that (5.15) implies By the conformal invariance of harmonic map and Theorem 3.6, w(x) is a nontrivial harmonic disk which can be seen as the second bubble. This contradicts the "one bubble" assumption. (b) w n has another energy concentration point p = 0. Without loss of generality, we may assume p is the only energy concentration point in D + r 0 ( p) for some r 0 > 0. Similar to the process of constructing the first bubble, there exist x n → p and r n → 0 such that By (5.4), we know r n t n ≥ r n . Then, passing to a subsequence we may assume lim n→∞ d n r n t n = d ∈ [0, a]. Moreover, there exists a nontrivial harmonic map v 2 (x) : R 2 d → N with free boundary v 2 (∂R 2 d ) on K satisfying, for any R > 0, where B n := {x ∈ R 2 |x n + r n x ∈ B n }. That is is also a bubble for the sequence u n . This is also contradiction to the "one bubble" assumption. Thus, we proved Claim (5.14). Let x n ∈ ∂ 0 D + be the projection of x n , i.e. d n = dist(x n , ∂ 0 D + ) = |x n − x n |. Firstly, we decompose the neck domain D + δ (x n )\D + r n R (x n ) as follows when n and R are large.
Since lim n→∞ d n r n = a, when n and R are large enough, it is easy to see that Moreover, for any 2r n R ≤ t ≤ 1 2 δ, there holds By assumption (5.14), we have E(u n ; 1 ) + E(u n ; 3 ) ≤ 2 (5.19) and |∇u n | 2 dx ≤ 2 for any t ∈ (2r n R, By a scaling argument, we may assume According to the small energy regularity theory Lemmas 2.1 and 4.1, we obtain for any t ∈ (2r n R, 1 2 δ). Thus, u n ( 2 ) ⊂ K δ 0 and we can extend the definition of u n to the domain 2 := D δ and satisfies Eq. (3.11) where we take F n (x) = τ (u n )(x) and define ϒ u n (·, ·), F n (x) as in (3.21). Define Then by (5.21), we have We have On the one hand, by Jessen's inequality, we have On the other hand, using Eq. (3.11), we get Thus, By the definition of u n (see (3.8)), we obtain 2 ∂ u n ∂r Note that where P is the matrix corresponding to the linear operator defined by (3.9) under the orthonormal basis of R N . Similarly, Noting that | K = I d, by the continuity of eigenvalues of P T P, we have that for any δ > 0, there exists a constant δ 1 = δ 1 (δ ) > 0, such that for any ξ ∈ R n and y ∈ K δ 1 , there holds P T (y)P(y)ξ, ξ ≤ (1 + δ )|ξ | 2 .
Step 2 We prove the energy identity for Case 2, i.e., lim sup n→∞ d n r n = ∞. The proof is similar to the Case 1. Firstly, we need to show the Claim (5.14) also holds in this case.
In fact, if (5.14) is not true, then we can find t n → 0, such that lim n→∞ t n r n = ∞ and Then passing to a subsequence, we may assume We set w n (x) := u n (x n + t n x) and B n := {x ∈ R 2 |x n + t n x ∈ D + }.
Then w n (x) lives in B n and 0 is an energy concentration point for w n . We have to consider the following two cases: Then B n tends to R 2 b as n → ∞. Here, we also need to consider two cases. (i) w n has no other energy concentration points except 0. It is almost the same as Case (a) in Step 1 where by passing to a subsequence, w n converges to a nontrivial harmonic map w(x) : R 2 b → N with free boundary w(∂R 2 b ) on K which can be seen as the second bubble. This is a contradiction to the "one bubble" assumption. (ii) w n has another energy concentration point p = 0. Similar to the process of Case (b) in Step 1, there exist x n → p and r n → 0 such that (5.17) that is u n (x n + t n x n + t n r n x) → v 2 (x) in W 1,2 loc (R 2 ).
In both cases, we will get the second bubble v 2 (x) or v 2 (x). This contradicts the "one bubble" assumption.
Then B n tends to R 2 as n → ∞. Again, we need to consider two cases. (iii) w n has no other energy concentration points except 0. By Lemma 2.1, Theorem 2.3 and (5.26), there exists v 2 (x) : R 2 → N is a nontrivial harmonic map such that Then, we get the second bubble v 2 (x) which contradicts the "one bubble" assumption.
(iv) w n has another energy concentration point p = 0. Similar to Case (b) in Step 1, there exist x n → p and r n → 0 such that (5.17) holds and passing to a subsequence, we have lim n→∞ d n r n t n = ∞.
Moreover, by the process of constructing the first bubble in Case 2, there exists a nontrivial harmonic map v 2 (x) : R 2 → N such that that is u n (x n + t n x n + t n r n x) → v 2 (x) in W 1,2 loc (R 2 ).
By assumption (5.14), we have E(u n ; 1 ) + E(u n ; 3 ) ≤ 2 (5.27) and |∇u n | 2 dx ≤ 2 for any t ∈ 2d n , 1 2 δ . (5.28) Noting that 4 = D + d n (x n )\D + r n R (x n ) = D d n (x n )\D r n R (x n ), by the well-known blow-up analysis theory of harmonic maps with interior blow-up points (also a sequence of maps with uniformly L p bounded tension fields for some p ≥ See [6,20,32] for details. Lastly, to estimate the energy concentration in 2 , we can use the same argument as in the previous Case 1 to get 2 |∇u n | 2 dx ≤ C(δ + ). (5.31) Combining (5.27), (5.29) with (5.31), it is easy to obtain (5.12). We proved the energy identity. Next, we prove the no neck property in Theorem 5.1, i.e., the base map and the bubbles are connected in the target manifold.

No neck property:
Here, we also need to consider two cases. But, for Case 2, we use the same argument as in the previous reasoning where we split the neck domain into two parts, an interior domain and a boundary domain. Then, with the help of the no neck results in [20,32] for a sequence of maps with uniformly L 2 -bounded tension fields, we just need to prove (5.13) for Case 1.
We may assume lim n→∞ d n r n = a and decompose the neck domain D + δ (x n )\D + r n R (x n ) = 1 ∪ 2 ∪ 3 , when n and R are large.
By assumption (5.14) and small energy regularity (Lemmas 2.1 and 4.1), we have and when n, R are large and δ is small. Without loss of generality, we may assume 1 2 δ = 2 m n (2r n R) where m n → ∞ as n → ∞. Inspired by a technique by Ding [5] for the interior bubbling case, we set Similar to the proof of (5.22) and (5.23), we have |τ n |dx (5.34) and where the last inequality follows from Corollary 4.4.
As for the boundary, by Poincaré's inequality, we have Similarly, we get Using these together, we have Taking and δ sufficiently small, we get Therefore, f (t) + C2 t 0 +t r n R. Integrating from 2 to L, we arrive at + C2 t 0 r n R2 (1−1/C)L . Now, let t 0 = i and L = L i := min{i, m n − i}. Then, we have Q(L i ) ⊂ D + δ/2 (x n )\D + 2r n R (x n ) ⊂ D + δ (x n )\D + r n R (x n ) and where we used the energy identity (5.12). By Lemmas 2.1 and 4.1, we obtain This inequality and (5.32), (5.33) imply (5.13) and we have proved there is no neck during the blow-up process. Now, we can prove Theorem 1.1.

Proof of Theorem 1.1
Combining the blow-up theory of a sequence of maps with uniformly L 2 -bounded tension fields from a closed Riemann surface (see [6,20,24,26,32]) and Theorem 5.1, we can easily get the conclusion of Theorem 1.1 by following the standard blow-up scheme in [6]. On the other hand, it is well known that harmonic spheres are minimal spheres and harmonic disks with free boundary on K are minimal disks with free boundary on K (see e.g. the proof of Theorem B in [27], page 300).

Application to the harmonic map flow with free boundary
In this section, we will apply the results in Theorem 1.1 to the harmonic map flow with free boundary and prove Theorem 1.2 and Theorem 1.3.
Firstly, we have Take the sequence u n = u(·, t n ), τ (u n ) = ∂ t u(·, t n ) in Theorem 1.1. Combining this with Lemma 6.3, the conclusion of Theorem 1.2 follows immediately.

Proof of Theorem 1.3
It is sufficient to consider the case that (x 0 , T 0 ) with x 0 ∈ ∂ M being the only singular point at time T 0 . For the case of an interior singularity x 0 ∈ M\∂ M, one can refer to [24]. Without loss of generality, we may assume M = D + 1 (0) and x 0 = 0. By Lemma 6.3, there exist sequences t n ↑ T 0 and λ n ↓ 0 such that lim n→∞ D + λn (0) |∇u| 2 (·, t n )dx = m.