Minimizing fractional harmonic maps on the real line in the supercritical regime

This article addresses the regularity issue for minimizing fractional harmonic maps of order $s\in(0,1/2)$ from an interval into a smooth manifold. H\"older continuity away from a locally finite set is established for a general target. If the target is the standard sphere, then H\"older continuity holds everywhere.


Introduction
In a series of recent articles [6,7], F. Da Lio and T. Rivière introduced the concept of 1/2-harmonic maps into a manifold. Given a compact smooth submanifold N ⊆ R d without boundary, such a map u : R → N is defined as a critical point of the nonlocal energy It satisfies the Euler-Lagrange equation where (−∆) 1 2 is the fractional Laplacian as defined in Fourier space. Obviously, this equation is in strong analogy with the standard harmonic map equation into N , and one main issue is to prove a priori regularity. This was achieved in [6,7], thus extending the famous regularity result of F. Hélein for classical harmonic maps on surfaces [11]. The notion of 1/2-harmonic maps has been then extended in [14,16] to higher dimensions, and partial regularity for minimizing or stationary 1/2-harmonic maps established (in analogy with the classical harmonic map problem [1,5,17]).
All these works pave the way to a more general theory for fractional harmonic maps where the energy E 1 2 is replaced by the Dirichlet form induced by the fractional Laplacian (−∆) s with exponent s ∈ (0, 1). As noticed in [15,Remark 1.7], the case s ∈ (0, 1/2) is in strong relation with the so-called nonlocal minimal surfaces introduced by L. Caffarelli, J.M. Roquejoffre, and O. Savin [2]. For this reason, we focus here on the case s ∈ (0, 1/2), and as first step toward such a theory, we shall consider minimizing s-harmonic maps in one space dimension. Before going further, let us give some details on the mathematical framework.
Definition. We say that u ∈ H s (ω; N ) is a minimizing s-harmonic map in ω if E s (u, ω) E s ( u, ω) 1 for every u ∈ H s (ω; N ) such that spt( u − u) ⊆ ω.
In terms of scaling, this equation turns out to be supercritical (since s ∈ (0, 1/2)), and one may expect that minimizing s-harmonic maps are singular, exactly as it happens for (classical) minimizing harmonic maps in dimensions greater than three [17].
The main objective of this paper is to provide a first partial regularity result for minimizing s-harmonic maps. At this stage, we should point out that existence is not an issue. Indeed, prescribing an exterior condition g ∈ H s (ω; N ), one can minimize the energy E s (·, ω) over all maps u ∈ H s (ω; N ) satisfying u = g a.e. in R\ω. Existence for this minimization problem easily follows from the Direct Method of Calculus of Variations, and it obviously produces a minimizing s-harmonic map in ω.
Our first main result concerns the case of a general (smooth) target N . Theorem 1.1. For s ∈ (0, 1/2), let u ∈ H s (ω; N ) be a minimizing s-harmonic map in ω. Then u is locally Hölder continuous in ω away from a locally finite set of points.
The proof of Theorem 1.1 follows somehow the general scheme for proving partial regularity of minimizing harmonic maps, or more precisely of minimizing harmonic maps with (partially) free boundary. Indeed, the problem can be rephrased as a degenerate regularity problem for harmonic maps with free boundary, once we use the so-called Caffarelli-Silvestre extension [3]. With this respect, our arguments ressemble to the ones in [4,9,10], except that they have to be suitably modified to deal with our degenerate setting. In view of this classical literature, one may wonder if Hölder continuity implies higher order regularity. We do not address this question here, as it will be the object of a future work. In a complementary direction, one can ask wether or not a (one dimensional) minimizing s-harmonic can be singular. We believe that, in general, Theorem 1.1 is optimal, but the question remains open. However, if the manifold N is a standard sphere, then there are no singularities at all. This statement (and proof) is in a sense an amusing fractional counterpart of the regularity result of R. Schoen & K. Uhlenbeck [18] for minimizing harmonic maps into spheres. Theorem 1.2. For s ∈ (0, 1/2) and d > 1, let u ∈ H s (ω; S d−1 ) be a minimizing s-harmonic map in ω. Then u is locally Hölder continuous in ω.
This article is organized as follows. In Section 2, we introduce the notion of harmonic maps with free boundary induced by the Caffarelli-Silvestre extension, together with some fundamental properties such as the monotonicity formula. In Section 3, we prove an ε-regularity theorem for those harmonic maps with free boundary. Section 4 is devoted to compactness properties of minimizing s-harmonic maps, and Theorems 1.1 & 1.2 are proved in Section 5.
Notation. We shall often identify R with ∂R 2 + = R × {0}. More generally, a set A ⊆ R can be identified with A × {0} ⊆ ∂R 2 + . Points in R 2 are written x = (x, y). We denote by B r (x) the open disc in R 2 of radius r centered at x = (x, y). For an arbitrary set Ω ⊆ R 2 , we write Ω + := Ω ∩ R 2 + and ∂ + Ω := ∂Ω ∩ R 2 + . If Ω ⊆ R 2 + is a bounded open set, we shall say that Ω is admissible whenever • ∂Ω is Lipschitz regular; is non empty and has Lipschitz boundary; The tangent and normal spaces to N at a point p ∈ N are denoted by Tan(p, N ) and Nor(p, N ), respectively.

Minimizing weighted harmonic maps with free boundary
The proof of our results relies on the already mentioned Caffarelli-Silvestre extension procedure [3] which allows to rephrase our fractional problem into a local one. Before going into details on the extension of minimizing s-harmonic maps, we briefly introduce the resulting local problem and its functional setting. and We refer to [15,Section 2] for the main properties of these spaces that we shall use. We simply recall that a map v ∈ H 1 (Ω; R d , y a dx) has a well defined trace on ∂ 0 Ω, and the trace operator from H 1 (Ω; R d , y a dx) into L 2 (∂ 0 Ω; R d ) is a compact linear operator.
Definition 2.1. Let Ω ⊆ R 2 + be a bounded admissible open set, and consider a map v ∈ H 1 (Ω; R d , y a dx) such that v(x) ∈ N a.e. on ∂ 0 Ω. We say that v is a minimizing weighted harmonic map in Ω with respect to the (partially) free boundary v(∂ 0 Ω) ⊆ N if for every competitor w ∈ H 1 (Ω, y a dx) satisfying w(x) ∈ N a.e. on ∂ 0 Ω, and such that spt(w − v) ⊆ Ω∪∂ 0 Ω. In short, we shall say u is a minimizing weighted harmonic map with free boundary in Ω.

2.2.
Extending minimizing s-harmonic maps. We now move on the extension procedure of [3]. Given a bounded open interval ω ⊆ R, we define the extension u e : R 2 This extension can be referred to as fractional harmonic extension of u (by analogy with the case s = 1/2) as it solves div(y a ∇u e ) = 0 in R 2 + , u e = u on R ≃ ∂R 2 + . (2. 3) It turns out that u e ∈ H 1 (Ω; R d , y a dx) for every bounded admissible open set Ω ⊆ R 2 + such that ∂ 0 Ω ⊆ ω. In addition, u e ∈ L ∞ (R 2 + ) whenever u ∈ L ∞ (R), and u e . We refer to [15,Section 2] for further details.
We shall make use of the following converse statement to control the fractional energy by the weighted Dirichlet energy.
for some constant C = C(s).
The following proposition draws links between minimizing s-harmonic maps and minimizing weighted harmonic maps with free boundary. Its proof follows exactly as in [14,Proposition 4.9] (see also [15,Corollary 2.13]), and we shall omit it.
2.3. The monotonicity formula. In this subsection, we consider a bounded admissible open set Ω ⊆ R 2 + , and a minimizing weighted harmonic map v ∈ H 1 (Ω; R d , y a dx) with free boundary. We present the fundamental monotonicity formula involving the following density function: for a point x 0 = (x 0 , 0) ∈ ∂ 0 Ω and r > 0 such that Proof. The proof follows classically from the stationarity implied by minimality.
To be more precise, let us consider a vector field X = (X 1 , X 2 ) ∈ C 1 (R 2 + ; R 2 ) compactly supported in Ω ∪ ∂ 0 Ω and such that X 2 = 0 on R × {0}. Then consider a compactly supported C 1 -extension of X to the whole R 2 , still denoted by X. We define {φ t } t∈R the flow on R 2 generated by X, i.e., for x ∈ R 2 , the map t → φ t (x) is defined as the unique solution of the differential equation Computing this derivative (see e.g. [19,Chapter 2.2] or [15]) leads to Corollary 2.6. For every x 0 ∈ ∂ 0 Ω, the limit exists, and the function Θ v : ∂ 0 Ω → [0, ∞) is upper semicontinuous. In addition, Proof. The existence of the limit defining Θ v as well as (2.7) are straightforward consequences of Lemma 2.5. Then Θ v is upper semicontinuous as a pointwise limit of a decreasing family of continuous functions.
1. An extension lemma and the hybrid inequality. This subsection is essentially devoted to the construction of comparison maps. We shall start with the construction of competitors from a boundary data satisfying a small oscillation condition. Testing minimality against such competitors leads to the so-called hybrid inequality (see [9,10]), a central estimate in the proof of the ε-regularity theorem. Let us start with an elementary lemma.
Then notice that the function d N is 1-Lipschitz, and by chain rule one derives |∇d 2 N | 2d N a.e. in R d . In turns, it implies that d 2 N (u) ∈ H 1 (∂ + B 1 , y a dx) and Since v(−1, 0) ∈ N , this estimate implies that for every x ∈ ∂ + B 1 , where ((−1, 0), x) denotes the arc in ∂ + B 1 going from (−1, 0) to x. The announced inequality then follows from Cauchy-Schwarz inequality.
Note that h ∈ L ∞ (B 1 ). Indeed, since v is absolutely continuous, it is bounded. Since h minimizes E s (·, B 1 ) over all maps equal to v on ∂B 1 , a classical truncation argument shows that |h| does not exceed v L ∞ (∂B 1 ).
Using (3.2) and (3.3), we infer from the divergence theorem and Cauchy-Schwarz inequality that Hence, by symmetry, By the fundamental theorem of calculus (and symmetry), we have We choose the point x 0 in such a way that x → |v(x) − ξ| achieves its minimum at x 0 . Then, and the maximum principle in [8] together with (3.5) implies that y a |∂ τ v| 2 dH 1 for every x ∈ B 1 . By our assumption, we thus have As a consequence, if ε 0 = ε 0 (N ) is small enough, h takes values in a small tubular neighborhood of N . In such a neighborhood, the nearest point retraction π N on N is well defined and smooth. Therefore, π N (h) belongs to H 1 (B 1 ; N , |y| a dx), and We shall now construct the extension w interpolating h and π N (h) near ∂ + B 1 . We proceed as follows. Consider the set and let ζ ∈ C ∞ (B + 1 ; [0, 1]) be a smooth cut-off function satisfying ζ = 1 in A ∩ B + 1 , and ζ = 0 on ∂ + B 1 . From the very definition of A, we can even find ζ in such a way that |∂ y ζ(x, y)| C and |∂ x ζ(x, y)| Cy where C = C(s). In particular, ζ ∈ H 1 (B + 1 ; [0, 1], y a dx). We finally define By construction, w(x) ∈ N for x ∈ ∂ 0 B + 1 , and w = h = v on ∂ + B 1 . Then we estimate  Let v ∈ H 1 (B + 1 ; R d , y a dx) be a minimizing weighted harmonic map with free boundary in B + 1 , and ξ ∈ R d . If for every λ ∈ (0, 1).
Proof. By a classical averaging argument, we can find r ∈ (1/2, 1) such that v restricted to ∂ + B r belongs to H 1 (∂ + B r ; R d , y a dx), and Setting v r (x) := v(rx) for x ∈ ∂ + B 1 , we deduce by scaling that v r satisfies (3.1) for ε 1 small enough. Denote by w r the extension of v r provided by Lemma 3.2, and set w(x) := w r (x/r) for x ∈ B + r . Scaling back, we discover that for every λ ∈ (0, 1).

3.2.
Small energy regularity. We shall now prove the aforementioned small energy regularity property. As usual, the cornerstone argument is an energy improvement under a small oscillation condition. This leads to an improved energy decay, which in turn implies Hölder continuity as in the classical Morrey's lemma.

Let us start with the following elementary lemma inspired from [4, Lemma 3.3].
Lemma 3.5. Let v ∈ H 1 (B + 1 ; R d , y a dx) be such that v(x) ∈ N for a.e. x ∈ ∂ 0 B + 1 . Settingv for some constant C = C(s).
Applying Poincaré's inequalities, and Hölder's inequality, where we have used again the fact that d N is 1-Lipschitz in the last inequality.
Proof of Theorem 3.4.
Step 1. We argue by contradiction assuming that for a given radius r 0 ∈ (0, 1/2) (to be chosen), there is a sequence {v n } in H 1 (B + 1 ; R d , y a dx) of minimizing weighted harmonic maps in B + 1 such that ε 2 n := E s (v n , B + 1 ) → 0 , (3.9) and 1 By Lemma 3.5, we have d N (v n ) Cε n → 0. Hence, for n large enough, there is a unique p n ∈ N such that d N (v n ) = |v n − p n |. Extracting a subsequence, there are p ∈ N and q ∈ R d such that p n → p ,v n → p , and p n −v n ε n → q .
Note that q ∈ Nor(p, N ) since p n −v n ∈ Nor(p n , N ).
Step 3. Set Since the embedding H 1 (B + 1 , y a dx) ֒→ L 2 (B + 1 , y −a dx) is compact (see e.g. [12]), we have Poincaré's inequalities telling us that 1 Here we have used the monotonicity formula in Lemma 2.5, the fact that the function d N is 1-Lipschitz, and d N (v n ) = 0 on ∂ 0 B + 1 . Changing variables, one discovers that the rescaled map x → v n (2r 0 x) satisfies the small oscillation condition in Corollary 3.3 with ξ = (v n ) 2r0 for n large enough, thanks to (3.9). Choosing λ = 1/8 in that corollary and scaling back, we infer that By Lemma 2.5 again, we have Then, By the two compact embeddings H 1 (B + 1 , y a dx) ֒→ L 1 (B + 1 ) and H 1 (B + 1 , y a dx) ֒→ L 2 (B + 1 , y −a dx), we have w n → w strongly in L 2 (B + 1 , y −a dx) and (w n ) 2r0 → (w) 2r0 . Hence, Next we decompose the map w as w =: w T + w ⊥ where w T takes values in Tan(p, N ), and w ⊥ takes values in Nor(p, N ). From (3.11) and (3.12), we derive that div(y a ∇w ⊥ ) = 0 in B + 1 , w ⊥ = q on ∂ 0 B + 1 . From the boundary condition, we can reflect oddly the map (w ⊥ − q) to the whole ball B 1 , so that the resulting w ⊥ belongs to H 1 (B 1 , y a dx) and satisfies div(|y| a ∇w ⊥ ) = 0 in B 1 .
Once again, [8] tells us that w T is α-Hölder continuous in B 1/2 , and thus (3.20) In view of (3.18), (3.19) and (3.20), we have proved that Finally, to estimate the last term in the right hand side of (3.16), we proceed as follows. First notice that d N (v n ) ε n |w n | + |v n − p n |, so that ε −1 . Up to a further subsequence, we also have v n (x) → a, w n (x) → w(x), and ε 1 n d N (v n (x)) → d(x) for a.e. x ∈ B + 1 . Given x ∈ B + 1 such that these convergences hold at x, we have On the other hand, for n large enough, v n (x) has a unique nearest point v n ∈ N , and v n → p. Nor(p, N ), taking a subsequence if necessary. In turn, it implies that ε −1 n (v n − p n ) is converging toward a vector t ∈ Tan(p, N ). Consequently, t + n = w(x) − q, so that n = w ⊥ (x) − q, and thus d(x) = |w ⊥ (x) − q|.
Arguing exactly as [9, Theorem 2.5], we infer from Theorem 3.4 the following decay estimate.
In turn, this last corollary implies Hölder continuity at the boundary as in Morrey's lemma.

Compactness of minimizing s-harmonic maps
This section is devoted to compactness of minimizing s-harmonic maps. As it will be clear in a few lines, the proof is here much simpler compare to classical harmonic maps, as minimality can be directly tested (as if the exterior condition were fixed). Consequences concerning the extensions and densities are then easy exercices. Proof. First we select a subsequence u k := u n k such that u k → u a.e. on R, and lim k→∞ E s (u k , ω) = lim inf n→∞ E s (u n , ω) < ∞ .
Since each u k takes values into N , we infer from the pointwise convergence that u(x) ∈ N for a.e. x ∈ R. Then, by Fatou's lemma, we have Let us now consider u ∈ H s (ω; N ) such that spt(u − u) ⊆ ω.
We select an open interval ω ′ such that spt(u − u) ⊆ ω ′ and ω ′ ⊆ ω. Define It is elementary to check that u k ∈ H s (ω; N ), and of course spt( u k − u k ) ⊆ ω. By minimality of u k , we have E s (u k , ω) E s ( u k , ω) which leads to Since u and u k are taking values in N , we have Hence E s ( u k , ω ′ ) → E s ( u, ω ′ ) by dominated convergence and the fact that u = u a.e. in R \ ω ′ . On the other hand, lim inf k E s (u k , ω ′ ) E s (u, ω ′ ), still by Fatou's lemma. Letting k → ∞ in (4.1), we can now conclude that E s (u, ω ′ ) E s ( u, ω ′ ). Once again, since u = u a.e. in R \ ω ′ , this yields E s (u, ω) E s ( u, ω). We have thus proved that u is a minimizing s-harmonic map in ω.
In addition, the argument above applied to u = u shows that E s (u k , ω ′ ) → E s (u, ω ′ ). In turn, again by dominated convergence. Hence, Since {u k } is bounded in H s (ω ′ ) and u k → u pointwise., we have u k ⇀ u weakly in H s (ω ′ ). Then (4.2) implies that u k → u strongly in H s (ω ′ ).   Proof. Without loss of generality we may assume that x = 0. For r > 0 small enough we have ∂ 0 B + 2r ⊆ ω. Setting r n := |x n |, we have r n < r for n large enough. Then, we infer from Corollary 2.6 that Θ u e n (x n ) Θ u e n (x n , r) 1 r 1−2s E s (u e n , B + r+rn ) .
By Theorem 4.2, we have u e n → u e strongly in H 1 (B + 2r ; R d , y a dx), and thus lim sup Letting now r ↓ 0 provides the desired conclusion.

Proof of Theorems 1.1 & 1.2
This section is devoted to the proof of Theorem 1.1 and 1.2. We consider for the entire section a bounded open interval ω ⊆ R, and u ∈ H s (ω; N ) a minimizing sharmonic map in ω. Both proofs rely on the analysis of tangent maps of u at a given point of ω. To define them, we fix a point x 0 ∈ ω, and for ρ > 0 we consider the rescaled map u x0,ρ (x) := u(x 0 + ρx) .
Tangent maps of u at x 0 are all possible weak limits of u x0,ρ as ρ ↓ 0, and this is is the purpose of the following proposition.
On the other hand, Therefore E s (u n , ω k ) C k for a constant C k depending only on s and k. In particular, {u n } is bounded in H s (ω k ) for each integer k 1. Hence, we can find a (not relabeled) subsequence such that u k ⇀ u 0 weakly in H s loc (R). From the compact embedding H s (ω k ) ֒→ L 2 (ω k ), we also deduce that u n → u 0 in L 2 loc (R). Applying Theorem 4.1 in each ω k , we derive that u 0 is a minimizing s-harmonic map in every bounded open interval. Next, Theorem 4.2 implies that Θ u e 0 (0, r) = lim n→∞ Θ u e n (0, r) = lim n→∞ Θ u e (0, ρ n r) = Θ u e (0) ∀r > 0 .
Remark 5.2. If u is continuous at x 0 , the limit u 0 obtained in Proposition 5.1 is obviously the constant map equal to u(x 0 ). As a consequence, if u is continuous at x 0 , then Θ u e (x 0 , 0) = 0.
Proof of Theorem 1.1. Let us consider the set where ε 2 > 0 is the constant given by Theorem 3.4. Since Θ u e is upper semicontinuous, S is a relatively closed subset of ω. Moreover, Corollaries 3.6 & 3.7 together with Corollary 2.6 implies that u is locally Hölder continuous in ω \ S. To prove The-orem1.1, it then remains to show that S has no accumulation point in ω. We argue by contradiction assuming that there is a sequence {x n } ⊆ S such that x n → x ∈ ω.
Without loss of generality, we may assume that x n > x. Setting ρ n := x n − x, we consider the sequence u n := u x0,ρn , and then apply Proposition 5.1 to find a (not relabeled) subsequence and a minimizing s-harmonic map u 0 of the form (5.1) such that u n → u 0 . In view of Corollary 4.3 we have On the other hand, by the explicit form (5.1), the map u 0 is continuous at 1. Hence, Θ u e 0 (1, 0) = 0 by Remark 5.2, contradiction.
Proof of Theorem 1.2. Recall that we assume now that N = S d−1 . In view of the proof of Theorem 1.1, it is enough to show that the set S defined in (5.2) is empty. Assume by contradiction that S = ∅. We may then assume without loss of generality that 0 ∈ S. Let u 0 be a s-minimizing harmonic map produced by Proposition 5.1, i.e., u 0 is the limit of the rescaled map u 0,ρn for some sequence ρ n → 0. Then Θ u e 0 (0) ε 2 > 0, so that u 0 is not constant. In other words, in the form (5.1) the two vectors a, b ∈ S d−1 are distinct. Upon working in the plane passing through a, b, and the origin, there is no loss of generality assuming that d = 2, that is N = S 1 . Moreover, rotating coordinates in the image if necessary, we can assume that a = (α, β) and b = (−α, β) , with 0 < α 1 and 0 β < 1 satisfying α 2 + β 2 = 1. Then set a * := (−β, α) and b * := (β, α) .