Plateau flow or the heat flow for half-harmonic maps

Using the interpretation of the half-Laplacian on $S^1$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$, we devise a classical approach to the heat flow for half-harmonic maps from $S^1$ to a closed target manifold $N$, recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author's 1985 results for the harmonic map heat flow of surfaces and in similar generality. When $N$ is a smoothly embedded, oriented closed curve $\Gamma$ the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces.


Background and results
1.1.Half-harmonic maps and their heat flow.Let N ⊂ Ê n be a closed sub- manifold, that is, compact and without boundary.The concept of a half-harmonic map u : S 1 → N ⊂ Ê n was introduced by Da Lio-Rivière [14], who together with Martinazzi in [12], Theorem 2.9, also made the interesting observation that the harmonic extension of a half-harmonic map yields a free boundary minimal surface supported by N , a fact which also was noticed by Millot-Sire [29], Remark 4. 28.
In his PhD-thesis, Wettstein [48], [49], [50], recently studied the corresponding heat flow given by the equation (1.1) dπ N (u) u t + (−∆) 1/2 u = 0 on S 1 × [0, ∞[, where u t = ∂ t u, and where π N : N ρ → N is the smooth nearest neighbor projection on a ρ-neighborhood N ρ of the given target manifold to N , and, with the help of a fine analysis of the fractional differential operators involved, he showed global existence for initial data of small energy.
Moser [32] and Millot-Sire [29] contributed important results to the study of half-harmonic maps by exploiting the fact that for any smooth u : S 1 → Ê n we can represent the half-Laplacian classically in the form where U : B → Ê n is the harmonic extension of u to the unit disc B 1 .Here, using the identity (1.2) we are able to remove the smallness assumption in Wettstein's work and show the existence of a "global" weak solution to the heat flow (1.1) Date: July 15, 2022. 1 The classical formula (1.2) is a special case of a much more general result, due to Caffarelli-Silvestre [4], who pointed out that many nonlocal problems involving fractional powers of the Laplacian can be related to a local, possibly degenerate, elliptic equation via a suitable extension of the solution to a half-space.
for data of arbitrarily large (but finite) energy, which is defined for all times and smooth away from finitely many "blow-up points" where energy concentrates, and whose energy is non-increasing.The solution is unique in this class in exact analogy with the classical result [42] by the author on the harmonic map heat flow for maps from a closed surface to a closed target manifold N ⊂ Ê n ; see Theorem 1.2 below.
In order to describe our work in more detail, let H 1/2 (S 1 ; N ) = {u ∈ H 1/2 (S 1 ; Ê n ); u(z) ∈ N for almost every z ∈ S 1 }.Interpreting S 1 = ∂B, where B = B 1 (0; Ê 2 ) and tacitly identifying a map u ∈ H 1/2 (S 1 ; N ) with its harmonic extension U ∈ H 1 (B; Ê n ), for a given function u 0 ∈ H 1/2 (S 1 ; N ) we then seek to find a family of harmonic functions u(t) ∈ H 1 (B; Ê n ) with traces u(t) ∈ H 1/2 (S 1 ; N ) for t > 0, solving the equation 1.2.Energy.The half-harmonic heat flow may be regarded as the heat flow for the half-energy of a map u ∈ H 1/2 (S 1 ; N ).Note that the half-energy of u equals the standard Dirichlet energy of its harmonic extension u ∈ H 1 (B; Ê n ).Indeed, integrating by parts we have where we use the Millot-Sire identity (1.2) and where the last identity easily follows from the representation of the operators (−∆) 1/2 and (−∆) 1/4 in Fourier space with symbols |ξ|, |ξ|, respectively, and Parceval's identity. 2Therefore, in the following for convenience we may always work with the classically defined Dirichlet energy.Moreover, we may interpret the half-harmonic heat flow as the heat flow for the Dirichlet energy in the class of harmonic functions with trace in H 1/2 (S 1 ; N ); see Section 2 below for details.Observe that the Dirichlet energy is invariant under conformal transformations, and we have E(u) = E(u • Φ) for any u ∈ H 1 (B; Ê n ) and any Φ ∈ M .
For smooth data we then have the following result.
2 Conversely, via Fourier expansion we also can prove (1.5) directly.Computing the first variations of E and E 1/2 , respectively, we then obtain (1.2).
Theorem 1.1.Let N ⊂ Ê n be a closed, smooth sub-manifold of Ê n , and suppose that the normal bundle T ⊥ N is parallelizable.Then the following holds: i) For any smooth u 0 ∈ H 1/2 (S 1 ; N ) there exists a time T 0 ≤ ∞ and a unique smooth solution u = u(t) of (1.3), hence of (1.1), with data (1.4) for 0 < t < T 0 .
iii) If T 0 = ∞, then, as t → ∞ suitably, u(t) smoothly converges to a halfharmonic limit map away from at most finitely many concentration points where non-constant half-harmonic maps "bubble off " as in ii).
By the Da Lio-Rivière interpretation of (1.6), the "bubbles" ū(i) as well as the limit u ∞ of the flow conformally parametrize minimal surfaces with free boundary on N , meeting N orthogonally along their free boundaries.
The hypothesis regarding the target manifold N in particular is fullfilled if N is a closed, orientable hypersurface of co-dimension 1 in Ê n , or if N is a smoothly embedded, closed curve Γ ⊂ Ê n .
It would be interesting to find examples of initial data for which the flow blows up in finite time, as in the work of Chang-Ding-Ye [5] on the harmonic map heat flow.
For data in H 1/2 (S 1 ; N ) the following global existence result holds, which is our main result.Theorem 1.2.For N ⊂ Ê n as in Theorem 1.1 the following holds: i) For any u 0 ∈ H 1/2 (S 1 ; N ) there exists a unique global weak solution of (1.3) with data (1.4) as in Definition 6.3, whose energy is non-increasing and which is smooth for positive time away from finitely many points in space-time where non-trivial half-harmonic maps "bubble off " in the sense of Theorem 1.1.ii).
ii) As t → ∞ suitably, u(t) smoothly converges to a half-harmonic limit map away from at most finitely many concentration points where non-constant halfharmonic maps "bubble off " as in Theorem 1.1.iii).
Note that uniqueness is only asserted within the class of partially regular weak solutions with non-increasing energy, as in the case of the harmonic map heat flow.It would be interesting to find out if the latter condition suffices, as in the work of Freire [18], [19], and, conversely, to explore the possibility of "backward bubbling" in (1.3), as in the examples of Topping [45] for the latter flow.1.4.Key features of the proof and related flow equations.In our approach, in a similar vain as Lenzmann-Schikorra [27], we uncover and exploit surprising regularity properties of the normal component dπ ⊥ N (u)∂ r u for the harmonic extension of u, likely related to the fractional commutator estimates for the normal projection in the work of Da Lio-Rivière [14] or the regularity estimates of Da Lio-Pigati [13], Mazowiecka-Schikorra [28], and others.
The use of the Dirichlet-to-Neumann map for the harmonic extension u : B → Ê n of u instead of the half-Laplacian, and the simple identity (3.2) as well as equation (3.5) allow to perform the analysis using only local, classically defined operators, avoiding fractional calculus almost entirely.Note that equation (1.3) is similar to the equation governing the (scalar) evolution problem for conformal metrics e 2u g Ê 2 of prescribed geodesic boundary cur- vature and vanishing Gauss curvature on the unit disc B, studied for instance by Brendle [2] or Gehrig [20].In contrast to the latter flows, due to the presence of the projection operator mapping u r to its tangent component, the flow (1.3) at first sight appears to be degenerate.However, surprisingly, within our framework we are able to obtain similar smoothing properties as in the case of the harmonic map heat flow of surfaces.
A different heat flow associated with half-harmonic maps, using the half-heat operator (∂ t − ∆) 1/2 instead of (1.1), was suggested by Hyder et al. [22], and they obtained global existence of partially regular, but possibly non-unique weak solutions for their flow, with a possibly large singular set of measure zero.
1.5.Applications to Plateau problem.In the case when N is a smoothly embedded, oriented closed curve Γ ⊂ Ê n the half-harmonic heat flow (1.3) may furnish an alternative gradient flow for the Plateau problem of minimal surfaces of the type of the disc, which has a long and famous tradition in geometric analysis.
Having been posed by Plateau in the 1890's, Plateau's problem was finally solved independently by Douglas [16] and Radó [33] in 1930/31.In order to analyse the set of all minimal surfaces solving the Plateau problem, including saddle points of the Dirichlet integral, thereby building on Douglas' ideas, in 1939 Morse-Tompkins [31] proposed a critical point theory for Plateau's problem in the sense of Morse [30], attempting to characterize non-minimizing solutions as "homotopy-critical" points of Dirichlet's integral.However, in the 1980's Tromba [47], [46] pointed out that it was not even clear that all smooth, non-degenerate minimal surfaces would be "homotopy-critical" in the sense of Morse-Tompkins [31].To overcome this problem, Tromba developed a version of degree theory that could be applied in this case and which yielded at least a proof of the "last" Morse inequality, which is an identity for the total degree.
In 1982, finally, this author [41] recast the Plateau problem as a variational problem on a closed convex set and he was able to develop a version of the Palais-Smale type critical point theory for the problem within this frame-work, which allowed him to obtain all Morse inequalities in a rigorous fashion; see the monograph [44] and the paper by Imbusch-Struwe [23] for further details.In the papers [43] by this author and [25] by Jost-Struwe the approach was extended to the case of multiple boundaries and/or higher genus.
A key element of critical point theory for a variational problem is the construction of a pseudo-gradient flow for the problem at hand.In [41] this was achieved in an ad-hoc way.However, starting with the work of Eells-Sampson [17] on the harmonic map heat flow, it is now an established approach in geometric analysis to study the (negative) (L 2 -)gradient flow related to a variational problem, similar to the standard heat equation.For Plateau's problem, such a flow was obtained by Chang-Liu [6] within the frame-work laid out by Struwe [41] in the form of a parabolic variational inequality, for which Chang-Liu obtained a solution of class H 2 by means of a time-discrete minimization scheme.Rupflin [35], Rupflin-Schrecker [36] studied the analogous parabolic variational inequality in the case of an annulus, which again had previously been studied by this author [43] by means of an ad-hoc pseudo-gradient flow.
In view of the much better regularity properties of the flow equation (1.3) it would be tempting to regard this as the correct definition of the canonical gradient flow for the Plateau problem, but an important issue still needs to be addressed.1.6.Monotonicity.Recall that in the classical Plateau problem u(t) is required to induce a (weakly) monotone parametrization of Γ for each t > 0.Even though it may seem likely that -at least for curves Γ on the boundary of a convex body in Ê 3 -this Plateau boundary condition will be preserved along the flow (1.3) whenever it is satisfied initially, at this moment even for a strictly convex planar curve Γ ⊂ Ê 2 it is not clear whether this actually happens.However, the results that we obtain also seem to be of interest if we drop the Plateau condition.In particular, our results motivate the study of smooth minimal surfaces with continous trace covering only a part of the given boundary curve Γ; dropping the monotonicity condition also brings the parametric approach to the Plateau problem closer to the approach via geometric measure theory or level sets.1.7.Plateau flow.It should be straightforward to extend our results to the case when the disc B is replaced by a surface Σ of higher genus with boundary ∂Σ ∼ = S 1 , if for given initial data u 0 ∈ H 1/2 (S 1 ; N ) we consider a family u = u(t) in H 1/2 (S 1 ; N ) solving the equation (1.3), that is, (1.1), where for each time we harmonically extend u(t) to Σ and denote as ∂ ν u the outward normal derivative of u along ∂Σ, as was proposed and analysed by Da Lio-Pigati [13] in the time-independent case.Similarly, one might study the flow (1.3) on a domain Σ with multiple boundaries.Of course, in order for the flow to converge to a minimal surface in the case of higher genus or higher connectivity it will be necessary to couple the flow (1.3) with a corresponding evolution equation for the conformal structure on Σ, as in the work of Rupflin-Topping [37] on minimal immersions.Note that on a general domain Σ the flow equations (1.1) and (1.3) no longer agree.In order to clearly distinguish the flow equation (1.3) from the equation (1.1) defining the half-harmonic map heat flow, we therefore propose to say that (1.3) defines the "Plateau flow".1.8.Outline.After a brief discussion of energy estimates in Section 2, in Section 3 we present the analytic core of the argument for higher regularity in Section 4 and for the blow-up analysis, later presented in Section 8.These tools are also instrumental in proving uniqueness of partially regular weak solutions in Section 7. The L 2 -bounds for higher and higher derivatives which we establish in Section 4, assuming that energy does not concentrate, may be of particular interest.These bounds either concern estimates for ∇∂ k φ u on B or on ∂B, and we view the latter bounds as stronger by an order of 1/2.These bounds may be used interlaced, as we later do in Section 6, to prove uniform smooth estimates, locally in time, for smooth flows with smooth initial data converging in H 1/2 (N ; S 1 ).Since the latter data are dense in H 1/2 (N ; S 1 ) we thus not only obtain existence of weak solutions for arbitrary data u 0 ∈ H 1/2 (N ; S 1 ) but also can show their smoothness for positive time and hence are able to derive Theorem 1.2 from Theorem 1.1.A peculiar feature is that one set of regularity estimates can only be obtained globally, that is on all of B, whereas the other set of estimates may be localized using cut-off functions.Similar estimates for a regularized version of (1.3) are employed in Section 5 to prove local existence of smooth solutions of (1.3) for smooth data (1.4).Finally, in Section 9 the large-time behavior of smooth solutions to (1.3) is discussed, finishing the proof of Theorem 1.1.
1.9.Notation.The letter C is used throughout to denote a generic constant, possibly depending on the "target" N and the initial energy E(u 0 ).
Finally, we fix a smooth cut-off function ϕ ∈ C ∞ c (B) satisfying 0 ≤ ϕ ≤ 1 with ϕ ≡ 1 on B 1/2 (0), and for any z 0 ∈ B, any 0 < R < 1 we scale Acknowledgement.I thank Amélie Loher and the anonymous referee for careful reading of the manuscript and useful suggestions.

Energy inequality and first consequences
The half-harmonic heat flow may be regarded as the heat flow for the Dirichlet energy in the class H 1/2 (S 1 ; N ).Indeed, let u(t) be a smooth solution of (1.3), (1.4) for 0 < t < T 0 .Then we have the following result.
Lemma 2.1.For any 0 < T < T 0 there holds Proof.Integrating by parts and using (1.3) we compute for any 0 < t < T 0 .The claim follows by integration.
Moreover, there holds a localized version of this energy inequality.
Lemma 2.2.There exists a constant C > 0 such that for any z 0 ∈ B, any 0 < R < 1, any ε > 0, and any 0 < t 0 < t 1 ≤ t 0 + εR < T 0 there holds Proof.Writing ϕ = ϕ z0,R for brevity, integrating by parts, and using Young's inequality, similar to the proof of Lemma 2.1 for any 0 < t < T 0 we have (2.1) then upon integration we find But with u = u(t) also u t = u t (t) is harmonic for each t.Expanding in a Fourier series, we see that the map r → ∂Br (0) with ds denoting the element of length along ∂B r (0), is non-decreasing.Thus for any z 0 ∈ B, any 0 < R < 1, and any t 0 < t < t 1 there holds and we may use Lemma 2.1 to conclude.

A regularity estimate
To illustrate the key ideas that later will allow us to prove higher regularity and analyze blow-up of solutions of (1.3), we first consider smooth solutions u ∈ H 1/2 (S 1 ; N ) of the equation where f ∈ L 2 (S 1 ).We prove the following a-priori estimate, where we use classical estimates similar to Wettstein's [48] Lemma 3.4, which in turn is a fractional version of an earlier result by Rivière [34].Note that with the truncated signed distance function dist N : Ê n → Ê m we have the orthogonal decomposition , where we recall that we use the shorthand notation Proposition 3.1.There exist constants C, δ 0 = δ 0 (N ) > 0 such that for any smooth solution u ∈ H 1/2 (S 1 ; N ) of (3.1) with E(u) ≤ δ 2 < δ 2 0 there holds Proof.Multiplying (3.2) with ∂ r u, we find the Pythagorean identity • ∇u, and there holds the equation where the basic L 2 -theory for the Laplace equation (3.5) yields the bound . Thus from (3.4) and (3.1) we have But Fourier expansion of the harmonic function u gives as well as the bound ) , and from (3.6) we obtain , which for sufficiently small δ > 0 by (3.7) yields the claim.
In particular, from Proposition 3.1 we obtain a positive energy threshold for non-constant solutions of (1.6).
0 , with δ 0 = δ 0 (N ) > 0 given by Proposition 3.1.Combining the ideas in the proof of the previous result with ideas from the classical proof of the Courant-Lebesgue lemma in minimal surface theory, we can obtain the following local version of Proposition 3.1.

Proposition 3.3.
There exists a constant δ > 0 with the following property.Given any smooth solution u ∈ H 1/2 (S 1 ; N ) of (3.1) with harmonic extension u ∈ H 1 (B), any z 0 ∈ ∂B, and any 0 < R ≤ 1/2 such that Proof.Fix any z 0 ∈ ∂B and 0 < R ≤ 1/2 such that (3.8) holds.For suitable We can bound the latter infimum by the average over ρ ∈ [R 2 , R] with respect to the measure with density 1/ρ to obtain the bound (3.9) Let Φ 0 : B → B be the conformal map fixing the circular arc C ρ and mapping the point z 0 to the point −z 0 , obtained as composition Φ . Note that by the variational characterization of harmonic functions and conformal invariance of the Dirichlet integral we have Moreover, for any smooth ϕ ∈ H 1 0 (B) by (3.9) we can estimate where [∂ ν v 1 ] denotes the difference of the outer and inner normal derivatives of v 1 along C ρ .Thus we have ∆w ∈ H −1/2 (B), and the basic L 2 -theory for the Laplace equation gives w ∈ H 3/2 ∩ H 1 0 (B) with and then also In view of (3.10), for sufficiently small δ > 0 from Proposition 3.1 we obtain the estimate (3.12) ) .Observe that since v 0 = v 1 on ∂B = S 1 and since we also have There exists a constant δ > 0 with the following property.Given any smooth solution u ∈ H 1/2 (S 1 ; N ) of (3.1), any 0 < R ≤ 1/2 with there holds 3 we obtain the claim.

Higher regularity
Again let u(t) be a smooth solution of the half-harmonic heat flow (1.3) for 0 < t < T 0 with smooth initial data (1.4).We show that as long as the flow does not concentrate energy in the sense of Theorem 1.1.ii)the solution remains smooth and can be a-priori bounded in any H k -norm in terms of the data.

H 2 -bound.
In a first step we show an L 2 -bound in space-time for the second derivatives of our solution to the flow (1.3).Recall that by harmonicity, writing u = u(t), ∂ φ u = u φ , and so on, for any 0 < t < T 0 we have (3.7), that is, as Fourier expansion shows, with similar identities for partial derivatives of u of higher order.Indeed, writing we see that also ∂ j φ u and then also ∇ k−j ∂ j φ u is harmonic for any j ≤ k in AE 0 , where ∇u = (u x , u y ) in Euclidean coordinates z = x + iy.Thus by induction we obtain (4.2) for any k ∈ AE.Similarly, for any 1/4 < r < 1 with uniform constants C > 0 we have Integrating, and using the mean value property of harmonic functions together with (4.2) to bound sup in particular, for any k ∈ AE we have the bound with an absolute constant C > 0.
The following lemma is strongly reminiscent of analogous results for the harmonic map heat flow in two space dimensions.Lemma 4.1.With a constant C > 0 depending only on N there holds where we use orthogonality On the other hand, by Young's inequality we can estimate and our claim follows.
Combining the previous result with a quantitative bound for the concentration of energy, we obtain a space-time bound for the second derivatives of u.Note that since u is smooth by assumption, for any δ > 0, any T < T 0 there exists a number Proposition 4.2.There exist constants δ = δ(N ) > 0 and C > 0 such that for any T < T 0 with R > 0 as in (4.4) there holds Proof.For given T < T 0 and δ > 0 to be determined we fix R > 0 such that (4.4) holds.Let B R/2 (z i ), 1 ≤ i ≤ i 0 , be a cover of B such that any point z 0 ∈ B belongs to at most L of the balls B R (z i ), where L ∈ AE is independent of R > 0. We then split Using the multiplicative inequality (10.2) in the Appendix for each i we can bound Summing over 1 ≤ i ≤ i 0 , we thus obtain the bound and for sufficiently small δ > 0 from Lemma 4.1 we obtain the claim.
With the help of Proposition 4.2 we can now bound u in H 2 (B) also uniformly in time.For this, we first note the following estimate, which also will be useful later for bounding higher order derivatives.
Proof.For any k ∈ AE we use harmonicity of where we split I = k j=0 k j I j with Hence for any 1 ≤ j ≤ k we can bound as claimed.It remains to bound the term . With the signed distance function we can express + II, where all terms in II can be dealt with as in the case 1 ≤ j ≤ k.Finally, we have . But by the chain rule we can bound Moreover, by (3.5) and elliptic regularity theory, there holds which gives the claim.
For k = 1, from Proposition 4.2 we now easily derive a uniform L 2 -bound for the second derivatives of the flow.Proposition 4.4.For any smooth u 0 ∈ H 1/2 (S 1 ; N ) and any T < T 0 with R > 0 as in Proposition 4.2 with a constant C 1 = C 1 (T, R, u 0 ) > 0 depending on the right hand side of (4.5) there holds Proof.For k = 1 by Lemma 4.3 we need to bound the term where J 1 contains all terms of lower order.By the maximum principle and Sobolev's embedding H 1 (∂B) ֒→ L ∞ (∂B) we can estimate where we have also used (3.7) and Proposition 4.2.Also bounding via (10.2), and again using (3.7) (and with similar, but simpler bounds for J 1 ), we arrive at the estimate With Lemma 4.3 and Young's inequality we then have Absorbing the first term on the right on the left hand side of this inequality and dividing by 1 + ∇u φ 2 L 2 (B) we obtain and from Proposition 4.2 we obtain the bound The claim then follows from (4.7).
4.2.H 3 -bounds.The derivation of a-priori L 2 -bounds for third derivatives of the solution u to the flow (1.3), (1.4) requires special care, which is why we highlight this case.
Proposition 4.5.For any smooth u 0 ∈ H 1/2 (S 1 ; N ) and any T < T 0 there holds where we denote as Proof.For k = 2 by Lemma 4.3 we need to bound the term and corresponding terms involving at most 3 derivatives in total, which we will omit.
In dealing with the first term, by the multiplicative inequality (10.2) and Sobolev's embedding H 2 (B) ֒→ L ∞ (B) we can estimate with a constant C 2 = C 2 (T, R, u 0 ) > 0 as in the statement of the proposition.
Similarly there holds Hence we can also bound Finally, we estimate to obtain But with the inequality for f ∈ H 2 (B) due to Brezis-Gallouet [1] (see also Brezis-Wainger [3] for a more general version) we have and Lemma 4.3 yields the differential inequality Simplifying, and recalling that 3), we then find that is, we have Arguing as in the proof of Proposition 4.4 we then obtain the claim.Proposition 4.6.For any k ≥ 3, any smooth u 0 ∈ H 1/2 (S 1 ; N ), and any T < T 0 there holds where we denote as 3) for 0 ≤ t < T we then have the uniform bounds with a constant of the type C k , as defined above.By Lemma 4.3 again we only need to bound the term Clearly we have We now distinguish the following cases: If k − 1 = k 0 ≥ 3 by (4.8) we can bound by our induction hypothesis (4.8) we have and we find In any case, inequality (4.3) and Lemma 4.3 now may be invoked to obtain + C k , and our claim follows.

4.4.
Local H 2 -bounds.The bounds established so far all require the initial data to be sufficiently smooth for the estimate at hand and do not yet allow to show smoothing of the flow.For the latter purpose we next prove a second set of "intermediate" estimates that in combination with the first set of estimates later will allow boot-strapping.Moreover, in contrast to the estimates established so far, the following estimates may be localized.This will be important for showing regularity of the flow at blow-up times away from concentration points of the energy on ∂B.
For the localized estimates, fix a point z 0 ∈ ∂B and some radius 0 Proof.Similar to the proof of Lemma 4.1, we compute 1 2 by Young's inequality.Finally, we can bound and our claim follows.
Likewise, as a substitute for the global non-concentration condition (4.4) we now suppose that z 0 ∈ ∂B is not a concentration point in the sense that for suitably chosen δ > 0 to be determined in the sequel and some 0 < R 0 < 1/4 as above there holds (4.10) sup We then obtain the following localized version of Proposition 4.2.
Proposition 4.8.There exist constants δ > 0 and C > 0 independent of R 0 > 0 such that whenever (4.10) holds then for any T ≤ T 0 we have Proof.With the help of the inequality (10.1) in the Appendix we can bound Thus, for sufficiently small δ > 0 our claim follows from Lemma 4.7.
The next lemma again prepares for a proposition that later will allow us to obtain higher derivative bounds by induction.Note the differences to Lemma 4.3.Lemma 4.9.For any k ≥ 2, with a constant C > 0 depending only on k and N , for the solution u = u(t) to (1.3), (1.4) for any 0 < t < T 0 there holds where we split For 1 ≤ j ≤ k we bound By the chain rule then for 1 ≤ j ≤ k we have By Cauchy-Schwarz and Young's inequality then we can bound as claimed.Finally, with as in the proof of Lemma 4.3, for j = 0 we can write where the terms in II and III involve products of at least two derivatives of orders between 1 and k of u.Thus we have + II 0 with a term II 0 that can be dealt with in the same way as the terms I j , 1 ≤ j ≤ k.
Using the divergence theorem and integrating by parts we can write the leading term as to see that this term may be bounded But by elliptic regularity we again have where from (3.5) we can bound the first term on the right Moreover, using that dist N (u)) = 0 on ∂B, with the help of Poincaré's inequality we find the bound The remaining terms for 1 ≤ j ≤ k can be estimated via the chain rule.Thus, finally, we obtain the bound By Cauchy-Schwarz and Young's inequality thus we can bound and together with our above estimate for the terms I j , j ≥ 1, our claim follows.
Proposition 4.10.There exists a constant δ > 0 independent of R 0 > 0 such that whenever (4.10) holds then for any T ≤ T 0 with a constant C 2 = C 2 (T, R, u 0 ) > 0 bounded by the terms on the right hand side in the statement of Proposition 4.8 there holds the estimate Proof.For k = 2 with the help of Young's inequality we can bound and Observing that ϕ 1 = 1 on the support of ϕ 2 , by (10.2) for the first term in J 1 we have Moreover, arguing as in (10.1) for the function |∇u| 6 ϕ 2 2 in place of |v| 4 ϕ 2 , we can bound where by Hölder's inequality we have so that with Young's inequality we obtain With Young's inequality for suitable ε > 0, and using (4.9), we then can bound where we also have estimated Similarly, with (10.2) we have Thus, from Lemma 4.7 we obtain From (4.11) we then obtain the differential inequality that is, and the right hand side is integrable in time by Proposition 4.8.The claim follows.
We continue by induction.
Proposition 4.11.There exists a constant δ > 0 independent of R 0 > 0 with the following property.Whenever (4.10) holds, then for any k ≥ 3, any smooth u 0 ∈ H 1/2 (S 1 ; N ), and any T < T 0 , there holds where we denote as C k = C k (T, R, u 0 ) > 0 a constant bounded by the terms on the right hand side in the statement of the proposition for k − 1.
Proof.By Proposition 4.10 the claimed result holds true for k = 2. Suppose the claim holds true for some k 0 ≥ 2 and let k = k 0 + 1.Note that by elliptic regularity, as in the proof of Proposition 4.10 we can bound By induction hypothesis and Sobolev's embedding H 2 (B) ֒→ W 1,4 ∩ C 0 ( B) for 0 ≤ t < T we then have the uniform bounds and it follows that . and set Suppose k 0 = 2. Recalling that ϕ k = ϕ k ϕ k0 , we can bound the listed terms respectively.Here we also have used (10.1),(10.2) to bound Similarly, we can bound the remaining terms and the terms in J 2 to obtain where the right hand side is integrable in time by Proposition 4.10.The claim for k = 3 thus follows.
For k ≥ 4 the analysis is similar (but simpler) and may be left to the reader.

Local existence
In order to show local existence we approximate the flow equation (1.3) by the equation (5.1) where ε > 0 and where we smoothly extend the nearest-neighbor projection π N , originally defined only in the ρ-neighborhood N ρ of N , to the whole ambient Ê n .
Our aim then is to show that for given smooth initial data u 0 the evolution problem (5.1), (1.4) admits a smooth solution u ε which remains uniformly smoothly bounded on a uniform time interval as ε ↓ 0. Fixing some 0 < ε < 1/2, we show existence for the problem (5.1) with data (1.4) by means of a fixed-point argument.
To set up the argument, fix smooth initial data u 0 : S 1 → N with harmonic extension u 0 ∈ C ∞ ( B; Ê n ) and some k ≥ 2. For suitable T > 0 to be determined let where R 0 = u 0 H k+1 (B) .We endow the space V with the metric derived from the semi-norm Note that this metric is positive definite on V in view of the initial condition that we impose.
Lemma 5.1.V is a complete metric space. Proof.
), and by weak lower semi-continuity of the norm there holds Moreover, we have ∆v(t) = 0 for all 0 ≤ t ≤ T and v(0) = u 0 by compactness of the trace operator Moreover, we have Lemma 5.2.There is T 2 > 0 such that for any T ≤ T 2 , any v ∈ V there is a solution u = Φ(v) ∈ V of the equation Proof.For v ∈ V we construct a solution u = Φ(v) ∈ X of (5.2) via Galerkin approximation.For this let (ϕ l ) l∈AE0 be Steklov eigenfunctions of the Laplacian, satisfying ∆ϕ l = 0 in B with boundary condition ∂ r ϕ l = λ l ϕ l on ∂B, l ∈ AE 0 .Note that the Steklov eigenvalues are given by λ 0 = 0 and λ 2l−1 = λ 2l = l, l ∈ AE.
In fact, we may choose ϕ 0 ≡ 1/ √ 2π and (5.3) to obtain an orthonormal basis for L 2 (S 1 ) consisting of these functions.Given Since for any m ∈ AE the coefficients λ j (ϕ l , (ε + dπ N (v))ϕ j L 2 (S 1 ) of this system are uniformly bounded for any v ∈ V , for any m ∈ AE there exists a unique global solution a (m) = (a Note that for any m ∈ AE and any j ∈ AE 0 the function and ∂ 2j φ u (m) is harmonic.In particular, for j = 0 we obtain 1 2 and we find the uniform (5.6) Moreover, for j = k ∈ AE as in the definition of X upon integrating by parts we find 1 2 where I = k j=1 k j I j with ) similar to the proof of Lemma 4.3.However, now we simply bound Note that by compactness of Sobolev's embedding H 1 (S 1 ) ֒→ L ∞ (S 1 ) and Ehrlich's lemma for any number 1 ≤ j ≤ k, any δ > 0 we can bound On the other hand, for any v ∈ V by the trace theorem we have ≤ CR 0 and we therefore also can bound Thus, for sufficiently small δ > 0 with a constant C > 0 depending on ε > 0 and R 0 there holds and from (5.7) with the help of (3.7) we obtain the inequality ), where we recall (5.6) for the last conclusion.
It follows that for suitably small T > 0 there holds u (m) 2 X ≤ 4R 2 0 for all m ∈ AE.Thus, there is a sequence m → ∞ such that u (m) ⇁ u weakly- * in Proof.Let T 2 > 0 be as determined in Lemma 5.2 and fix some 0 Multiplying with w r and integrating we obtain , where with u 2,r L ∞ (S 1 ) ≤ C u 2 H 3 (B) ≤ CR 0 we can bound Thus, with a constant C = C(ε) > 0 we find (5.9) Similarly, from (5.8) we can bound (5.10) . Integrating over 0 ≤ t ≤ T and observing that we have which we may use together with (5.10) to bound For sufficiently small T > 0 then our claim follows.
Thus, by Banach's fixed point theorem, for any ε > 0, any smooth u 0 ∈ H 1/2 (S 1 ; N ) there exists T > 0 and a solution u = u(t) ∈ V of the initial value problem (5.1), (1.4).We now show that the number T > 0 may be chosen uniformly as ε ↓ 0. Indeed, we have the following result.
Lemma 5.4.There exists a constant C > 0 such that for any k ≥ 2, any smooth u 0 ∈ H 1/2 (S 1 ; N ), and any 0 < ε ≤ 1/2 for the solution u to (5.1) with u(0) = u 0 there holds Proof.Similar to the proof of Lemma 5.2, for given 2 ≤ k ∈ AE we compute 1 2 where we now drop the term ε ∂ k φ u r 2 L 2 (S 1 ) from (5.7).Again we split but now we bound these terms as in the proof of Lemma 4.3 via Using that for any k ≥ 2 by Sobolev's embedding H 2 (B) ֒→ W 1,4 ∩ C 0 ( B) be can bound and also using (4.3), we obtain the claim.
We now are able to conclude.
Proof.In view of Lemma 5.4, there exists a uniform number T > 0 such that, with V as defined above, for any 0 < ε ≤ 1/2 for there exists a solution u ε ∈ V to (5.1).By definition of V , as ε ↓ 0 suitably, we have But this suffices to pass to the limit ε ↓ 0 in (5.1), and u ∈ V solves (1.3) with u(0) = u 0 .
Proof of Theorem 1.1.i).By Proposition 5.5 for any smooth u 0 ∈ H 1/2 (S 1 ; N ) and any k ≥ 2 there exists T > 0 and a solution u ∈ V of (1.3), (1.4) for 0 < t < T .Alternatingly employing Propositions 4.11 and 4.6, we then obtain smoothness of u for 0 < t ≤ T , including the final time T .(This argument later appears in more detail in Section 6 after Lemma 6.2.) Iterating, the solution u may be extended smoothly until some maximal time T 0 where condition (4.4) ceases to hold.
Uniqueness (even within a much larger class of competing functions) is established in Section 7.

Weak solutions
Given u 0 ∈ H 1/2 (S 1 ; N ), there are smooth functions u 0k ∈ H 1/2 (S 1 ; N ) with u 0k → u 0 in H 1 (B) as k → ∞.Indeed, similar to an argument of Schoen-Uhlenbeck [39], Theorem 3.1, with a standard mollifying sequence (ρ k ) k∈AE for the mollified functions v 0k := u 0 * ρ k we have dist N (v 0k ) → 0 uniformly, and Let u k be the corresponding solutions of (1.4) with initial data u k (0) = u 0k , defined on a maximal time interval [0, T k [, k ∈ AE.We claim that each function u k can be smoothly extended to a uniform time interval [0, T [ for some T > 0. To see this, we first establish the following non-concentration result.
Lemma 6.1.For any δ > 0 there exists a number R > 0 and a time T 0 > 0 such that Proof.Given δ > 0, by absolute continuity of the Lebesgue integral and In view of Proposition 3.3, from Lemma 6.1 and Lemma 2.1 we obtain the following bound for u k in H 1 (S 1 ).Lemma 6.2.There exist a time T 0 > 0 and constants C > 0, From Lemma 6.2 we obtain locally in time uniform smooth bounds for (u k ) for t > 0 by iteratively applying our previous regularity results.More precisely, Fatou's lemma and Lemma 6.2 first yield the bound Thus for almost every 0 < t 0 < T 0 there holds For any such 0 < t 0 < T 0 , if δ > 0 is sufficiently small, from Proposition 4.2 with another appeal to Fatou's lemma we may conclude for some C 1 > 0, so that now we even have lim inf for almost every t 0 < t satisfying uniform smooth bounds on ]t 0 , T 0 ] for any t 0 > 0. Passing to the limit k → ∞ for this subsequence we obtain a weak solution to (1.3), (1.4) of energy-class in the following sense.
3) is satisfied in the weak sense, that is, if there holds for all ϕ ∈ C ∞ c (S 1 ×]0, T 0 [), and if there holds the energy inequality (6.2) for any 0 < T < T 0 , with the initial data u 0 ∈ H 1/2 (S 1 ; N ) being attained in the sense of traces.
We then may summarize our results, as follows.
Proof.For any open U ⊂ S 1 ×]0, T 0 [ we have uniform smooth bounds for u k on U ; thus a suitable sub-sequence u k → u smoothly locally as k → ∞.The equation (6.1) follows from the corresponding identites for u k .Moreover, (6.2) follows from the energy identity, Lemma 2.1, for u k in view of H 1 -convergence u 0k → u 0 as well as weak lower semi-continuity of the energy and of the L 2 -norm.

Uniqueness
With the help of the tools developed in Section 3 we can show uniqueness of partially regular weak energy-class solutions as in Proposition 6.4.
Proof.Using the identity (3.2) for u and v, respectively, for the function w = u − v for almost every 0 < t < T 0 we have on ∂B = S 1 .From equation (3.5), moreover, we obtain , integrating by parts, and using Young's inequality, for any ε > 0 we obtain ). (7.3) On the other hand, for any 0 < t 0 < T ≤ T 0 , multiplying the equation (7.1) with w and integrating by parts on S 1 × [t 0 , T ], upon letting t 0 ↓ 0 we find We first estimate the term we can bound for each t.Also using (3.5), we can moreover estimate for almost every 0 < t < T to obtain . Similarly, we estimate the term Noting that with (7.3) we can bound and that with (7.2) we have ) we find the estimate ) for almost every 0 < t < T .But Sobolev's embedding H 1/2 (B) ֒→ L 4 (B) and Fourier expansion give the bound and similar bounds for ∇u as well as ∇v.Moreover, since by the energy inequality (6.2) we have u(t), v(t) → u 0 strongly in H 1 (B) as t ↓ 0, there exist a radius 0 < R ≤ 1/2 and a time 0 < T < T 0 such that condition (3.13) in Proposition 3.3 holds true on [0, T ] for both u and v, allowing to bound with the help of (3.7), and similarly for |∇v|.Choosing ε = 1/4, for sufficiently small 0 < T < T 0 by absolute continuity of the integral we thus can estimate and it follows that w = 0, as claimed.
Proof of Theorem 1.2.Existence for short time and uniqueness of a partially regular weak solution to (1.3), (1.4) for given data u 0 ∈ H 1/2 (S 1 ; N ) follow from Proposition 6.4 and Theorem 7.1, respectively.Since by Proposition 6.4 our weak solution is smooth for t > 0, the remaining assertions follow from Theorem 1.1.Note that at any blow-up time T i−1 , i ≥ 1, of the flow as in Theorem 1.1.ii)there exists a unique weak limit u i = lim t↑Ti−1 u(t) ∈ H 1/2 (S 1 ; N ), and we may uniquely continue the flow using Proposition 6.4.

Blow-up
Preparing for the proof of part ii) of Theorem 1.1 suppose now that for the solution constructed in part i) of that theorem there holds T 0 < ∞.Then, as we shall see in more detail below, by the results in Section 4 condition (4.4) must be violated for T = T 0 and there exist δ > 0 and points z k ∈ B as well as radii r k ↓ 0 as k → ∞ such that for suitable t k ↑ T 0 there holds We may later choose a smaller constant δ > 0, if necessary.Moreover, for later use from now on we consider local concentrations in the sense that for some z 0 ∈ B and some fixed radius r 0 > 0 for a sequence of points z k ∈ B with z k → z 0 and radii r k ↓ 0 for suitable t k ↑ T 0 as k → ∞ there holds Passing to a sub-sequence we may assume that the domains Ω k exhaust a limit domain Ω ∞ ⊂ Ê 2 , which either is the whole space Ê 2 or a half-space H.
By the energy inequality Lemma 2.1 for t ∈ I k there holds (8.2) and for any t 0 < 0 and sufficiently large k ∈ AE we have as k → ∞, where ds is the element of length and where ν k is the outward unit normal along ∂Ω k .Expressing the harmonic functions ∂ t u k (t) in Fourier series for each t < 0, it then also follows that ∂ t u k → 0 locally in L 2 on Ω ∞ ×] − ∞, 0[.Finally, again using the fact that u k (t) for each t is harmonic, by the maximum principle we have the uniform bound |u k | ≤ sup p∈Γ |p| as well as uniform smooth bounds locally away from the boundary of Ω ∞ .Hence we may assume that as k → ∞ we have u k → u ∞ weakly locally in H 1 on Ω ∞ ×] − ∞, 0[, where u ∞ (z, t) = u ∞ (z) is independent of time, harmonic, and bounded.Moreover, we have smooth convergence away from ∂Ω ∞ .Thus, if we assume that Ω ∞ = Ê 2 by (8.1) it follows that But any function v : Ê 2 → Ê which is bounded and harmonic must be constant, which rules out this possibility.Hence Ω ∞ can only be a half-space.
After a suitable rotation of the domain B and shift of coordinates in Ê 2 ∼ = we may then assume that z k = (0, −y k ) with 1 − y k ≤ M r k for some M ∈ AE and that Ω ∞ = {(x, y); y > y 0 } for some y 0 .Finally, replacing r k > 0 with (M + 1)r k and z k with z k = (0, −1), if necessary, we may assume that Ω k ⊂ Ê 2 + = {(x, y); y > 0} is the ball of radius 1/r k around the point (0, 1/r k ) with 0 ∈ ∂Ω k , while from (8.1) with a uniform number L ∈ AE we have and by (8.4) with a uniform number L 1 ∈ AE there holds where B + r (z) = B r (z) ∩ Ê 2 + for any r > 0 and any z = (x, y) ∈ Ê 2 .Moreover, from (8.3) for any t 0 < 0 and any R > 0 for the integral over ] − R, R [×{0} ⊂ ∂Ê 2 + we obtain For a suitable sequence of times t 0 < s k < 0, we then also have locally weak Thus, for sufficiently small δ > 0 by Proposition 3.3, applied to the functions + is a suitable conformal map, we also have uniform local L 2 -bounds for ∂ x w k on ∂Ê 2 + , and we may assume that w k → w ∞ locally uniformly and weakly locally in H 1 on ∂Ê 2 + as k → ∞.Since w k is harmonic, we then also have locally strong To see that w ∞ is non-constant, let ϕ k = ϕ z0,4r k , k ∈ AE.Integrating the identity (2.1) from the proof of Lemma 2.2 in time, with error o(1) → 0 and suitable numbers With the help of (2.2) and (8.3) for suitable ε k ↓ 0 we can bound Since for any choice t 0 < s k < 0 we also can estimate Finally, in view of locally uniform convergence w k → w ∞ and weak local L 2convergence of the traces ∇w k → ∇w ∞ on ∂Ê 2 + , we may pass to the limit k → ∞ in (8.8) to conclude that (8.11) dπ N (w ∞ )∂ y w ∞ = 0 on ∂Ê 2 + .Since w ∞ is harmonic, the Hopf differential With stereographic projection Φ : B → Ê 2 + from a point z 0 ∈ ∂B define the map ū = w ∞ •Φ ∈ H 1/2 (S 1 ; N ).By conformal invariance, ū again is harmonic with finite Dirichlet integral and satisfies (1.6) on ∂B \ {z 0 }; since the point {z 0 } has vanishing H 1 -capacity, ū then is stationary in the sense of [21].Moreover, ū is conformal.For such mappings, smooth regularity on B was shown by Grüter-Hildebrandt-Nitsche [21]; thus condition (1.6) holds everywhere on ∂B in the pointwise sense, and ū parametrizes a minimal surface of finite area supported by N which meets N orthogonally along its boundary.
Proof of Theorem 1.1.ii).For given smooth data u 0 ∈ H 1/2 (S 1 ; N ) let u be the unique solution to (1.3), (1.4) guaranteed by part i) of the theorem, and suppose that the maximal time of existence T 0 < ∞.Then condition (4.4) must fail as t ↑ T 0 ; else from Propositions 4.11 and 4.6 we obtain smooth bounds for u(t) as t ↑ T 0 and there exists a smooth trace u 1 = lim t↑T0 u(t).But by the first part of the theorem there is a smooth solution to the initial value problem for (1.3) with initial data u 1 at time T 0 , and this solution extends the original solution u to an interval [0, T 1 [ for some T 1 > T 0 , contradicting maximality of T 0 .
Let z (i) ∈ B, 1 ≤ i ≤ i 0 , such that for some number δ > 0 and suitable t By the argument following (8.9)thus for a suitable sequence of radii 0 < r (0) k → 0 such that r k )∩B |∇u(t for all T 0 − r (0) k < t < T 0 , uniformly in 1 ≤ i ≤ i 0 .For sufficiently large k ∈ AE such that r k < inf i<j |z (i) − z (j) |/4 it follows that i 0 ≤ E(u 0 )/δ, and we may fix r 0 > 0 and redefine t for the concentration energy quanta, which gives the claimed upper bound for the total number of concentration points.Finally, with the help of Proposition 4.11 we can smoothly extend the solution u to B \ {z (1) , . . ., z (i0) } at time t = T 0 .

Asymptotics
Suppose next that the solution u to (1.3), (1.4) exists for all time 0 < t < ∞.Then u either concentrates for suitable t k ↑ ∞ in the sense that condition (4.4) does not hold true uniformly in time, or u satisfies uniform smooth bounds, as shown in Section 4.
In the latter case, the claim made in Theorem 1.1.iii)easily follows.
Proposition 9.1.Suppose that for any δ > 0 there exists R > 0 such that condition (4.4) holds true for all 0 < t < ∞.Then there exists a smooth solution u ∞ ∈ H 1/2 (S 1 ; N ) of (1.6) such that u(t) → u ∞ smoothly as t → ∞ suitably, and u ∞ parametrizes a minimal surface of finite area supported by N which meets N orthogonally along its boundary.
Proof.For sufficiently small δ > 0, for any j ∈ AE by iterative reference to Propositions 4.2, 4.4 -4.6, and 4.10, 4.11, respectively, as in Section 6 we can find constants C j > 0 such that u(t) H j (B) ≤ C j for all t > 1, Moreover, by the energy inequality Lemma 2.1 for a suitable sequence t k → ∞ there holds u t (t k ) → 0 in L 2 (∂B) as k → ∞.Then for any j ∈ AE a subsequence u(t k ) → u ∞ in H j (B), and a diagonal subsequence converges smoothly, where u ∞ solves (1.6).By the argument after (8.11) in Section 8 then u ∞ is conformal and u ∞ parametrizes a minimal surface with free boundary on N which meets N orthogonally along its boundary.
In the remaining case that for some δ > 0 condition (4.4) fails to hold, there exists a sequence k ↑ ∞ and points z (1) , . . ., z (i0) such that for sequences z as k → ∞, where ν k is the outward unit normal along ∂Ω k .Just as in Section 8 for suitable t 0 < s k < 0 we then obtain local uniform and H 1 -convergence of a subsequence of the conformally rescaled maps w k = u k (s k ) • Φ k ∈ H 1 loc (Ê 2 + ) to a smooth, harmonic and conformal limit w ∞ with finite energy and continuously mapping ∂Ê 2 + to N , inducing a solution ū∞ = w ∞ • Φ ∈ H 1/2 (S 1 ; N ) of (1.6) corresponding to a minimal surface with free boundary on N .This ends the proof of Theorem 1.1.iii)

Lemma 4 . 3 .
For any k ∈ AE, with a constant C > 0 depending only on k and N , for the solution u = u(t) to (1.3), (1.4) for any 0 < t < T 0 there holds bounded by the terms on the right hand side in the statements of Propositions 4.2 and 4.4.

4. 3 .
H m -bounds, m ≥ 4. In view of Proposition 4.5 we can now use induction to prove the following result.
bounded by the terms on the right hand side in the statement of the proposition for k − 1. Proof.By Proposition 4.5 the claimed result holds true for k = 2. Suppose the claim holds true for some k 0 ≥ 2 and let k = k 0 + 1.Note that by Sobolev's embedding H 2 (B) ֒→ W 1,4 ∩ C 0 ( B) and (4.