Regular 1-harmonic flow

We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target manifold has non-positive sectional curvature or in the case that the datum is small, solutions are shown to exist globally and to become constant in finite time. We also consider the case where the domain is a compact Riemannian manifold without boundary, solving the homotopy problem for 1-harmonic maps under some assumptions.


Introduction
Let (N , g) be a complete, connected smooth n-dimensional Riemannian manifold (without boundary). Throughout the paper, without loss of generality [32,20], we will treat it as an isometrically embedded submanifold in the Euclidean space R N . Given an open, bounded Lipschitz domain Ω ⊂ R m we consider the formal steepest descent flow with respect to the L 2 distance of the functional TV N Ω : the total variation functional constrained to functions taking values in N , given for smooth u by Following the L 2 -steepest descent flow is one way of controlled decreasing TV N Ω , which is a problem appearing in image processing. Besides the case N ⊆ S N −1 , which appears in denoising of optical flows [36] or color images [37], other examples of targets appearing in applications include the space of isometries SO(3) × R 3 [26], the cylinder R 2 × S 1 (LCh color space) [41] and the space of positive definite symmetric matrices (diffusion tensors) Sym + (3) [41]. All of these examples are homogeneous spaces, and therefore have natural invariant metrics. Our main goal in this paper is to develop a well-posedness theory for the flow in a generality encompassing these cases. As some of these manifolds are non-compact, we refrain from the unnecessary (although convenient) assumption of compactness of N .
Given a point p ∈ N , we denote by the orthogonal projection onto the tangent space of N at p, T p N . Similarly, π ⊥ p will denote the orthogonal projection of R N onto the normal space T p N ⊥ . The centered dot will denote the Euclidean scalar product on R m or R N , while k stacked dots will denote the induced scalar product on a Cartesian product of any k-tuple of these spaces. Calculating the first variation of (1) at u, one obtains that the flow in a time interval [0, T [ starting with initial datum u 0 is formally given by the system u t = π u div ∇u |∇u| in ]0, T [×Ω, (2) ν Ω · ∇u |∇u| = 0 in ]0, T [×∂Ω, u(0, ·) ≡ u 0 .
The meaning of the expression ∇u |∇u| in (2, 3) deserves a clarification even for smooth u: we understand ∇u |∇u| as a multifunction ∇u |∇u| : (t, x) → and require that (2,3) are satisfied for an appropriate selection. This is formalized in the following definition, which is an adapted version of [3,Definition 2.5]. Here and in the following we will use the notation X(U, N ) = {w ∈ X(U, R N ) : w(y) ∈ N for a. e. y ∈ U }, where U is any domain in R l (or a compact l-dimensional Riemannian manifold), l = 1, 2, . . . and X(U, R N ) is a subspace of L 1 loc (U, R N ).
Definition 1. Let T ∈]0, ∞]. We say that is a (regular) solution to (2) u t = π u (div Z) L 1+m − a. e. in ]0, T [×Ω. We say that a regular solution u to (2) satisfies (homogeneous) Neumann boundary condition (3) if ν Ω · Z = 0 (9) Remark. Due to Morrey embedding theorem, any regular solution to (2) has a representative that is locally Hölder continuous on [0, T [×Ω [21,Theorem 5]. We will identify it with this representative. In particular, the initial condition (4) can be understood pointwise. On the other hand, ν Ω · Z in (9) has to be understood as the normal trace of an L ∞ vector field with integrable divergence, as defined in [38,4].
If conditions in Definition 1 are satisfied, we will often say that the pair (u, Z) is a (regular) solution to (2) and/or (3). We will often use equivalent (see e. g. the proof of Lemma 2) form of (8): where A p denotes the second fundamental form of N at p ∈ N . Here and throughout the paper, we use Einstein's summation convention. The adjective regular in Definition 1 is justified by the following considerations. Firstly, W 1,∞ (Ω) is the highest Sobolev regularity that is preserved by the scalar total variation flow [24,5]. Secondly, such attribute distinguishes the class of solutions in Definition 1 from weak (energy) solutions, whose natural spatial regularity is BV (Ω). However, we note that in the constrained case, even defining a proper concept of solution is non-trivial in the BV setting, the crucial issue being an appropriate identification of the right-hand side of (8) or of (10). In this regard, the only case considered so far is N ⊆ S n , in which (10) drastically simplifies due to the isotropy of the sphere: Suitably defined solutions to (2,3) have been obtained in [14] when the initial datum is contained in an hyper-octant of S n [14]. When n = 1, the assumption on u 0 may be relaxed and uniqueness results are available too [13]. A notion of solution extending the one in [13,14] to (N − 1)-dimensional manifolds with unique geodesics has been proposed in [8]. Existence of solutions for a discretized Dirichlet problem for (2) in the case N = S n , m = 2 has been obtained in [19].
The validity of Definition 1 is supported by the well-posedness results that we obtain. First of all, regular solutions are unique.
Theorem 1. Suppose that u 1 , u 2 are two regular solutions to (2,3) The proof Theorem 1 is different from the proofs of analogous results for p-harmonic flow in [22,11] in that we do not use strict monotonicity of the p-Laplace operator (since it does not hold for p = 1).
Provided that Ω is convex, we are able to construct local-in-time Lipschitz solutions to (2,3). We need the assumption of convexity, as we are forced to use global L p estimates for ∇u. Localization of these estimates is not available due to the strong degeneracy of the 1-Laplace operator div ∇u |∇u| . The assumption of convexity is not very restrictive from the point of view of image processing, as typical domains in applications are rectangles (or boxes of different dimensions).
The existential theory depends on the sectional curvature K N of N or, equivalently, on the Riemannian curvature tensor R N of N . We denote by K N the supremum of sectional curvature over N , i. e.
Recall that K SO(n)×R n is positive (and finite) and K S 1 ×R n , K Sym + (n) are non-positive.
Theorem 2. Suppose that Ω is convex, the embedding of N in R N is closed and K N < ∞. Given u 0 ∈ W 1,∞ (Ω), there exists T = T (N , ∇u 0 L ∞ ) and a regular solution u to (2,3,4) in [0, T [ satisfying the energy inequality ess sup This theorem bears similarity to [16,Theorem 3.4], where Lipschitz local-in-time solutions to (2) are constructed in the case where Ω is a flat torus, i. e. a box with periodic boundary conditions. However, aside from the choice of boundary condition, there are differences between these results -most importantly, in [16], smallness of ∇u 0 in L 1+ε (Ω) is assumed. This is because in [16], global solutions to p-harmonic flows constructed in [12] for small intial data are used as an approximation. In our case a different approximation scheme is proposed. In fact we cannot use the results in [12] as non-trivial boundary conditions are not handled there.
At least in the case of Dirichlet boundary data, regular solutions to (2) can blow up in finite time, as shown by explicit examples in [7,15]. In our case, we prove that solutions exist globally in time, provided that the range of the initial datum is contained in a small enough ball in N . In fact, in this case they become constant in finite time, similarly as for the scalar total variation flow [17]. Note that in the case of inhomogeneous Dirichlet boundary conditions, the evolution of generic initial data under 1-harmonic flow does not stop in finite time [18], in contrast to what is observed in the scalar total variation flow, at least in 1dimensional domains [24]. Let us denote by B g (p, R) the ball centered at p ∈ N of radius R > 0 with respect to the metric induced by g on N .
Theorem 3. Let p 0 ∈ N , u 0 ∈ W 1,∞ (Ω, N ) and u be a regular solution to (2,3,4) In the particular case K N ≤ 0 no blow-up occurs for any Lipschitz datum, and we can obtain a stronger result of global existence.
Theorem 4. Suppose that Ω is convex and K N ≤ 0. Let u 0 ∈ W 1,∞ (Ω, N ). There exists a regular solution u to (2,3,4) in [0, ∞[ satisfying the energy inequality (12). There exists We remark that in the scalar case the preservation of the W 1,∞ bound follows from [6,Corollary 5.6]. However, the methods there are not readily adaptable to vectorial problems.
In the present paper we consider regular 1-harmonic flows which are continuous, and hence capable of generating homotopy. For this reason we find it appropriate to discuss in detail the case where the domain is a compact Riemannian manifold (M, γ). In this setting, the total variation functional takes form where, in local coordinates The expression for div γ acting on a 1-form ϑ on M in coordinates is Observe that (14) is a formal limit as p → 1 + of systems u t = π u (div γ (| du| p−2 du)) in ]0, T [×M (16) corresponding to p-harmonic map flows between Riemannian manifolds. These were first considered in the case p = 2 in connection with the homotopy problem for harmonic maps, i. e. the problem of finding a harmonic map homotopic to a given one. The problem was solved in [10] under the condition that K N ≤ 0 by constructing the harmonic map flow. Analogous result was later obtained in [11] for any p > 1. We note that there are several nonequivalent notions of p-harmonic maps, among them weakly p-harmonic maps, i. e. stationary weak solutions to (16). We introduce the notation Measurable selections of du | du|γ (t, ·) can be seen as L ∞ sections of the bundle T * M × R N over M for a. e. t ∈]0, T [, see [33] for reference. As in [33], we let We are ready to introduce a concept of solution to (14).
is a (regular) solution to (14) The strength of our result in this case depends on the sign of Ricci curvature Ric M of M. We denote Theorem 5. Let (M, γ) be a compact, orientable Riemannian manifold and let (N , g) be a compact submanifold in the Euclidean space R N . Given u 0 ∈ W 1,∞ (M, N ), there exists T ∈]0, ∞] and a unique regular solution to (14,4) As u is continuous and the sequence (u(t k , ·)) converges to u * in C(M, N ), u * and u 0 are homotopic. Thus, we have solved the homotopy problem for (weakly) 1-harmonic maps assuming that M is orientable with Ric M ≥ 0 and K N ≤ 0.
The plan of the paper is the following one: Firstly, in section 2, we prove Theorem 1. In section 3, we obtain well-posedness of an approximating system to (2,3,4) and we obtain some a priori estimates (independent of the parameter of approximation) for their solutions. This permits us to prove Theorem 2, to which section 4 is devoted. The asymptotic behaviour is treated in the next sections: in section 5, we prove Theorem 3 while in section 6, we treat the case of nonnegative curvature; i.e Theorem 4. Section 7 is devoted to the case where the domain is a compact Riemannian manifold, in which we prove Theorem 5. The last part of the paper is an appendix where some technical lemmata are stated and proven.

Uniqueness
In this section, we give the proof of Theorem 1.
Let (u 1 , Z 1 ), (u 2 , Z 2 ) be two regular solutions to (2,3). For i = 1, 2 there holds Here and in the rest of this section, u i,x j and Z i,j denote, respectively, the derivative of u i in direction of x j and the j-th component of Z i , i = 1, 2, j = 1, . . . , m. We calculate In the first term on the r. h. s. of (22) we integrate by parts, yielding which is non-positive as |Z i | ≤ 1, i = 1, 2. Next, we note that for any p 1 ,

Uniform bounds
In this subsection, we prove essential a priori estimates for u ∈ C The basic energy estimate reflects the gradient flow structure behind (23,24). (23,24). Then Proof. The estimate follows from the equality In order to derive further uniform bounds, our main tool is the following version of Bochner's identity (see [27,Chapter 1.] for the case of harmonic maps).
Using this frame, we express where X, Y ∈ T p N and D p N k : which allows us to rewrite (23) as Using (30), we obtain where in the last line we used that A u is orthogonal to u x j ∈ T u N . Next, again using (28), we perform the following calculations: Hence, (31) may be rewritten as Finally, we recall the Gauss-Codazzi equation for any quadruple of vectors X, Y , Z, W ∈ T p N , p ∈ N , which finishes the proof.
We are now ready to derive uniform Lipschitz bounds.
Proof. Given a finite p ≥ 1, using (27) and integrating by parts, we calculate We have N and j, k = 1, . . . , m. Thus, we can rewrite (we use the notation I l for the l-dimensional identity matrix). On the other hand, From (35), (36) and the fact that |Z| ≤ 1|, it is clear that, provided p ≥ 2, the first two terms on the r. h. s. of (34) are non-positive. To treat the remaining boundary term, we extend ν Ω to a normal tubular neighbourhood of ∂Ω in such a way that it is constant in the fibers, and calculate (at points in ∂Ω) where by A ∂Ω we denoted the second fundamental form of hypersurface ∂Ω. As Ω is convex, ν Ω · A ∂Ω is non-negative. This ends the proof of (33) in the case K N ≤ 0. Now, assume that K N ∈]0, ∞[. By virtue of previous calculations and (11), we have d dt Passing to the limit p → ∞ we obtain, at least in a weak sense, which implies (32).

Existence for the approximate system
In order to prove existence of solutions to the approximate system we proceed similarly as in [22,Section 3.]. The assumption that the embedding of N into R N is closed enables us to construct a metric h on R N such that (N , g) is a totally geodesic Riemannian submanifold of (R N , h) (see Lemma A.1 in the appendix), i. e., • the restriction of h to T N coincides with g, that is h p | TpN ×TpN ≡ g p for p ∈ N , • there is a tubular neighborhood T of N in R N such that the involution τ : T → T given by multiplication by −1 in the fibers of T is an isometry.
The gradient flow of the unconstrained functional Ω |∇u| h defined for any regular enough function u : Ω → R N is expressed by the system ν Ω · ∇u i = 0, where i = 1, . . . , N and Γ i jk are the Christoffel symbols of (R N , h). As h restricted to T N coincides with g, the system (37, 38) is identical to (23,38) as long as the range of u is contained in N . In order for C 3+α 2 ,3+α loc (Ω [0,T [ , N ) solutions to the system (37, 38) with initial datum u 0 to exist, the following compatibility conditions on ∂Ω for i = 1, . . . , N need to be satisfied.
Proposition 1. Suppose that K N < ∞ and α ∈]0, 1[. Let u 0 ∈ C 3+α (Ω, N ) satisfy (39,40). Then for any ε > 0 the system (23-25) has a unique solution Note that T † in Proposition 1 does not depend on ε. The expressions on the right hand side of (37) make sense without assuming a priori that the range of u is contained in N . This fact enables us to obtain a local-in-time solution using known results on existence for parabolic systems. For that purpose, the authors in [22] or in [11] combine a general existence result from [28] with sectoriality estimates from [40]. On the other hand, in [30] the author employs estimates from [35] and [31]. However, both [40] and [35] can only be applied to the system with Dirichlet boundary condition, or to the case with no boundary. As we are dealing with homogeneous Neumann boundary condition, we appeal instead to the existence result of Acquistapace and Terreni [1, Theorem 1.1.] for quasilinear systems with general boundary conditions.
To justify its applicability to our problem, let us briefly check the assumptions. In our case, defines a locally uniformly strongly elliptic operator (see e. g. [2]) and therefore satisfies assumption (0.2) from [1]. It is easy to check that (38) satisfies the complementarity condition (0.3) from [1], and that the system satisfies regularity condition (0.4) from [1].

Local existence
In this section we prove Theorem 2.
Step 1. We assume that Ω is smooth and the initial datum u 0 ∈ C 3+α (Ω) satisfies the compatibility conditions (39), (40). We want to pass to the limit ε → 0 + in (23)(24)(25). Owing to Lemmata 1 and 3, we have uniform bounds on u ε t in L 2 (]0, T [×Ω) and on ∇u ε in L ∞ (]0, T [×Ω) for any T < T † . Consequently, we also have uniform bound on u ε in C 1 n+1 (]0, T [×Ω) [21]. All these imply that we can extract a sequence (u k ) = (u ε k ) from (u ε ) such that Due to definition of Z ε , we have Z ε L ∞ ≤ 1, hence for a sequence (Z k ) = (Z ε k ). Furthermore, by virtue of the strong convergence of u k , Next, note that due to the Hölder bound, the family u ε is contained in a compact subset of N . Rewriting (23) as we deduce a uniform bound on div Z ε in L 2 (]0, T [×Ω). By a standard div-curl reasoning, A simple calculation shows that Hence, by lower semicontinuity of | · | with respect to weak convergence, we get Collecting (43, 44, 46, 48) we obtain that ∇u and Z satisfy (7). Boundedness of div Z ε in L 2 (]0, T [×Ω) together with strong convergence of u k is enough to pass to the limit in (23,24), obtaining that ∇u and Z satisfy (8,9).
Step 2. Now, we relax the regularity assumption on the initial datum to u 0 ∈ W 1,∞ (Ω, N ). Take a sequence (u 0,j ) ⊂ C ∞ (Ω, N ) such that u 0,j converges uniformly to u 0 , satisfies compatibility conditions (39,40) and Such a sequence is produced in Lemma A.2. By the previous step, there exists a regular solution (u j , Z j ) to (2, 3) with initial datum u 0,j . Recall that due to the form of estimates in Lemmata 1 and 3 the norms of u j,t in L 2 (]0, T [×Ω, R N ) and of ∇u j in L ∞ (]0, T [×Ω, R m·N ) are controlled by ∇u 0,j L ∞ . By virtue of (49), this control is uniform with respect to j. Hence, we can extract a subsequence converging to a regular solution to (2,3,4) following the same argument as in the previous step, with (u ε , Z ε ) replaced by (u j , Z j ), except that now we have ∇u j . . Z j = |∇u j | instead of (47).
Step 3. Next, we lift the smoothness assumption on the domain. A convex domain Ω can be approximated with respect to Hausdorff distance by smooth convex domains Ω k ⊂ Ω, k = 1, 2, . . .. For a proof of this result using the signed distance function of Ω, see Lemma A.3 in the appendix. The reasoning in the previous paragraph yields a sequence of pairs (u k , Z k ), with k-th one satisfying (7,8,9) in ]0, T [×Ω k with initial datum u 0 | Ω k . The estimates provided by Lemmata 1 and 3 are uniform with respect to k. Hence, we can use them as before together with a diagonal argument to extract subsequences of (u k ), (Z k ) that converge on compact subsets of [0, T [×Ω to a regular solution (u, Z) to (2,4) in ]0, T [×Ω.

Finite extinction time
In order to prove Theorem 3 we will work directly with regular solutions to (2,3,4) in local coordinates p → (p 1 , . . . , p n ) on N , in which (8) is expressed [10] as where Γ i jk are the Christoffel symbols of the chosen coordinate system. For p 0 ∈ N we denote where [K Bg (p 0 ,R) ] + is the supremum of sectional curvature over B g (p 0 , R) (compare with (11)) or +0 if the supremum is negative, ℓ(p 0 ) is the infimum of lengths of maximal closed geodesics in N passing through p 0 , and π is the length of a circle of radius 1 2 . R * (p 0 ) is positive and lower than both the convexity radius and the injectivity radius of N at p 0 [34, section 6.3.2].
First, we prove A unique center of mass exists for any Radon measure on B g (p 0 , R) and we have where we identified elements of T * p c N and T p c N via g [23, Section 1]. For p 0 ∈ N , we denote We are ready to state Let p c (t) be the center of mass of the pushforward measure µ(t) = u(t, ·) # L m on B g (p 0 , R). There exists C = C(Ω, N , p 0 ) such that for t > 0.
Proof of Theorem 3. First of all, by Lemma 4, we obtain the bound u(t, Ω) ⊂ B g (p 0 , R) if u 0 (Ω) ⊂ B g (p 0 , R) for R < R * (p 0 ) and any t ∈ [0, T [. Next, we deduce the estimate on extinction time from (57) by solving the ordinary differential inequality, which yields 6 Non-positive sectional curvature of the target This section is entirely devoted to the proof of Theorem 4. Let T > 0 and suppose that Ω is convex and N is a complete Riemannian manifold with K N ≤ 0. In order to prove Theorem 4 without the assumption that there is a closed embedding of N into R N , we introduce a universal cover γ : N → N of N with a Riemannian manifold ( N , g). As a simply-connected Riemannian manifold of non-positive curvature, N is diffeomorphic to R n via the exponential map (this is the content of Cartan-Hadamard theorem [9]). In other words, there is a global coordinate system on N , p → exp −1 As Ω is topologically trivial, any function u 0 ∈ C(Ω, N ) can be lifted preserving any Sobolev or Hölder regularity to u 0 ∈ C(Ω, N ) such that u 0 = γ • u 0 . Then, assuming that Ω and u 0 are of class C 3+α and u 0 satisfies the compatibility conditions (39,40) for i = 1, . . . , n, we consider the system i = 1, . . . , n. This system satisfies the assumptions of the Aquistapace-Terreni existence theorem (see subsection 3.2), hence unique solution exists for some T * > 0. Vector lengths | u ε t | g and |∇ u ε | g are invariant under local isometries of the target manifold, and any Riemannian manifold is locally isometric to a submanifold in a Euclidean space. Therefore, we can repeat the proofs of Lemmata 1, 2 and 3 performing the computations in a neighbourhood of any point, obtaining bounds on u ε t L 2 (]0,T * [×Ω) and ∇ u ε L ∞ (]0,T * [×Ω) independent on T * . Reasoning as in subsection 3.2, the solution can be prolonged up to the arbitrary given T . Then, taking u ε = γ • u ε , we obtain a solution to (23)(24)(25). Using the uniform bounds, we pass to the limit as in section 4 obtaining a regular solution u to (2)(3)(4) with any u 0 ∈ W 1,∞ (Ω) in any convex Ω.
Finally, we consider any lifting u : Ω → N of u with u t ∈ L 2 (]0, T [×Ω, R N ), ∇ u ∈ L ∞ (]0, T [×Ω, R N ). As R * = +∞ for N , arguments from section 5 imply that u becomes constant in finite time (if we take large enough T ), and consequently the same holds for u = γ • u.

The case where the domain is a Riemannian manifold
Using Hölder inequality, Passing to the limit p → ∞,  Proof of Theorem 5. The proof of uniqueness follows along the lines of the proof of Theorem 1. An important point is that integration by parts is allowed because M is orientable.
The first item in (72) can be rewritten as hence (70) implies that the sequence div γ Z(t k , ·) is uniformly bounded in L 2 (M, R N ). The second item in (72) is equivalent to Hence, there exists Z ⋆ ∈ L ∞ (T * M × R N ) satisfying div γ Z * ∈ L ∞ (M, R N ) and (possibly decimating the sequence (t k )) Using a standard div-curl reasoning and weak-star convergence of u(t k , ·) in W 1,∞ (M, N ) we also obtain This together with (76) yields the second item of (20). The first item of (20) is produced by passing to the limit in the first item of (72) using (71, 75).

Appendix: Technical lemmata
Lemma A.1. Let (N , g) be a closed embedded Riemannian submanifold in the Euclidean space R N . There exists a Riemannian metric h on R N such that (N , g) is a totally geodesic Riemannian submanifold of (R N , h).
Proof. Let R > 0. As N is a closed submanifold of R N , N ∩ B(0, R) is compact. Hence, there is a non-increasing function R → δ R ∈]0, 1[ such that N R,δ = {y + n : y ∈ N ∩ B(0, R), n ∈ T y N ⊥ , |n| < δ} is a tubular neighborhood of N ∩ B(0, R) in R N that does not intersect N \ B(0, R) for δ ∈]0, δ R [. Identifying T y+n N R,δ R with T y N × R N −n , we define a Riemannian metric h R on N R,δ R as follows: is an open cover of R N . Indeed, if z / ∈ R N \ T , i.e. z ∈ T , then letting k 0 be the smallest integer bound of |z|, we have z ∈ T ∩ B(0, k 0 + 1) ⊂ Here, we used the fact that U ∩ ∞ k=1 A k ⊂ ∞ k=1 U ∩ A k for any sequence of sets A k and open set U . Hence, by definition of k 0 , z ∈ N k 0 , 1 2 δ k 0 +1 . Therefore z = y + n with |n| ≤ 1 2 δ k 0 +1 < δ k 0 +1 and y ∈ B(0, k 0 ) ⊂ B(0, k 0 + 1), that is, z ∈ N k+1,δ k+1 . We take a smooth partition of unity {ϕ 0 , ϕ 1 , ϕ 2 , . . .} subordinate to this cover (a construction of a partition of unity subordinate to an infinite open cover can be found in [39, Appendix C]) and define for y ∈ R N . It is easy to check that (N , g) is a totally geodesic submanifold in (R N , h).
Proof. Let d denote the signed distance function of Ω, i. e.
This function is convex and satisfies |d(x) − d(y)| ≤ |x − y| for x, y in R N .
Recall that a critical point of a smooth convex function on R m is necessarily its global (possibly improper) minimum. Hence, by virtue of (81, 82), Ω ε does not contain critical points of d ε , and so it is a smooth hypersurface. Finally, (81) implies the Hausdorff convergence of Ω ε to Ω as ε → 0 + .