On a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain and its computation

We introduce a capillary Chambolle type scheme for mean curvature flow with prescribed contact angle. Our scheme includes a capillary functional instead of just the total variation. We show that the scheme is well-defined and has consistency with the energy minimizing scheme of Almgren-Taylor-Wang type. Moreover, for a planar motion in a strip, we give several examples of numerical computation of this scheme based on the split Bregman method instead of a duality method.


Introduction
In this study, we consider the mean curvature flow equation with prescribed contact angle condition of the form V = − div Γt n on Γ t ∩ Ω for t ≥ 0, ∠(n, n Ω ) = θ(t, •) on ∂Γ t ∩ ∂Ω for t ≥ 0, (MCFB) where Ω ⊂ R d is a smooth bounded domain and n Ω is the unit normal velocity vector field on ∂Ω.Here, {Γ t } t is a time evolving hypersurface to be determined and n represents the outward unit normal vector field of Γ t ; V denotes the velocity of Γ t in the direction of n, which is the outward unit normal vector to Γ t , and θ is a given function on [0, T ] × ∂Ω that describes the contact angle between Γ t and ∂Ω for each t ≥ 0. Here, div Γt denotes the surface divergence so that − div Γt n becomes the (d − 1 times) mean curvature of Γ t in the direction of n.
For the mean curvature flow equation in R d , Almgren, Taylor and Wang [1] introduced a time discrete approximation of the solution which is often called the Almgren-Taylor-Wang scheme; a similar scheme is given by Luckhaus and Sturzenhecker [25].However, one has to minimize a non-convex problem for each time step.In [11], Chambolle introduced another scheme which is based on a strict convex problem and the solution of each step chooses one of minimizers of the Almgren-Taylor-Wang's functional.
One of goals of this paper is to extend Chambolle's scheme to the problem (MCFB) which includes the prescribed contact angle condition.We call our scheme the capillary Chambolle type scheme.We shall show that the scheme is well-defined, and it chooses one of minimizers of the corresponding Almgren-Taylor-Wang's functional.We shall explain them more explicitly.
Let us first introduce the capillary Almgren-Taylor-Wang functional.Given a Caccioppoli set F 0 ⊂ Ω, the capillary Almgren-Taylor-Wang functional is defined by for each Caccioppoli set F ⊂ Ω, where F ∆E 0 = (E 0 \F ) ∪ (F \E 0 ).Here, C β denotes the capillary functional defined by where H d−1 denotes the d − 1 dimensional Hausdorff measure.The function β ∈ L ∞ (∂Ω) will be formally taken as β = cos θ on the intersection of ∂Ω and the boundary of F .One observes that the functional defined in ( 1) is reduced to the functional introduced by Almgren, Taylor and Wang [1] when β ≡ 0. The capillary scheme is as follows.Let h = 1/λ be a given time step.For a given Caccioppoli set E 0 find a minimizer (E 0 ) h which minimizes E → A β (E, E 0 , λ).We repeat this process and find a sequence of sets which is expected to approximate the solution of (MCFB).However, the functional A β is not convex so it is a priori not easy to find its minimizer.To overcome this inconvenience, we introduce the capillary Chambolle type scheme.Let Ω be a smooth bounded domain.Then, for each u ∈ L 2 (Ω), we define Here, h > 0 is a time step that discretizes an interval [0, T ] for some time horizon T > 0; d Ω,E 0 denotes the geodesic signed distance function to E 0 ⊂ Ω with respect to Ω.The capillary functional C β is defined by The functional E β h is a strictly convex functional on L 2 (Ω), so there exists a unique minimizer once we know E β h is lower semi-continuous in L 2 (Ω).In fact, Modica [26] proved that C β (u) is lower semi-continuous in L 1 (Ω) when Ω is a C 2 bounded domain and β ∞ ≤ 1.
The assumption β ∞ ≤ 1 is natural since it is given as a cosine functional.As well-known, the condition β ∞ ≤ 1 is a necessary condition for the lower semi-continuity; see e.g.[19].Although his lower semi-continuity result is enough for our purpose, we give a proof of the semi-continuity in L 1 (Ω) which works for any uniformly C 2 domain not necessarily bounded.We first prove it for β ∞ < 1 and using an argument by contradiction as in [9, Proof of Lemma 2] to prove the case β ∞ = 1.See Proposition 4.
We shall explain the capillary Chambolle type scheme.Given a set E ⊂ Ω, consider the minimizing problem of the energy E β h (u) in the Lebesgue space L 2 (Ω).Since E β h with E 0 = E is lower semi-continuous and convex in the topology L 2 (Ω), we see that E β h has a unique minimizer w h E ∈ L 2 (Ω).We set It turns out that this T h (E) is a minimizer of the capillary Almgren-Taylor-Wang functional.
In the case β ≡ 0, this was proved by Chambolle [11,Proposition 2.2].More precisely, we have Moreover, characterization of the subdifferential of the capillary functional C β seems important because Chambolle used this characterization in implementation of his scheme.Though we do not adopt this direction in numerical experiment, we state its rigorous form.Here D (Ω) is the space of Schwartz's distributions and [z • ν] is the normal trace.We prove Theorem 1 by a duality argument for positively homogeneous function due to Alter; see e.g.[9].However, C β (u) may not be positive for some u.We add a linear functional and characterize its subdifferential.Unfortunately, the characterization by a duality argument is more involved because of addition of a linear function.
We also give a numerical simulation of our scheme.In other words, for given initial data E 0 , we define a discrete solution In [12], Chambolle gave a way to calculate the minimizer of E β h with β ≡ 0 based on duality.Although this idea applies several problems including higher-order total variation flow [18], we do not use his approach.Instead, we adapt a split Bregman method as applied by [27] to calculate a planar crystalline curvature flow.This method was introduced by Goldstein and Osher [23] to calculate energy minimizer including total variation.It applies the fourth order total variation flow [20].Our domain Ω is a strip, and we calculate various examples including translative soliton [2].Since in Theorem 1, we are forced to assume that the average of β is equal to zero, at each time step, we redefine β outside contact points.
Let us remark a few related preceding works to our discrete solution E h (t).In [6], Bellettini and Kholmatov considered A β (F, E 0 , λ) when Ω = R d + , the half space.In their case, C β (u) ≥ 0 and the lower semi-continuity is easy to prove, though they invoked a flat version of the inequality (Corollary 1).Bellettini and Kholmatov [6] showed the convergence of their scheme to a time evolution of sunsets (which is called a generalized minimizing movement, GMM for short).They proved, under conditional assumption similar to [25], that GMM is a "distributional" solution of (MCFB).
In the case β ≡ 0, Chambolle [11] proved that E h (t) actually converged to the level-set flow [14], [16], see also [17] of the mean curvature flow equations provided no fattening occurs; see e.g.[15] for a generalization to general anisotropic flow for unbounded sets.Recently, Chambolle, Gennaro and Morini [arXiv:2212.05027]established a convergence result of a proposed energy minimizing scheme for the anisotropic mean curvature flow with a forcing term and a mobility which depends on both the position and the direction of the normal vector.The minimizing movement constructed in their scheme turned out to converge to a distributional solution à la Luckhaus-Sturzenhecker.The family of time step functions whose upper-level sets are equal to the minimizing movement was shown to converge to the viscosity solution of the corresponding level-set equation.
In the case β ≡ 0, the unique existence of the level-set flow has been already established by [24] and [5].We expect that our discrete solution converges to the level-set flow, although we do not try to prove it in this paper.
This paper is organized as follows.In Section 2, we prepare several notions and notations which will be used frequently throughout the paper.Topics include basic convex analysis, functions of bounded variation, and geodesic distance.In Section 3, we present our proof of the lower semi-continuity of C β .In Section 4, we recall a method to characterize the subdifferential of a functional proposed by Alter.This method will give a concrete form of the subdifferential of the capillary functional (see Theorem 1).In Section 5, we will prove that the capillary Chambolle type scheme implements the capillary Almgren-Taylor-Wang type scheme in some sense (see Theorem 2).After that, we shall carry out a numerical experiment to confirm that our scheme works well and its outcome is as expected in Section 6.The employed scheme is the Split Bregman method.Note that we cannot apply the method directly without any modification due to contribution of the boundary energy.No convergence result is given in this paper.

Preliminaries
In this section, we recall several basic notions and notations without proofs which are important to investigate properties of the capillary functional and the capillary Chambolle type scheme.

Convex analysis
Let E be a normed (real vector) space and E * be its conjugate (dual) space, that is the set of all bounded linear functionals on E.Then, for each function f : E → R ∪ {∞} and u ∈ E, the subdifferential ∂f (u) of f at u is defined as the set of all elements p in E * such that holds for every v ∈ E, where , denotes the duality pair.Note that f is allowed to take the value as ∞.If f ≡ ∞, then f is called proper.The domain D(f ) ⊂ E of f is defined the set of all elements u in E for which f (u) < ∞.Given a function f on E, we define another function for each p ∈ E * .The function f * is called Fenchel conjugate of f .Let us recall one important characterization of the subdifferential in terms of Fenchel conjugate as follows; see e.g.[28].
Then, p ∈ ∂f (u) if and only if the following identity is valid: Fenchel identity yields another characterization of the subdifferential when f is positively homogeneous of degree 1, i.e., f satisfies f (λu) = λf (u) for all λ > 0, u ∈ E.
Proposition 2. Assume that f : E → R ∪ {∞} is proper and u ∈ D(f ).Suppose that f is positively homogeneous of degree 1.Then, it holds that Proposition 3 ([8], Proposition 1.10).Suppose that f : E → (−∞, ∞] is lower semicontinuous and convex with ϕ ≡ ∞.Then, f is bounded from below by an affine continuous function.In other words, there exist p ∈ E * and b ∈ R such that Thus, in the case that f is positively homogeneous of degree 1, b can be taken as zero.

Function of bounded variation
Let Ω be a smooth, bounded, and connected domain.Then, the total variation of f : Ω → R is defined by: In other words, the weak derivative of u is a Radon measure in Ω.Now, we let For every z ∈ X 2 (Ω) and u ∈ BV (Ω), the Radon measure (z, Du) is defined by: (z, Du) is often called the Anzellotti pair (see [3,Definition 1.4]).Moreover, there exists a linear operator [ ).The following Green's formula related to (z, Du) and [z • ν Ω ] is important for our study:

Geodesic distance
In this section, we always assume that a domain Ω ⊂ R d is smooth, bounded and connected.
Definition 1 (Path).Let x, y ∈ Ω be distinct points.Then, a Lipschitz continuous function l : [0, 1] → Ω is called a path between x and y if and only if l(0) = x and l(1) = y.
Definition 2 (Geodesic distance between two points).Let x, y ∈ Ω be distinct points.Then, the geodesic distance dist Ω (x, y) between x and y is defined by: Remark 2. The infimum of (PPGD) can be attained, namely a minimizer exists.This fact is shown in terms of the Ascoli-Arzerá theorem and the lower semi-continuity of l → 1 0 |l (t)|dt.See Section Minimal geodesics in [13].
Definition 4 (Geodesic signed distance function).Let E ⊂ Ω.Then, the geodesic signed distance function to E is defined by: If Ω is convex, then d Ω,E corresponds to the ordinary signed distance function d E defined by It is easy to see that F ⊂ E does not necessarily imply that d E ≤ d F unless Ω is convex.In other words, d E is not monotonous with respect to E. This is a reason we introduce d Ω,E .Indeed, by definition we have Lemma 1.The geodesic signed distance is monotonous with respect to E. In other words,

Lower semi-continuity of capillary functional
In study of the energy (2), the lower semi-continuity of C β is crucial.Due to the boundary integral term, this property is not straightforward.Nevertheless, it was already shown by Modica [26] in the case where Ω is a C 2 bounded domain in R d and C β is of the form: where τ : ∂Ω × R → R is a Borel function which is 1-Lipschitz continuous with respect to the second variable.Note that (ModicaCF) includes (CF) as a special case (set τ (x, s) := β(x)s for each (x, s) ∈ ∂Ω × R and τ turns out to be β ∞ -Lipschitz continuous).
For the proof, he invoked a trace inequality for BV functions derived by [4].The inequality is of the form for f, g ∈ BV (Ω) with c independent of f, g and t > 0, where This yields It turns out that this is enough to prove the lower semi-continuity of C β .
In [6], the lower semi-continuity of C β (u) is proved when Ω is the half space by using an inequality when Ω is the half space In [6], neither the paper [4] nor [26] was not mentioned.This type of the inequality (4) is found in [22, Proof of Proposition 2.6, (2.11)], where ∂Ω = B R and Γ t = B R × (0, t).In [22], it is used that trace is continuous with respect to the strict convergence in BV.In this paper, we establish a curved version of (4) and prove the lower semi-continuous of C β (u) defined by (CF).It works even for unbounded domains provided that it is uniformly C 2 ; for the definition, see e.g.[7].

Lemma 2.
Let Ω be a uniformly C 2 domain in R d and let κ 1 , • • • , κ d−1 be the (inward) principal curvatures of ∂Ω.Let R 0 be its reach, i.e., the largest number such that the projection π : Γ t → ∂Ω is well-defined for t < R 0 , where |x − π(x)| = d(x).Then, for every f, g ∈ BV (Ω), it holds that Here, ∇ ν = ∇d • ∇ denotes the directional derivative in the direction of ∇d so that |∇ ν f | is well-defined as a Radon measure.
Proof.Since ∂Ω is uniformly C 2 , the reach R 0 can be taken positive.We take t ∈ (0, R 0 ) to see that the normal coordinate system is available in Γ t .Precisely, for each x ∈ Γ t , there exists a unique y ∈ ∂Ω such that x = y + ν(y)d(x), where ν(y) denotes the inward unit normal vector to ∂Ω at y.Then, we are able to use a C 1 change of variables between Γ t and ∂Ω × (−t, 0) defined by Φ : where J denotes the Jacobian of Φ.For the difference f t − g t , we obtain |f − g| dx.
Recall the exact form of the Jacobian J(Φ(x)) (see e.g.[21, Chapter 14, Appendix]): Therefore, the desired inequality follows by the triangle inequality.

Corollary 1.
Let Ω be a uniformly C 2 domain in R d and µ ∈ (0, ∞] be the supremum of radius of inscribed circles of ∂Ω.Then, Proof.By the selection of µ, it follows that κ i (π(x)) ≤ 1/µ for all 1 ≤ i ≤ d − 1 and x ∈ Γ t .Moreover, we have d(x) < t.Thus, we can estimate as follows: Hence, the desired inequality is immediately derived from Lemma 2 since |∇ ν f | ≤ |∇f | by the Schwarz inequality and |∇d| = 1.

Proposition 4.
Let Ω be a uniformly C 2 domain in R d .Then, C β is lower semi-continuous with respect to L 1 (Ω) whenever β ∞ ≤ 1. (if Ω has a finite measure, L 1 can be replaced by Proof.First, we prove the assertion when β ∞ < 1.Let {u i } i be a sequence in L 1 (Ω) and u ∈ L 1 (Ω).Suppose that u i → u in L 1 (Ω) as i → ∞.Then, it suffices to prove that lim sup i→∞ {C β (u) − C β (u i )} ≤ 0. For simplicity, set δ µ,t := β ∞ (µ/(µ − t)) d−1 .Then, we can take t > 0 so small that δ µ,t < 1.For such a t > 0 and for each i ∈ N, we compute Here, we have used the lower semi-continuity of u → Ω |∇u| with respect to L 1 (Ω)topology to obtain the last inequality.Since χ Γt converges to 0 as t → 0 pointwise and u ∈ BV (Ω), the Lebesgue convergence theorem gives the desired inequality by letting t → 0.
We next treat the case where β ∞ = 1.We cannot apply Proposition 3 directly for C β (u) since we do not know if it is lower semi-continuous.We know C β/2 is lower semi-continuous by the above argument.By Proposition 3 and Remark 1, there is g ∈ L ∞ (Ω) such that C β/2 (u) + Ω gu dx is non-negative for all u ∈ L 1 (Ω).We set Let us argue by contradiction.Suppose that C β,f is not lower semi-continuous.Then, there exist a δ > 0, a u ∈ L 1 (Ω), and a sequence Here, the second inequality is derived by C β,f ≥ 0. The estimate (5) leads Since the above estimate is valid for all λ ∈ [λ 0 , 1), letting λ 1 yields a contradiction.Therefore, we conclude that C β,f is lower semi-continuous.The proof is complete.

Subdifferential of capillary functional
The unique solution w h E to the minimizing problem min u∈L 2 (Ω)∩BV (Ω) E h (u) has been comuputed as w h E = d E − π hK d E where K = ∂C 0 (0) and π hK denotes the orthogonal projection of L 2 (Ω) onto hK (see the discussion in [12]).Because of this formula, it will be useful to characterize the set K β := ∂C β (0) for capturing the behaviour of the solution to the minimizing problem min u∈L 2 (Ω)∩BV (Ω) E β h (u).To this end, let us recall an approach to the characterization due to an unpublished work by Alter explained in the book by Caselles et al. [10].
Let H be a Hilbert space and Φ : H → [0, ∞].For this Φ, one can define a function Φ : Remark 3. Suppose that D(∂Φ) = ∅ and Φ is positively homogeneous of degree 1.Then, it is easy to see that Φ * (p) = I K (p) where K = ∂Φ(0) and Here, Φ * denotes the Fenchel conjugate defined by (FC).The function Φ is the support function of v Φ(v) ≤ 1 and it is positively homogeneous of degree 1.The set p Φ(p) ≤ 1 equals to K which equals to the set p Φ * (p) < ∞ .Along the line with the discussion of [10, p. 15], we are able to characterize the subdifferential ∂C β with the setting H := L 2 (Ω) and Φ := C β .However, our definition of C β may allow itself to take a negative value.If so, we cannot directly apply Alter's method.We add a linear functional to reduce the problem for positive functionals.This is possible by Proposition 3.
We are now in the position to prove Theorem 2, which characterizes the subdifferential ∂C β (u).
Proof of Theorem 2. Since we know by Proposition 3 that C β (u) is convex, lower semicontinuous and positively homogeneous of degree 1 in L 2 (Ω), there is f ∈ L 2 (Ω) such that C β,f (u) = C β (u) + Ω f u dx ≥ 0 for all u ∈ L 2 (Ω) by Proposition 3 and Remark 1.To prove Theorem 2, we introduce a functional A key step is to prove that C β,f = Ψ β,f which is rigorously stated as follows.
Remark 5.It is not clear that the infimum in the definition of Ψ β,f is attained.This causes extra technical difficulty compared with the case β = 0.
We continue to prove Theorem 2 admitting Lemma 3. Since C β,f = Ψ β,f by Lemma 3, Proposition 7 yields For a moment, we pretend that the infimum in the definition of Ψ β,f is attained.In this case, Ψ β,f (q) ≤ 1 is equivalent to saying that z 0 ∞ = Ψ β,f (q) ≤ 1, where z 0 satisfies if one uses the Anzellotti pair (z 0 , Du).This implies which together with (7) implies Thus, unless C β,f (u) = 0, then z 0 ∞ = 1.In this case, by (7) we have Since the infimum in the definition of Ψ β,f may not be attainable, the argument is more involved.Let {z i } ⊂ X 2 (Ω) be a minimizing sequence of the infimum in the definition of Ψ β,f (q).We may assume that z we conclude that (z i , Du) (z, Du) as measures by [3,Theorem 4.1].Integration by parts yields (7) with z 0 = z i .As in the previous paragraph, we obtain We take any ϕ ∈ C ∞ 0 (Ω).Testing q = − div z i + f z i ∞ by ϕ and sending i → ∞, we see that Here, the second equality follows from [10,Proposition C.4] and the last convergence is deduced from z i z * −weakly in L ∞ (Ω, R d ).Hence, we have q = − div z + f in D (Ω).Meanwhile, for any ϕ ∈ C ∞ (Ω), we again test both q = − div z i + f z i ∞ and q = − div z + f by ϕ.Then, sending i → ∞ yields The converse is easy to prove.We thus conclude that for all α ∈ R. Thus, and As we observed, Ψ β,f (q) = 1 with Ω qu dx = C β,f (u) is equivalent to saying that there exists Thus, we obtain the desired characterization of ∂C β (u) in Theorem 2.
Proof of Lemma 3. The proof is similar to the case β = 0, f = 0 in [10, Proposition 1.9].If Ψ β,f (q) = ∞, then C β,f (q) ≤ Ψ β,f (q) so we may assume that Ψ β,f (q q) now follows by taking the infimum of z ∞ .For the converse inequality, it suffices to prove that C β,f (u) ≤ Ψ β,f (u) by Proposition 5 and Proposition 6.We may assume that u ∈ L 2 (Ω) ∩ BV (Ω).We proceed where the last supremum is taken for z ∈ X 2 (Ω) satisfying Since |∇u| is a Radon measure in Ω, for ε > 0 there is δ > 0 such that ) which is compactly supported in Ω δ , we are able to extend z to Ω such that the extended z satisfies z ∈ X 2 (Ω) with We know that where the second supremum is taken over Since ε is arbitrary, we now conclude the desired inequality.Thus, we have proved that Ψ β,f (u) ≥ C β,f (u).

Capillary Chambolle type scheme
We will show that Chambolle's scheme is a concrete way to implement Almgren-Taylor-Wang's scheme.In other words, we shall prove Theorem 1.
Proof of Theorem 1. Existence and uniqueness of the minimizer w of E β h follows from strict convexity and the lower semi-continuity of the energy with respect to L 2 (Ω).The Euler-Lagrange inclusion of (2) reads We set p := (w − d Ω,E 0 )/h for simplicity.Then, we have −p ∈ ∂C β (w).There exists an M > 0 such that |d Ω,E 0 | ≤ M .Then, we deduce from the maximal principle that |w| ≤ M .Set The first equality is derived by −p ∈ ∂C β (w).Since −p ∈ ∂C β (0), it holds that C β (u) ≥ − Ω pudx for every u ∈ L 2 (Ω).In particular, substituting 1, −1 ∈ L 2 (Ω) into this inequality gives Here the average-free assumption on β is invoked.We next observe that To get (12), we have used the co-area formula with respect to BV functions: Moreover, since χ Fs ≡ 1 for all s ∈ (M, ∞) and χ Fs ≡ 0 for all s ∈ (−∞, −M ), we deduce For the term containing β, we have Thus, the formula (12) follows.
Combining (10) and ( 12) yields Since −p ∈ ∂C β (w), we see that p ∈ ∂C −β (−w) ⊂ ∂C −β (0).Thus, it follows that C −β (χ Fs ) ≥ Ω pχ Fs dx.Therefore, the identity ( 14 This leads We set E s := {d Ω,E 0 < s} for each s ∈ R. Noting that F s turns out to be a minimizer of F → C −β (χ F ) + Ω∩(F Es) |d Ω,E 0 − s|/h.We can take a decreasing sequence s i → 0 as i → ∞ such that p ∈ ∂C −β (χ Fs i ) for every i ∈ N.Then, since χ Fs i → χ F 0 in L 2 (Ω), the lower semi-continuity of C −β implies Here, the last equality follows from pointwise convergence of χ Fs i to χ F 0 and the Lebesgue convergence theorem.The converse inequality is derived from p ∈ ∂C −β (0).Therefore, we conclude p ∈ ∂C −β (χ F 0 ) which leads In Theorem 1, β cannot be constant owing to the restriction Ω βdH d−1 = 0.Moreover, we are not sure that the given β exactly describes the desired contact angle condition because we do not know the position of ∂ * E 0 ∩ ∂Ω.Thus, we have to define β as − cos θ(0, •) in a neighbor of the boundary ∂E 0 ∩ ∂Ω of the hypersurface, and we set β as a constant so that Ω βdH d−1 = 0. Subsequently, we rigorously state how to implement Chambolle's scheme with capillary functional.
Let w h,0 ∈ L 2 (Ω) ∩ BV (Ω) be the unique minimizer of the energy E h .Then, we set T h (E 0 ) := {w h,0 < 0}.Next, β h,1 is defined in terms of T h (E 0 ) as follows: h .By the inductive step, we define N h,i , β h,i and T i+1 h (E 0 ) for 0 ≤ i ≤ N − 1 with T 1 h (E 0 ) := T h (E 0 ) assuming that at each step N h,1 can be taken with the property that ∂Ω\N h,1 has positive H d−1 -measure on ∂Ω.
Under these notations, we define a time discrete evolution E h (t) of sets in Ω by: The definition of E h depends not only on the choice of h, but also on N h,i .If one tends to prove the convergence of the proposed discrete scheme, then independence of the choice of N h,i should be shown as well.Though we expect that E h (t) will converge to a time evolution E(t) in some sense, we do not provide any convergence result and leave it for future works.Alternatively, we shall carry out several numerical experiments to confirm that the proposed scheme works well and behaves as desired.

Numerical experiment
In this section, we show how the discrete scheme works through some examples.Our scheme consists of the following parts.
1. Given an initial data E ⊂ Ω, compute the signed distance d E in terms of fast marching algorithm.
2. Solve the isotropic TV denoising problem with the initial data d E by the Split Bregman method.Let w E be a solution of it.
3. Compute the zero sublevel set E of w E and set E := E .
4. Repeat the process from 1 to 3.
Our goal in this section is to modify the scheme mentioned above so that it also works in the case where TV is replaced by the capillary functional C β for some β ∈ L ∞ (∂Ω) and to verify its accuracy through some strong solutions to the mean curvature flow with contact angle condition.Nevertheless, we begin with a classical case to get in touch with the basic idea of this method.By the classical case, we mean that E is sequentially compact in Ω, that is E ⊂⊂ Ω.

How to derive distance function?
To compute the signed distance function d E numerically, we put collocation points X i (1 ≤ i ≤ N ) and regard E as the polygon ∪ N i=1 [X i−1 , X i ] with solid.For short, this polygon will be still denoted by E.Then, we can judge whether each point x in Ω is included in the polygon or not by investigating the winding number of the polygon around x.In this way, we have the discrete function d Then

Split Bregman method
Let us recall the Split Bregman method first proposed by Goldstein and Osher [23].Their scheme aims to solve problems that are categorized in the class of L 1 regularized optimization problem.Before applying their scheme to our problem, let us briefly review the proposed scheme.Let f be a given data and µ > 0, they considered the following energy minimizing problem: Note that this quantity is nothing but the energy to be minimized in Chambolle's scheme if one chooses µ := 1 h and f := d E for some given initial data E ⊂ Ω.In our problem, this corresponds to the case where β ≡ 0. The idea of the Split Bregman method is to divide the variable u of (15) into two portions u and d = (d x , d y ) := ∇u and to solve alternatively the following problem: The last two terms are regarded as a constraint d = ∇u and ( 16) is an unconstrained problem.Note that the problem under consideration is represented as the sum of L 1 and L 2 terms.The minimizer u is approximated by a sequence {u (k) } k of functions that are generated an iterate step.To this end, setting u (0) := f and d k) and b y (k) (k ∈ N) are determined by the following equality:

Closed curves
We begin with classical cases, namely the case where an initial data with solid is fully included in Ω.In this case, we always impose the periodic boundary condition to minimizers u to be determined through our scheme.Precisely speaking, we assume that u 1,j = u Ny,j (1 ≤ j ≤ N x ), u i,1 = u i,Nx (1 ≤ i ≤ N y ).Curves presented below are zero level lines of minimizers which are derived through our scheme.We borrow the initial data of the pi curve from the website "https://ja.wolframalpha.com/".Since its parameterization is quite complicated, we do not cite it.We have multiply coordinates by 1 240 to be included our Ω.The hyperparameters used in this case are same as in the previous section, namely star shaped curve.Following the idea of the method, we prefer to split this problem into the following two sub-problems: To this end, we recall the derivation of the equality (20).This equality is nothing but discrete version of the following equality:  We shall explain how to treat the boundary part of the system, namely (24).It is convenient to be able to get the discrete values of u on the boundary ∂Ω in the formula (19).However, to calculate the quantities u i,Nx+1 which is fictional.It is a common way to consider that this kind of imaginary points exist and to compute these values in terms of an imposed Neumann boundary condition, that is to say the formula (24).At first, consider the case where mesh points (x j , y i ) are on the edge of ∂Ω.Then, By the central difference method, (u x ) i,j and (u y ) i,j are computed as follows: (u x ) i,1 ≈ u i,2 − u i,0 ∆x , (u x ) i,Nx ≈ u i,Nx+1 − u i,Nx−1 ∆x , (u y ) 1,j ≈ u 2,j − u 0,j ∆x , (u y ) Ny,j ≈ u Ny+1,j − u Ny−1,j ∆x .

Figure 1 :
Figure 1: Evolution of star shaped curve

Figure 2 :
Figure 2: Evolution of Pi shaped curve

Figure 3 :
Figure 3: Contact angle problem for 1 ≤ j ≤ N x and u (k) i,1 , u (k) i,Nx for 1 ≤ i ≤ N y , we need the values u

Figure 5 :
Figure 5: Convergence to a straight line [11] obtain the approximate values of d E at mesh points near ∂E by applying the argument done in §3.1 in[11].As stated in[11], we are now in the position to start the fast marching algorithm to determine the approximate values of d E far from ∂E.To this end, we have utilized "FastMarching.jl", a library of Julia developed by Hellemo [Github; hellemo/FastMarching.jl; accessed; 2023May 21].FastMarching.jlaccepts coordinate of mesh points nearby ∂E and returns distance between each mesh point and ∂E.After that, we finally update the sign of each d E (i, j) by checking the sign of w E (i, j).In the second iteration, we do not have to calculate the winding number of E because we already know the level set function w E .