A caricature of a singular curvature flow in the plane
aa r X i v : . [ m a t h . A P ] M a r A caricature of a singular curvature flow in the plane
Piotr B. Mucha and Piotr RybkaInstitute of Applied Mathematics and Mechanics, Warsaw Universityul. Banacha 2, 02-097 Warszawa, PolandE-mail: [email protected], [email protected] author: Piotr Rybka, [email protected], fax: +48 22 55 44 300November 19, 2018
Abstract.
We study a singular parabolic equation of the total variation type in one dimension. Theproblem is a simplification of the singular curvature flow. We show existence and uniqueness of weaksolutions. We also prove existence of weak solutions to the semi-discretization of the problem aswell as convergence of the approximating sequences. The semi-discretization shows that facets mustform. For a class of initial data we are able to study in details the facet formation and interactions andtheir asymptotic behavior. We notice that our qualitative results may be interpreted with the help of aspecial composition of multivalued operators.
AMS subject classification: keywords: singular parabolic equations, singular curvature flow, monotone operators, facet forma-tion, facet interaction
Many free boundary problems involving the Gibbs-Thomson relation may be considered as a drivenweighted mean curvature flow coupled through the forcing term to a diffusion equation (see [CR],[Ra], [L] [AW]). We have a considerable body of literature concerning this problem for the Euclideancurvature of the interface, including the question of precise regularity of solutions treated by Escher,Prüss, Simonett and Mucha, see [EPS], [ES], [Mu]. On the other hand, less is known if the curvatureappearing in the Gibbs-Thomson relation is singular, see e.g. [Ry]. This line of research has beeninitiated by Taylor, [T], and independently by Gurtin, [AG]. However, just solvability of equations ofthe singular curvature flow is interesting. Existence of the flow was obtained by Bellettini, Novaga,Paolini [BNP1], [BNP2] and by Chambolle [Ch]. Driven singular curvature flow was studied byM.-H.Giga, Y.Giga and Rybka, see [GG], [GR1], [GR2].In fact, the existence and properties of solutions to the singular weighted mean curvature flow V = κ on Γ( t ) , (1.1)are interesting in itself even in the plane and without forcing, especially when the anisotropy function(also called ‘energy density function’) is singular, i.e. just convex. Here, Γ( t ) is the unknown curveand κ denotes the weighted mean curvature related to the underlying anisotropy function and V is the1elocity of surface Γ( t ) . Our ultimate goal would be to study existence and behavior of solutions to(1.1).In its full generality problem (1.1) for an arbitrary initial curve is rather difficult. One source ofdifficulties is the geometry of the system, it is already present in the two-dimensional setting. Here,we want to concentrate only on the purely analytical difficulties appearing in (1.1). This is why wewill restrict our attention to a simplified equation, which retains the singular character of the originalproblem.Here is our postulated equation Λ t = ∂∂s ddφ J ( s + Λ s ) in S × (0 , T ) , Λ( s,
0) = Λ ( s ) on S, (1.2) Λ(2 π, t ) = Λ(0 , t ) , t ≥ , here S is the unit circle parameterized by interval [0 , π ) and Λ is the sought function. Comparedwith (1.1) our new system has one analytical advantage. Namely, the domain of definition of Λ( · , t ) is independent of time.We present a justification of this equation in the Appendix. Here, we explain our notation. Thevariable s plays the role of the arclength parameter, the subscript s denotes the differentiation withrespect to s . We frequently refer to ϕ = Λ s + s as the angle between the x axis and the outer normalto the curve. Such an interpretation helps drawing pictures, but the relation to the actual angle is ratherloose.We make a specific choice of J corresponding to the surface energy density functions. We wantto study a situation which is already very singular yet tractable. In many instances of a great physicalinterest an anisotropy appears, which is merely convex, not even strictly convex (understood in aproper sense). As a result, we choose J , which is convex and piecewise linear. This is an independentsource of difficulties. In order to avoid further technical troubles we will choose J correspondingto the situation where that curve minimizing the surface energy (which is the Wulff shape of theanisotropy function) is a square. We must stress again that the correspondence is at the level of ideas,because (1.2) is not a curvature flow, but its caricature. However, the obtained behavior of solutionsto (1.2) is almost the same as for the equation (1.1) with the anisotropy function corresponding to asquare, [Ch].Thus, we pick J which suffers jumps of equal height π at the equi-spaced angles A = (cid:26) α k = − π k ∆ α : k = 0 , , , , with ∆ α = π (cid:27) . (1.3)Specifically, we put J ( ϕ ) = π (cid:18) | ϕ − π | + | ϕ − π | + | ϕ + π | + | ϕ + 3 π | (cid:19) . (1.4)Since Λ is defined over the unit circle its graph over S is a closed curve. The meaning of the spacingbetween α k ’s can be explained by looking at the equation ∂∂s ddϕ J ( s + Λ s ) = 1 , considered in [MRy] – see subsection 3.2, too. Roughly speaking, the spacing between α k and α k +1 corresponds to the length of facets having the normal vector n with the normal angle α k . The size2f the jump of ddϕ J ( s + Λ s ) corresponds to the angle between the normals to the curve, which is asolution to the above equation, at a corner.The chosen anisotropy function (1.4) is nowhere regular, hence we can expect nonstandard effectsrequiring new analytical tools. This has been observed by researchers working on the total variationflow, whose simplification is u t − δ ( u x ) u xx = 0 (1.5)augmented with initial and boundary data. Here, δ a is the Dirac measure concentrated at a .We noticed so far two main types of motivation to study the total variation flow, u t − div (cid:18) ∇ u |∇ u | (cid:19) = 0 . (1.6)The first one is the image denoising and reconstruction introduced by Rudin and Osher, [RO], [ROF].The second one is evolution of the facets of crystals. The bulk of the papers (see [ABC1], [ABC2],[BCN], [GK], [GGK], [ACM], [Mo]) uses the theory of nonlinear semigroups to establish existence.The last paper is particularly interesting, because it deals with the anisotropic total variation flow.Moreover, the notion of entropy solutions was introduced to deal with uniqueness of the total variationflow (see [ABC1], [BCN]). The tools of convex analysis were useful to make sense out of (1.5).The authors, mentioned above, paid special attention to piecewise constant initial data and they wereinterested in the asymptotic behavior, in particular the asymptotic shape was identified. M.-H.Giga,Y.Giga and R. Kobayashi, [GK], [GGK], also calculated the speed of flat facets. No matter what isthe approach, it is apparent that the most important information is located in sets { u x = 0 } , where thesingular dissipation starts to play a role and where the classical multivalued theory of function losesthe meaning.Our approach differs in many aspects. We prove existence by a regularizing procedure and passingto the limit with the regularizing parameter, this approach was used, e.g. by Feng and Prohl, see [FP].The main difficulty is associated with studying the limit of the non-linear terms. We present a moredetailed analysis of regularity of solutions permitting us to call them ‘almost classical’. For genericdata, our solutions are twice differentiable with respect to s , except a finite number of points (for fixedtime). This will be explained in detail below. We mention here that we use the tools of the convexanalysis, in particular we rely on the fact that for a convex function the subdifferential is well-definedeverywhere. However, the classical theory of multivalued functions is not sufficient. We have tointroduce a new definition of the composition of two multivalued functions to describe the meaningand qualitative properties of solutions to system (1.2) as well a class of the J - R functions, whereregularity is described from the point of view of the properties of the function J . In our opinion theresults we prove contribute to better understanding parabolic systems with measure coefficients.Our technique requires a new look at the regularity of functions. We will generalize the meaningof the convexity defining a class of J -regular functions preserving some important properties of theconvexity. Our main qualitative result says that any sufficiently regular initial curve evolving accord-ing to system (1.2), will eventually reach a minimal solution, which is called the asymptotic profilein the area of the total variation flow. The geometric interpretation is that the solution reaches itsasymptotic shape, i.e. the square in our case. This may happen in infinite of finite time dependingupon initial data. If this event occurs in finite time, then subsequently, the solution shrinks to a point.This behavior can be illustrated by the pictures below. The precise meaning is contained in Theorem5.1. 3Time t = 0 . ❘✒ ■✠ ✠✒ Time t = t The evolution is determined by motion of facets defined by singularities of the J -function (thearrows show the direction of the evolution). In finite time we obtain a convex domain, which becomesa square converging to a point in finite time.. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❅❅ ❅❅❅❘✒ ■✠ ✠✒ Time t = t . (cid:0)(cid:0)(cid:0) ❅❅❅(cid:0)(cid:0)(cid:0)❅❅❅ ✠■❘✒ Time t = t .Time t = t end We have to underline that the illustrated evolution hides the novel idea of definition of singularterm δ ( u x ) u xx being a multiplication of two Dirac deltas (as in (1.5)), however the nonlocal characterwill allow us to define this object. Additionally, by the uniqueness of solutions to our system we showthat our novel definition is the only admissible. Formally, the dissipation caused the Dirac deltacoefficient is so strong that the changes of regularity (i.e. appearance of the facets) happen instantly.We will state our results in the Section below, the proofs will be presented in the further Sections.Here, we present the outline of the rest of the paper. We show the existence of weak solutions inSection 2, uniqueness is the content of that Section, too. The qualitative analysis is based on thesemi-discretization which is performed in Section 3. Our goal is to make some of the properties moreapparent. Namely, we want to show that facets (i.e. intervals where ϕ = Λ s + s has a constant valueequal to one of the α i ’s) form instantaneously. In Section 5, we show further geometric properties ofsolutions, namely the curve becomes convex (i.e. the angle ϕ becomes monotone) in finite time. Inaddition, we show that solutions become fully faceted in finite time, i.e. the solution is composed onlyof facets. These two events are not correlated in time. Finally, we show that our solutions converge toa special solution which we call minimal. 4 The main results
Here, we present our results. We begin by noticing that if J is given by (1.4), then the meaning of(1.2) is not clear at all because its right-hand-side formally becomes Λ t = π X k =0 δ k π − π ( s + Λ s )Λ ss . Hence the above equation can be viewed as a generalization of equation (1.5).We will use the tools of the convex analysis to interpret it. Due to convexity of J its subdifferentialis always well-defined. Since in general ∂ φ J ( φ ) is not a singleton it is necessary to find its properselection, in particular (1.2) takes the form, Λ t ∈ ∂∂s ∂ ϕ J (Λ s + s ) , in S × (0 , T ) , Λ( s,
0) = Λ ( s ) , on S, Λ(2 π, t ) = Λ(0 , t ) , for t ≥ , (2.1)where S is the unit circle.In other words, we have to find (weakly) differentiable selections of ∂ φ J (Λ s + s ) . Thus, we arelead to the following notion of a weak solution to (1.2). Definition 2.1.
We say that Λ ∈ C ([0 , T ]; L ( S )) , such that Λ s ∈ L ∞ (0 , T ; TV ( S )) is a weaksolution to (2.1), if there exists a function Ω ∈ L (0 , T ; W ( S )) such that Ω( s, t ) ∈ ∂I (Λ s + s ) a.e.,and for any function h in C ∞ ( S ) it holds Z S Λ t h = − Z S (Ω − s ) h s + Z S h. With this definition we can show the following existence result.
Theorem 2.1.
Let us suppose that J is defined by(1.4), Λ ∈ L ( S ) and Λ ,s ∈ TV ( S ) , then thereexists Λ ∈ C α (0 , T ; L ( S )) with α > ,additionally Λ s ∈ L ∞ (0 , T ; TV ( S )) and Λ t ∈ L (0 , T ; L ( S )) such that it isaunique weak solution to(2.1).The proof will be achieved through an approximation procedure, it is performed in Section 3.Moreover, we show uniqueness of the solution constructed here, this is the content of Theorem 3.1 inSection 3.However, our main goal is to describe precisely qualitative properties of solutions to (2.1). As amotivation, we present a special type of solutions, which we will call minimal solutions , which aregiven explicitly, one of them is given here, (see also §3.2), ¯Λ( s, t ) = Z s ¯ ϕ ( u ) du + t, where ¯ ϕ ( s ) = π χ [0 , π ) ( s ) + 3 π χ [ π ,π ) ( s ) + 5 π χ [ π, π ) ( s ) + 7 π χ [ π , π ) ( s ) . (2.2)5t is a matter of an easy exercise to see that ¯Λ defined above with ¯Ω( x, t ) = x is indeed a weak solutionto (2.1). In fact, this an asymptotic profile, which can be reached in finite time.We will keep in mind this example while developing the proper class of regular solution. The ideais that we want to extend properties of convex solutions to a more general class, hence we introduce aclass of J-regular function, where restrictions on regularity depend on function J from (1.4).Firstly, we define the space of functions which are helpful to describe the regularity of the deriva-tive of our solutions. We recall that any function φ ∈ T V is a difference of two monotone functions.Thus, we shall call a multifunction φ : [0 , π ) → R a maximal T V function if it is a difference oftwo maximal monotone multifunctions and one of them is continuous.
Definition 2.2.
We say that a maximal
T V multivalued function φ : [0 , π ) → R is J-regular, i.e. φ ∈ J - R[0 , π ) , provided that the set Ξ( φ ) = { s ∈ [0 , π ) : φ ( s ) ∋ α k for k = 0 , , , } consists of a finite number of connected components, i.e. we allow only isolated intervals or iso-lated points. Additionally, on any connected subset [0 , π ) \ Ξ function φ takes its values in interval ( α k , α k + π ) for some k = 0 , ...,3,modulo π –see (1.3).Foreach φ ∈ J - R[0 , π ) wedefine afunction K : J - R[0 , π ) → N bythe formula K ( φ ) = the number of connected components of theset Ξ( φ ) . Additionally weput || φ || J - R[0 , π ) = || φ || T V [0 , π ) + K ( φ ) . Let us note that the J - R class does not form a Banach space. It is not a linear space. In order toformulate the meaning of solutions, first we define the composition of J - R functions with ∂J . Becauseof the complex structure the definition is long. Definition 2.3.
We define the composition ∂J ¯ ◦ A , ∂J ¯ ◦ A : [ a, b ] → [ e, f ] , where A : [ a, b ] → [ c, d ] is an J - R function and ∂J : [ c, d ] → [ e, f ] as follows:To begin with, we decompose the domain [ a, b ] into three disjoint parts [ a, b ] = D r ∪ D f ∪ D s ,where D s = { s ∈ [ a, b ] : A ( s ) = [ c s , d s ] and c s < d s } ; D f = { S k ( a k , b k ) : A | ( a k ,b k ) = c k , where c k is a constant } ; D r = [ a, b ] \ ( D s ∪ D f ) . (2.3)Then, the composition is defined in three steps:1. For each s ∈ D r the set A ( s ) is a singleton, thus the composition is given in the classical way ∂J ¯ ◦ A ( s ) = ∂J ( A ( s )) for s ∈ D r . (2.4)2. In the case s ∈ D f the definition is “unnatural”. For a given set ( a k , b k ) ⊂ D f we have A | ( a k ,b k ) = c k . If ∂J ( c k ) is single-valued, then for s ∈ ( a k , b k ) we have, ∂J ¯ ◦ A ( s ) = { dJdφ ( c k ) } . ∂J ( c k ) is multivalued, i.e. ∂J ( c k ) = [ α k , β k ] , then the definition is not immediate.We have to consider four cases related to the behavior of multifunction A in a neighborhood of interval ( a k , b k ) . The regularity properties of the J - R class imply the necessity to consider the following fourcases (for small ǫ > ):(i) A is increasing, i.e. A ( s ) < c k for s ∈ ( a k − ǫ, a k ) and A ( s ) > c k for s ∈ ( b k , b k + ǫ ) ;(ii) A is decreasing, i.e. A ( s ) > c k for s ∈ ( a k − ǫ, a k ) and A ( s ) < c k for s ∈ ( b k , b k + ǫ ) ;(iii) A is convex, i.e. A ( s ) > c k for s ∈ ( a k − ǫ, a k ) and A ( s ) > c k for s ∈ ( b k , b k + ǫ ) ;(iv) A is concave, i.e. A ( s ) < c k for s ∈ ( a k − ǫ, a k ) and A ( s ) < c k for s ∈ ( b k , b k + ǫ ) .The case (i) we put ∂J ¯ ◦ A ( t ) = x k ( t − b k ) + y k ( t − a k ) for t ∈ ( a k , b k ) , (2.5)where x k = α k a k − b k and y k = β k b k − a k .For case (ii) we put ∂J ¯ ◦ A ( t ) = x k ( t − b k ) + y k ( t − a k ) for t ∈ ( a k , b k ) , (2.6)where x k = β k a k − b k and y k = α k b k − a k .When we deal with case (iii) we set ∂J ¯ ◦ A ( t ) = β k for t ∈ ( a k , b k ) . (2.7)Finally, if (iv) holds, then we put ∂J ¯ ◦ A ( t ) = α k for t ∈ ( a k , b k ) . (2.8)3. In the last case, if s ∈ D s our definition is just a consequence of first two steps. Since set D s consists of a countable number of points we consider each of them separately. We have A ( d k ) =[ e k , f k ] with e k = f k , then ∂J ¯ ◦ A ( d k ) = [lim sup t → d − k ∂J ¯ ◦ A ( t ) , lim inf t → d + k ∂J ¯ ◦ A ( t )] . (2.9)Definition 2.3 is complete.Thanks to the J - R regularity of A , the above limits are well defined. As a result, we are able toomit point from D s in (2.3). We note that the above construction guarantees that ∂J ¯ ◦ A : [ a, b ] → [ e, f ] is a J - R function.After having completed the definition we make additional comments on step 2. Formulae (2.5)-(2.8) are immediate consequences of the pointwise approximation of the considered function bysmooth functions. The presented composition agrees with the results from [MRy], where a stationaryversion of the problem has been considered. In particular, our definition follows from a requirement:if A is maximal monotone then we expect A − ¯ ◦ A = Id.
Moreover, the composition of two maximal increasing functions is maximal increasing. Anotherpoint, which should be emphasized, is the nonlocal character of the above definition. Step 3 dependson step 2, so steps 1 and 2 should be performed at the very beginning.Now we are prepared to introduce the main definition.7 efinition 2.4.
We say that a function
Λ : S → R is an almost classical solution to system (1.2) iff Λ is a weak solution with Ω = ∂J ¯ ◦ [Λ s + s ] , Λ s + s ∈ L ∞ (0 , T ; J - R[0 , π )) and Λ t = dds ∂J ¯ ◦ [Λ s + s ] in [ S × ((0 , T ) \ N )] \ S Let Λ be such that Λ ,s + s ∈ J - R[0 , π ) , then there exists a unique almost classicalsolution tosystem (2.1)conforming to Definition 2.4.In fact this is a statement about the regularity of weak solutions. Theorem 2.2 is a result of thesemi-discretization of system (1.2). At this level, we will be able to show that facets must appear,as suggested by the pictures in the Introduction. The semi-discretization will determine the RHS of(2.10) on sets where the solution falls into the singular part of ∂J . We will obtain that on these setsthe term ∂J is constant on each connected part (or time dependent for the evolutionary system). Next,by the elementary means we will show that the semi-discretization tends uniformly to the solutionsobtained by Theorem 2.1. However, performing a rigorous proof that we indeed constructed an almostnormal solution requires more work on the structure of weak solutions, which is the content of Section5. Thus, it will be postponed until the end of this part.At the end, in Section 5, we deeply go into the qualitative analysis of the evolution showing theconvexification effect and convergence to the minimal solutions. Since we know that facets mustappear and the solutions are unique we are in a position to construct quite explicit solutions. Weare able to follow their qualitative changes. This is made precise in Theorem 5.1. In particular weshow instantaneous creation of facets. For the sake of this study we show a comparison principle insubsection 5.1. Moreover, we show that the evolution of facets is governed by a system of ODE’swhich are coupled if the facets interact, this is explained in Section 5. A conclusion from our analysisis existence of a sequence of instances at which our solution gets simplified before it gets the finalform of the asymptotic profile, i.e. the minimal solution. In this Section we show an existence and uniqueness of weak solutions of (1.2). We use the tools ofthe convex analysis to interpret it. In particular, we shall make the gradient flow structure of (1.2)transparent. However, the existence is shown by the method of regularization. Some of the statementsare easier to interpret if they are written in the language of the ‘angle’ ϕ = Λ s + s . Here, ϕ playsthe role of the angle between the normal to the curve and the x –axis. Thus, for convex closed curves ϕ must be increasing, but we shall not require that, instead we admit ϕ being a functions of boundedtotal variations, i.e., ϕ ( · , t ) ∈ T V ( S ) , in particular ϕ ∈ L ∞ ( S ) and it may be discontinuous though.8 .1 The proof of the general existence result We present a proof of our existence result, Theorem 2.1. It will be achieved through an approximationprocedure. For any ǫ > we set J ǫ ( x ) := J ⋆ ρ ǫ ( x ) + ǫ x , (3.1)where ρ ǫ is a standard mollifier kernel, with support in ( − ǫ, ǫ ) . Let us note properties of the approxi-mation J ǫ :(a) J ǫ ∈ C ∞ ( R ) ;(b) ddx J ǫ is strictly monotone;(c) d dx J ǫ ≥ ǫ ;(d) ddx J ǫ ( x ) − ǫx = ddx J ( x ) for x such that | x − α k | > ǫ for k = 0 , , , .We start with existence of the regularized system. Lemma 3.1. Let us suppose that J ǫ is defined by (3.1) and Λ ǫ is smooth and π -periodic. Then, forany T > there exists aunique, smooth solution tothe regularized problem, Λ ǫt = ∂∂s ddϕ J ǫ (Λ ǫs + s ) , in S × (0 , T ) , Λ ǫ ( s, 0) = Λ ǫ ( s ) , on S, (3.2) Λ ǫ ( s + 2 π, t ) = Λ ǫ ( s, t ) , for t > . Proof. By properties (a), (b) (c) and (d) of J ǫ , see (3.1), the existence and uniqueness of smoothsolutions to (3.2), is guaranteed by the standard theory of parabolic systems, see [LSU].We now study properties of established solutions. Lemma 3.2. Letus suppose that Λ ǫ is asmooth solution to(3.2).(a) Iffor a, b ∈ R and theinitial datum satisfies a ≤ (Λ ǫ ,s ( s ) + s ) ≤ b ,then, for all t < T wehave a ≤ (Λ ǫs ( s, t ) + s ) ≤ b. (b) Ifmoreover, (Λ ǫ ,s ( s ) + s ) s ∈ L (0 , π ) , then, forall t < T wehave (Λ ǫs ( s, t ) + s ) s ∈ L ∞ (0 , T ; L (0 , π )) . Proof. We use the maximum principle. First of all, we differentiate (3.2) with respect to s , Λ ǫst = dds (cid:18) ∂ J ǫ ∂ϕ ( s + Λ ǫs )( s + Λ ǫs ) s (cid:19) . We notice ( s + Λ ǫs ) t = Λ ǫst . We set w = ( s + Λ ǫs ) , hence we obtain the equation for w , w t = dds ( a ( s, t ) w s ) , (3.3)9here by (3.1) we have a ( s, t ) = ∂ J ǫ ∂ϕ ( s + Λ ǫs ) ≥ ǫ > . Hence, by the maximum principle we obtain ( a ) .To prove (b) we note that from (3.3) we obtain w st = d ds ( a ( s, t ) w s ) . (3.4)By Lemma 3.1 our solutions are smooth. In oder to finish the proof of (b) it is enough to integrate(3.4) over sets { w s > } and { w s < } to reach, ddt Z { w s > } w s dx ≤ and ddt Z { w s < } w s dx ≥ . (3.5)Having established this Lemma, we will obtain L ∞ estimates for the spatial derivative of solution Λ . Corollary 3.1. Thereis aconstant M independent of ǫ and T such that k ϕ ǫ k L ∞ ( S × (0 ,T )) ≤ M, k ϕ ǫ ( · , t ) k L ∞ (0 ,T ; T V [0 , π )) ≤ M. Proof. The first part follows from Lemma 3.2 (a) directly, because ϕ ǫ = Λ ǫs + s . The second part is theresult of Lemma 3.2 (b), combined with the properties of approximation of T V functions in L .We want to show that the estimates for Λ ǫ will persist after passing to the limit with ǫ . Lemma 3.3. Let us suppose that Λ ǫ converges weakly in L ( S × (0 , T )) to Λ . If (Λ ǫs + s ) s ≥ in D ′ ( S ) ,then (Λ s + s ) s ≥ aswell in D ′ ( S ) . Proof. Indeed, if h ∈ D ( S ) is positive, then ≤ R S (Λ ǫs + s ) h s . The inequality holds after taking thelimit. Lemma 3.4. Thereis aconstant independent of ǫ such that Z T Z π (Λ ǫ ) dxdt ≤ M, Z T Z π [(Λ ǫx ) + (Λ ǫt ) ] dxdt ≤ M. Proof. The bound on R T R π (Λ ǫ ) is trivial, due to L ∞ estimates established in previous lemmas.Similarly, the bounds in Corollary 3.1 imply that R T R π (Λ ǫx ) ≤ M . We shall calculate the lastintegral with the help of integration by parts, Z T Z π (Λ ǫt ) dsdt = − Z T Z π Λ ǫst ddϕ J ǫ (Λ ǫs + s ) dsdt + Z T Λ ǫt ddϕ J ǫ (Λ ǫs + s ) | s =2 πs =0 dt = Z π J ǫ ( ϕ ( s )) ds − Z S J ǫ ( ϕ ( s, T )) ds + Z T Λ ǫt (0 , t ) (cid:18) ddϕ J ǫ (Λ ǫs (0 , t ) + 2 π ) − ddϕ J ǫ (Λ ǫs (0 , t )) (cid:19) dt, Here, we also exploited periodicity of Λ . We notice that the difference ddϕ J ǫ (Λ ǫs (0 , t ) + 2 π ) − ddϕ J ǫ (Λ ǫs (0 , t )) equals exactly π . Hence, Z T Z π (Λ ǫt ) dsdt ≤ Z π J ǫ ( ϕ ( s )) ds + 2 π (Λ ǫ (0 , T ) − Λ ǫ (0 , ≤ M Remark. We want to stress that the above estimate on Λ t is one of the most important differencesbetween (1.1) and (2.1).Now, we have enough information to select a weakly convergent subsequence, with propertiesannounced in the theorem. Proposition 3.1. Thereexists asubsequence { ǫ k } converging to zero, such that(a) Λ ǫ k ⇀ Λ in W ( S × (0 , T )) ; ϕ ǫ k s ⇀ ϕ s as measures in S × (0 , T ) .(b) Λ ∈ C ([0 , T ) , L ( S )) . Proof. The first part of (a) is implied by Lemma 3.4. The second part of (a) follows from ϕ ǫ = Λ ǫs + s ,and Lemmas 3.2, 3.4. Part (b) follows from Lemma 3.2 and 3.4 and the embedding theorem (we havealready proved Λ ǫ ∈ L (0 , T ; W (0 , π )) ∩ W (0 , T ; L (0 , π )) ).The next step is to show that that the limit is indeed a solution. In particular, we have to passto the limit in the non-linear term. First of all, we shall change the notation in order to make moretransparent what we are doing. We want to find w ( s, t ) such that w s ( s, t ) = ϕ ( s, t ) . By a simpleintegration of this formula and the definition of ϕ , we can see w ( s, t ) = 12 s + Λ( s, t ) , where we set w (0 , t ) = Λ(0 , t ) . Hence, w s = ϕ and we can re-write the evolution problem as agradient system w t ∈ dds ∂J ( w s ) , in S × (0 , T ) ,w ( s, 0) = 12 s + Λ ( s ) , for s ∈ S, (3.6) w ( s, t ) − s is periodic for t ∈ (0 , T ) . If ϕ ( · , is increasing, then due to Lemma 3.2 (b) and Lemma 3.3 ϕ ( · , t ) is increasing as well,hence w ( · , t ) is convex. Obvious changes are required to write the system for the regularization w ǫ ( s, t ) = s + Λ ǫ ( s, t ) . Proposition 3.2. For any fixed t ≥ and a sequence { ǫ k } converging to zero there exists its subse-quence { ǫ k } (not relabeled), such that for each x ∈ [0 , π ) the limit lim ǫ → ddϕ ( J ǫ )( ϕ ǫ )( x, t ) = Ω( x, t ) exists. Moreover, Ω( x, t ) ∈ ∂J ( ϕ ( x, t )) for almost every x ∈ [0 , π ) . Remark. It is important for us to make the selection of the subsequence independently of t . Proof. Indeed, once we fix t > , we may recall that ϕ ǫ ( · , t ) ∈ T V as well as ddϕ J ǫ ( ϕ ǫ )( x, t ) ∈ T V . Hence, by Helly’s convergence theorem there exists a subsequence ǫ k such that these sequencesconverge. Using the new notation, we write, lim ǫ → ϕ ǫ ( x, t ) = w x ( x, t ) , lim ǫ → ddϕ J ǫ ( w ǫx ( x, t )) = Ω( x, t ) . x the number Ω( x, t ) belongs to ∂J ( w x ( x, t )) . Since thefunctions J ǫ are convex, we have the inequality Z π J ǫ ( w ǫx ( x, t ) + h x ( x )) − J ǫ ( w ǫx ( x, t )) dx ≥ Z π ddϕ J ǫ ( w ǫx ( x, t )) h x ( x ) dx, for each h ∈ C ∞ (0 , π ) . We know that w ǫ and ddϕ J ǫ ( w ǫx ( x, t )) have pointwise limits, which arebounded, hence after passing to limit our claim will follow, Z π J ( w x ( x, t ) + h x ( x )) − J ( w x ( x, t )) dx ≥ Z π Ω( x, t ) h x ( x ) dx. We finish the proof of Theorem 2.1. By previous Lemmas there exists a sequence Λ ǫ whichconverges weakly in W ( S × (0 , T )) . In particular, if h ∈ C ∞ (0 , π ) , t > and τ > is arbitrary,then we see Z t + τt − τ Z S Λ ǫt h dsdt ′ = Z t + τt − τ Z S ∂∂s ∂∂ϕ J ǫ (Λ ǫs + s ) h dsdt ′ = − Z t + τt − τ Z S ∂∂ϕ J ǫ (Λ ǫs + s ) h s dsdt ′ . Since ∂∂ϕ J ǫ (Λ ǫs + s ) is bounded, it converges weak- ∗ in L ∞ ((0 , π ) × (0 , T )) to Ω . We have to showthat Ω( s, t ) ∈ ∂J (Λ s + s ) . First we notice that we may pass to the limit in the above integral identity, Z t + τt − τ Z S Λ t ( s, t ′ ) h ( s ) dsdt ′ = − Z t + τt − τ Z S Ω( s, t ′ ) h s ( s ) dsdt ′ . By the Lebesgue differentiation theorem we deduce, Z S Λ t ( s, t ) h ( s ) ds = − Z S Ω( s, t ) h s ( s ) ds (3.7)for a.e. t ∈ [0 , T ] for h ∈ W ( S ) (we used the fact that is not distinguished on S ). In principle, theset G = { t ∈ [0 , T ] : (3.7) holds } depends upon h , i.e. G = G ( h ) . We shall see, that in fact wecan choose G independently of h . Let us recall that W ( S ) is separable and let us suppose that D isa dense, countable subset of W ( S ) . Of course, G = T ∞ h ∈ D G ( h ) is a set of full measure. Let us thentake t ∈ G and h ∈ C ∞ ( S ) . Let us suppose that { h n } is a sequence in C ∞ ( S ) converging to h in the W ( S ) -norm. Then, Z S Λ t ( s, t ) h n ( s ) ds = − Z S Ω( s, t )( h n ) s ( s ) ds for all t ∈ G . We may pass to the limit with n on both sides, thus we reach, Z S Λ t ( s, t ) h ( s ) ds = − Z S Ω( s, t ) h s ( s ) ds. In other words, (3.7) holds for all h ∈ C ∞ ( S ) and all t ∈ G .If we now fix t ∈ G , we next apply Proposition 3.2 to deduce that Ω( s, t ) ∈ ∂J (Λ s ( s, t ) + s ) .Hence the limit, Λ , is indeed a weak solution.Now, we are going to prove uniqueness. Theorem 3.1. If Λ i , i = 1 , are two solutions with Λ ( s, 0) = Λ ( s, , then Λ ( s, t ) = Λ ( s, t ) ,for t ≤ T . 12 roof. If Λ i , i = 1 , , are weak solutions, then by the definition of weak solutions we have Z S Λ it h ds = − Z S (Ω i − s ) h s ds + Z S h ds, where w i ∈ − ∂J and h is in H . We subtract these two identities for Λ and Λ , then we take (Λ − Λ ) as a the test function. Finally, the integration over (0 , ¯ t ) , ¯ t < T yields Z ¯ t Z S ddt (Λ − Λ ) dsdt = − Z ¯ t Z S (Ω − Ω )(Λ − Λ ) s dsdt. Monotonicity of ∂J implies that k Λ − Λ k L ( S ) (¯ t ) ≤ . Hence, k Λ − Λ k L ( S ) (¯ t ) = 0 for any ¯ t < T . It is well-known that important information about the studied system is provided by special solutions,like traveling waves, self-similar solutions and other symmetry solutions. We can not talk about self-similar solutions because our systems lacks direct geometrical interpretation, however we may lookfor special ones, which we named minimal solutions.In the theory of curvature flows it is natural to anticipate existence of curves such that their cur-vature is constant, but may change in time. Here, we ask if there exists such a solution ¯ ϕ to (1.2)that dds ∂J ( ¯ ϕ ) ∋ k, hence ∂J ( ¯ ϕ ) ∋ ks + s ∗ , (3.8)where s ∗ is appropriately chosen, e.g. s ∗ = π and | k | = 1 . The last restriction is of geometric nature,namely we want that for any a ∈ R the image of S by ∂J be contained in an interval no longer than π . In fact, we may come up with explicit formulas. One for k = 1 is provided by formula (2.2). It isthen obvious that ¯ ϕ ( s ) := ( ∂J ) − ( s + s ∗ ) , (3.9)as in [MRy] and in Section 2. Moreover, R π ¯ ϕ ( s ) ds = 2 π = R π s ds. By the reversal of theorientation, we immediately obtain the solution for k = − , ¯ ϕ − ( s ) = 7 π χ [0 , π ) ( s ) + 5 π χ [ π ,π ) ( s ) + 3 π χ [ π, π ) ( s ) + π χ [ π , π ) ( s ) . We choose ¯ ϕ ( s ) := ¯ ϕ ( s ) , which is given by (2.2), because we prefer to have ¯ ϕ an increasing function.As a result, ¯Λ defined by ¯Λ( s, t ) = R s ¯ ϕ ( u ) du + F ( t ) is indeed π periodic in s and it is a solutionto (1.2). Here, we must take F ( t ) = A + t . One can check in a straightforward manner that indeed ¯Λ solves (3.9). This is indeed so, because we have found ¯ ϕ ( s ) = Λ s ( s ) + s and Ω is a section of ∂I ( ¯ ϕ ) ,namely, Ω( s, t ) = s , which satisfies (3.9). If we take A = 0 , then ¯Λ satisfies the initial condition: ¯Λ( s, 0) = R s ¯ ϕ ( u ) du . In this part we examine the semi-discretization of (2.1). Our goals are not only to establish existencefor the presented scheme, but also to show qualitative properties of the obtained solutions. In partic-ular our considerations will explain the appearance of facets. Finally, we prove the convergence ofsolutions of the semi-discretization to the solutions obtained in Section 3.13e define the semi-discretization in time of system (2.1) as follows λ kh ( s ) − λ k − h ( s ) h ∈ dds ∂J [ λ kh,s ( s ) + s ] (4.1)and λ kh (0) = λ kh (2 π ) and ( λ h ) s = φ for k = 1 , . . . , [ T /h ] ; or equivalently equation (4.1) can bestated λ kh ( s ) − h dds ∂J [ λ kh,s ( s ) + s ] ∋ λ k − h ( s ) . (4.2)We establish existence of solution to this problem. Lemma 4.1. Let us suppose that an absolutely continuous function v is such that v s = ϕ ∈ T V [0 , π ) , then there exists u ∈ AC ([0 , π )) such that u s ∈ T V , which is a solutions to (4.1),i.e. u − v ∈ h dds ∂J ( u s ) (4.3)with u (0) + (2 π ) = u (2 π ) and the following bound isvalid || u s || T V ≤ || v s || T V . (4.4) Remark. Our understanding of (4.3) is the same as that of (2.1), i.e., there exists ω ∈ W ([0 , π )) ,such that ω ( x ) ∈ ∂J ( u s ) and u − v = h dds ω .We also note that u and v appearing in this Lemma need not be periodic, on the other hand Λ( · , t ) and λ kh ( · ) are periodic. Proof. Let us notice that if u is a solution to (4.3), then belongs to the subdifferential of the functional J ( u ) = Z π [ hJ ( u s ) + 12 ( u − v ) ] , for u ∈ AC ([0 , π ) , u s ∈ T V. i.e. u is a minimizer of J . To be precise, we define J on L ( S ) by the above formula for u ∈ AC ([0 , π with u s ∈ T V and we put J ( u ) = + ∞ for u belonging to the complement of this set.In order to solve (4.3), we consider a family of regularized problems, J ǫ ( u ) = Z π [ hJ ǫ ( u s ) + 12 ( u − v ) ] , where J ǫ is the same regularization of J that we used in (3.1).The functional J ǫ is well-defined, convex and coercive on the standard Sobolev space W (0 , π ) ,thus it possesses a unique minimizer u ǫ . Now, we apply again the methods used in Section 3.1 toshow existence of a weak solution of the evolution problem (1.2). The regularization of system (4.3)leads to the following equation u ǫss − d ds ( ∂J ǫ ∂ϕ ( u ǫs ) u ǫss ) = v ǫss . By repeating the argument for (3.4),we get || u ǫss || L ≤ || v ǫss || L . Passing to the limit with ǫ → yields (4.4).In addition we have the following bounds R π ( u ǫ ) dx ≤ M , R π ( u ǫx ) dx ≤ M . In order toprove them we follow the lines of reasoning of Corollary 3.1 and Lemma 3.4. These bounds suffice14o show existence of a subsequence { ǫ k } converging to zero, such that(a) u ǫ k ⇀ u in W (0 , π ) ; u ǫ k ss ⇀ u ss as measures.Subsequently, by Helly’s theorem we conclude existence of the pointwise limits (for another sub-sequence { ǫ k } , not relabeled) lim t →∞ ϕ ǫ ( x ) = u x ( x ) , lim t →∞ ddϕ J ǫ ( u ǫx ( x )) = Ω( x ) . Moreover, Ω( x ) ∈ ∂J ( u x ( x )) for each x ∈ [0 , π ) .Now, we show uniqueness of solutions, constructed in Lemma 4.1. Lemma 4.2. Let v ∈ AC ([0 , π )) , v s ∈ T V ([0 , π )) , then there exists at most one weak solution u ∈ AC ([0 , π )) , u s ∈ T V ( S ) to problem (4.3) Proof. Let us suppose that there are two solutions to (4.3), u i , i = 1 , . By the definition, thereare two functions ω i ∈ ∂J ( u is ) , i = 1 , , such that u i − v = h dds ω i , i = 1 , . After subtracting these two equations and multiplying them by u − u and integrating over [0 , π ) we see k u − u k − Z π h ( dds ω − dds ω )( u − u ) ds = 0 . The integration by parts leads us to k u − u k + Z π h ( ω − ω )( u s − u s ) ds ≥ k u − u k ≥ . As a result u = u .In order to finish our preparations, we introduce the sets of preferred orientation which dominatethe behavior of solutions. Let us suppose, that w is absolutely continuous and w s ∈ T V , then at anypoint s , the left derivative w − s , as well as the right derivative w + s are well-defined, hence we may set ∂w ( s ) = { τ w − s + (1 − τ ) w + s : τ ∈ [0 , } . (4.5)If w is convex, then ∂w is the well-known subdifferential of w .Now, for each l = 0 , , , , we set Ξ l ( w s ) = { s ∈ [0 , π ] : w is differentiable at s and w s ( s ) = α k or α k ∈ ∂w ( s ) } (4.6)Furthermore, we set Ξ( w s ) = S l =0 Ξ l ( w s ) .The result, delivering the main properties of solutions, is the following. Theorem 4.1. Let φ = λ h,s + s ∈ J - R[0 , π ) . Then a solution { λ hk } to problem (4.1) exists, it isunique and itsatisfies the following bound || λ hk,s + s || J - R[0 , π ) ≤ || λ ,s + s || J - R[0 , π ) . (4.7)Moreover, wehave Ξ( λ hk − ,s + s ) ⊂ Ξ( λ hk,s + s ) and K ( λ kh,s + s ) ≤ K ( λ k − h,s + s ) (4.8)15nd sup k sup l =0 , , , | Ξ kl \ Ξ k − l | ≤ C ( V ( h ) + h / ) . (4.9)where V ( s ) → as s → and V is determined by the initial datum φ . Moreover, on connectedcomponents of theset Ξ k − \ ( S l =0 Ξ kl \ Ξ k − l ) dds ∂J [ λ kh,s + s ] is constant. (4.10) Proof. By Lemmas 4.1 and 4.2 we conclude existence of the sequence of solutions to the semi-discretization, the solutions are such that λ hs belong to T V ( S ) . It is enough to restate the equation(4.2) as follows: u − h dds ∂J [ u s ] = v (4.11)with u = λ kh + s and v = λ k − h + s , and boundary condition u (0) + 2 π = u (2 π ) .The set, where function J [ u s ] is singular, i.e. Ξ( u s ) , plays the key role. Our first task is to provethe inclusion from (4.8). Note that in a neighborhood of any point s / ∈ Ξ( u s ) function ∂J [ u s ( · )] isconstant, hence we get u ( s ) = v ( s ) . Thus, we point the first feature of solutions to (4.11) u ( s ) = v ( s ) for s ∈ (0 , π ) \ Ξ( u s ) . (4.12)From (4.12) we deduce that if s / ∈ Ξ( v s ) , then s / ∈ Ξ( u s ) . Subsequently, we get Ξ( v s ) ⊂ Ξ( u s ) which proves the inclusion from (4.8). Thus, the isolated elements stay isolated or merge with otherelements. From this we obtain that K ( v s ) ≥ K ( u s ) what ends the proof of line (4.8).By properties (4.12), (4.8) and the estimate from Lemma (4.1), we immediately deduce estimate(4.7). In particular, what we gain is a uniform bound in L ∞ ( S ) on { λ kh,s } .The set Ξ( u s ) is defined as the sum of S l =0 Ξ l ( u s ) , thus without loss of generality we can con-centrate our attention on one of them, e.g. on the set Ξ ( u s ) – see (4.6). From the J - R -regularity of u s set Ξ ( u s ) is a sum of closed intervals, so we take one of them, say, [ a − , a + ] ⊂ Ξ ( u s ) and u s | ( a − ,a + ) = π . (4.13)Recalling the required regularity of the functions in the J - R -class, we find ǫ > such that one of thefour following possibilities holds: ( i ) u s | ( a − − ǫ,a − ) > π , u s | ( a + ,a + + ǫ ) < π , ( ii ) u s | ( a − − ǫ,a − ) < π , u s | ( a + ,a + + ǫ ) > π , ( iii ) u s | ( a − − ǫ,a − ) > π , u s | ( a + ,a + + ǫ ) > π , ( iv ) u s | ( a − − ǫ,a − ) < π , u s | ( a + ,a + + ǫ ) < π . (4.14)Subsequently, we integrate (4.11) over ( a − − ǫ, a + + ǫ ) to get Z a + + ǫa − − ǫ uds − h ( ∂J [ u s ] | a + + ǫa − − ǫ ) = Z a + + ǫa − − ǫ vds. (4.15)After passing with ǫ → + , we obtain – according to the above four cases (4.11) – the followingidentities ( i ) R a + a − uds − h π = R a + a − vds (convexity), ( ii ) R a + a − uds + h π = R a + a − vds (concavity), ( iii ) and ( iv ) R a + a − uds = R a + a − vds (monotonicity). (4.16)16n our present analysis, we essentially use the fact that the energy density function J is definedby a square. Due to the definition of J , see (1.4), formula (4.16) exhausts all the possibilities of thebehavior of u s . For more complex polygons, we would have to discuss more possible types of facets– here, there are just four of them.We keep considering the interval [ a − , a + ] ⊂ Ξ ( u s ) , see (4.13). Let us introduce a set Π = ([ a − , a + ] ∩ Ξ( v s )) \ (Ξ ( u s ) ∪ Ξ ( u s ) ∪ Ξ ( u s )) , (4.17)then by the properties of sets Ξ , we deduce that ( u − v ) | Π = C h is constant . (4.18)The sign of constant C h is determined by the geometrical properties of cases in (4.16). We have C h > for ( i ) , C h < for ( ii ) and C h = 0 for ( iii ) and ( iv ) . (4.19)Also identity (4.18) and equation (4.11) yield dds ∂J [ u s ] (cid:12)(cid:12)(cid:12)(cid:12) Π = C h h and dds ∂J [ u s ] (cid:12)(cid:12)(cid:12)(cid:12) (0 , π ) \ Ξ( u s ) = 0 . (4.20)Thus, we proved (4.10).Next, we are going to study (4.9). From the analysis of (4.11), we conclude that || u − v || L ( S ) ≤ h π K ( φ ) . (4.21)Additionally, from (4.8) we have also that u, v ∈ W ∞ ( S ) , thus simple considerations lead us to thefollowing bound || u − v || L ∞ ( S ) ≤ h / C ( φ ) . (4.22)In order to measure the set Ξ l ( u s ) \ Ξ l ( v s ) we split it into two parts Ξ l ( u s ) \ Ξ l ( v s ) = [(Ξ l ( u s ) \ Ξ l ( v s )) ∩ Ξ( v s )] ∪ (Ξ l ( u s ) \ Ξ( v s )) = Π ∪ Π . (4.23)Let us consider Π . On this set we watch the evolution of the intersection of facets. Thanks to thefull information about the direction of this facet, we deduce immediately that | Π | ≤ C ( φ ) h / (4.24)The number of possible intersections is controlled by K ( φ ) .To estimate Π , let us note that this set is a subset of Ξ( λ ,s + s ) , thus in the general case we cansay only | Π | ≤ V ( h ) , (4.25)where V ( s ) → as s → and V is determined by the initial datum. Assuming strict convexity ofinitial domain we would obtain V ( h ) ∼ h / – see the example at the end of subsection 5.2.Theorem 4.1 is proved.Next, we show that sequences { λ kh } converge to solutions of the original problem. We will com-pare solutions given by Theorem 2.1 and Lemma 4.1, in particular, all assumptions of Theorem 2.1are not required. We follow the standard procedure which is valid for parabolic operators (see [MRa]).Our next task is to show the following lemma.17 emma 4.3. Let Λ and { λ kk } besolutions to problems (2.1)and (4.1)respectively, then || Λ( s, t ) − [ T/h ] X k =0 λ kh ( s ) χ [ k,k +1) ( t ) || L (0 ,T ; L ( S )) → as h → + . (4.26)Ifthe initial datum fulfills the assumptions of Theorem 2.1,i.e. Λ ,s ∈ T V ( S ) ,then || Λ( s, t ) − [ T/h ] X k =0 λ kh ( s ) χ [ k,k +1) ( t ) || L p (0 ,T ; W − ǫ ( S )) → as h → + (4.27)for any < p < ∞ and ǫ > . Proof. From the properties of solutions to problem (2.1), we know that Λ t ∈ L (0 , T ; L ( S )) . Itfollows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Λ( s, t ) − Λ( s, t − h ) h − Λ t ( s, t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( h,T ; L ( S )) → as h → + . (4.28)For fixed h > we denote R h ( s, t ) = Λ( s, t ) − Λ( s, t − h ) h − Λ t ( s, t ) , (4.29)then the equation (2.1) can be restated as follows Z S Λ( s, t ) − Λ( s, t − h ) h πds = − Z S Ω( s, t ) π s + R h ( s, t ) πds (4.30)for each π in C ∞ ( S × (0 , T )) and each selection Ω( s, t ) of multivalued function ∂J [Λ s ( s, t ) + s ] .We want to compare the above system with the semi-discretization given in Section 4. Z S λ kh ( s ) − λ k − ( s ) h πds = − Z S ω ( s, t ) πds (4.31)where t ∈ [ kh, ( k + 1) h ) and ω ( s, t ) is any section of ∂J [ λ kh,s ( s ) + s ] .Let us define A k ( s, t ) = Λ( s, t ) − λ kh ( s, t ) , (4.32)provided t ∈ [ kh, ( k + 1) h ) , then from (4.30) and (4.31) we deduce Z S A k ( s, t ) − A k − ( s, t − h ) h πds = − Z S { Ω( s, t ) − ω ( s, t ) } π s + R h ( s, t ) πds. (4.33)Taking in (4.33) as a test function A k ( t, s ) , we get R (0 , π ) | A k ( s, t ) | ds = R (0 , π ) A k ( s, t ) A k − ( s, t − h ) ds − h R (0 , π ) (Ω( s, t ) − ω ( s, t )) (cid:16) Λ s ( s, t ) − λ kh,s ( s, t ) (cid:17) ds + h R (0 , π ) R h ( s, t ) A k ( s, t ) ds, (4.34)18ut the monotonicity of ∂J implies Z (0 , π ) (Ω( s, t ) − ω ( s, t )) (cid:16) Λ s ( s, t ) − λ ks ( s, t ) (cid:17) ds ≥ . (4.35)So, defining α k ( t ) = || A k ( · , t ) || L ( S ) , by the Schwarz inequality, we get from (4.29) the followinginequality α k ( t ) ≤ α k − ( t − h ) + hr kh ( t ) for t ∈ [ kh, ( k + 1) h ) , (4.36)where r kh ( t ) = || R h ( · , t ) || L (0 , π ) . Thus (4.36) yields α k ( t ) ≤ α ( t − kh ) + k X l =1 hr lh ( t − ( k − l ) h ) with t ∈ [ kh, ( k + 1) h ) . (4.37)Integrating (4.37) over t ∈ [ kh, ( k + 1) h ) , we get Z ( k +1) hkh α k ( t ) dt ≤ Z h α ( τ ) dτ + h Z T || R h ( · , t ) || L (0 , π ) dt (4.38)for T > ( k + 1) h . Introducing function ˜ α ( t ) = P Ll =0 α l ( t ) χ [ lh, ( l +1) h ) ( t ) with L = [ T /h ] , from(4.38) we get Z T ˜ α ( t ) dt ≤ h / T || Λ t || L (0 ,T ; L (0 , π )) + T || R h || L (0 ,T ; L (0 , π )) , (4.39)because the first term of the right-hand-side (RHS) of (4.39) is a consequence of the following estimate h Z h α ( t ) dt ≤ h Z h || Z t Λ t ( · , τ ) dτ || L (0 , π ) dt ≤ h ( Z h t dt ) / ( Z h || Λ t || L (0 , π ) dt ) / ≤ Ch / || Λ t || L (0 ,T ; L (0 , π )) . (4.40)Hence, from (4.39) and (4.28) we conclude || ˜ α h || L (0 ,T ; L (0 , π )) → as h → + and we get(4.26). From the interpolation estimates and the results of Theorem 2.1, we deduce that for any p < ∞ and ǫ > the convergence (4.27) is valid. Lemma 4.3 is proved. The semi-discretization process proves that the set Ξ( u kh,s ) grows with k . One can show that theset S t ≥ Ξ( u h,s ( · , t )) may be estimated from below to show that it survives the limiting process as h → . This may be achieved by the analysis of the semi-discretization procedure, but this seemstedious. We propose an alternative approach by the construction of an explicit solution to (1.2) fordata in ϕ ∈ J - R . By uniqueness result, see Theorem 2.1, this is the solution.We shall assume in this Section that ϕ ≡ w s belongs to J - R and this is the case for the initial data ϕ of system (2.1). As a result of the definition of the J - R class we see Ξ( ϕ ) = [ l =0 Ξ l ( ϕ ) = N [ k =1 [ ξ − k , ξ + k ] , (5.1)19here ξ − k ≤ ξ + k and ξ + k ≤ ξ − k +1 , k = 1 , . . . , N , (5.2)(with the understanding ξ − N +1 = ξ − + 2 π ). Moreover, each interval [ ξ − k , ξ + k ] is a connected com-ponent of one of the sets Ξ l ( ϕ ) , l = 0 , , , . We shall also adopt the convention that ≤ ξ − andpossibly ξ + N > π , but ξ + N − π ≤ ξ − .If [ ξ − k , ξ + k ] is one of the connected components of Ξ l ( ϕ ) , then we will call by a facet the set F = F k ( ξ − k , ξ + k ) = { ( x, y ) ∈ R : y = w ( x ) , x ∈ [ ξ − k , ξ + k ] } . The interval [ ξ − k , ξ + k ] will be calledthe pre-image of facet F k . Let us stress that we admit ξ − k = ξ + k , i.e. a facet degenerated to a point aswell as ξ + k − = ξ − k , i.e. we expect interaction of facets. We shall see that the generic initial data leadto the facet creation (from the degenerate ones) and their interaction. We show that facets are formedinstantaneously from the data. At this point we mention that creation of interacting facets leads toadditional difficulties and this process is handled separately.We will come up with an explicit formula. Once we check that indeed this formula yields asolution to equation (2.1), we will be assured that this is the unique solution we seek. Subsequently,we shall see that solutions get convexified, i.e. after some finite time the angle becomes increasing,hence w becomes convex. Finally, we study interaction of facets. We will prove that w ( · , t ) becomesa minimal solution at the limit time.It will be also convenient to say that a facet F k ( ξ − k , ξ + k ) , has zero curvature , if [ ξ − k , ξ + k ] is aconnected component of Ξ( ϕ ) and there exists an open interval ( A, B ) , containing [ ξ − k , ξ + k ] suchthat w s is not monotone on any interval ( a, b ) , satisfying [ ξ − k , ξ + k ] ⊂ ( a, b ) ⊂ ( A, B ) . Furthermore, we say that a facet F k = F k ( ξ − k , ξ + k ) is regular if ξ − k < ξ + k . Otherwise, we say that F k is degenerate . If w s ∈ J - R is such that the graph of w contains degenerate facets, then we say thatfacets are created in solutions to (2.1).Finally, we say that facets F l , . . . , F l + r for r > , interact (or are interacting ) if F k ∩ F k +1 , k = l, . . . , l + r − , is a singleton. We call a single facet F k non-interacting , if it is not true that itinteracts with any other facet.Thus, we have the total of eight combinations, we will treat each case separately. We are going to establish that solutions to equation (2.1) enjoy the expected comparison principle.This result is interesting for its own sake but also it is a useful tool analysis. We will apply it to showcreation of interacting facets.We first recall the basic result (see, e.g. [S]). Proposition 5.1. Letus suppose that u , u are smooth solutions to astrongly parabolic equation u t = ( a ( x, u x )) x in S × (0 , T ) and u ( x, ≥ u ( x, , then u ( x, t ) ≥ u ( x, t ) forall t ∈ (0 , T ) .With this result we may deduce the following comparison principle. Proposition 5.2. Let us suppose that Λ , Λ are weak solutions to (2.1) and Λ ( x, ≤ Λ ( x, ,then Λ ( x, t ) ≥ Λ ( x, t ) for all t ∈ (0 , T ) . 20 roof. Since Λ ( x, ≤ Λ ( x, , we deduce that Λ ǫ ( x, ≤ Λ ǫ ( x, , where Λ ǫi , i = 1 , aresolutions to the regularized system (3.2). Application of the preceding result yields Λ ǫ ( x, t ) ≤ Λ ǫ ( x, t ) . Since the point-wise limit exists we conclude that our proposition holds.We stress that no information about Ω i , i = 1 , is needed in the proof of the above result. We shall see below that the evolution of a facet F k separated from other facets is governed by an ODEfor its end-points, see (5.14) below. In the case of interacting facets their evolution is described by asystem of ODE’s (5.19).As we mentioned we admit facets F k degenerated to a single point at the initial instance t = 0 .In this case the single ODE (5.11) and system ODE (5.19) become singular. While we can resolvesatisfactorily the singularity of the single ODE, the analysis of the system is more difficult. In fact,we circumvent this problem by using the comparison principle to show creation of interacting facets.We shall use the notions and notation introduced above. In addition, in order to facilitate ourconstruction we shall write x α k ( x − s k ) + τ k =: l k ( x, s k , τ k ) , where α k ∈ A , s k ∈ Ξ( ϕ ) , τ k ∈ R . Theorem 5.1. Let us assume that ϕ = w ,s ∈ J - R and w is the unique solution to (2.1). We alsoassume that the set Ξ( w ,s ) = S N k =1 [ ξ − k , ξ + k ] fulfills conditions (5.1) and (5.2). Then, there exists afinite sequence of time instances ≤ t < t < . . . < t M < ∞ and a finite sequence of continuousfunctions ξ ± k : [ t i , t i +1 ] → R , i = 0 , . . . t M − , k = 1 , . . . N i ,ξ ± k : [ t M , ∞ ) → R , k = 1 , . . . N M = 4 , where N ≥ N ≥ . . . ≥ N M = 4 .Thefunctions ξ − k ( · ) , ξ + k ( · ) satisfying (5.2)have the following properties:(a) ξ ± k (0) = ξ ± k ;(b) ≤ ξ − ( t ) ≤ ξ +1 ( t ) ≤ ξ − ( t ) ≤ . . . ≤ ξ − N ( t ) ≤ ξ + N ( t ) ≤ ξ − + 2 π , t ∈ [ t i , t i +1 ) ;(c) for t ∈ [ t i , t i +1 ) we have Ξ( ϕ ( · , t )) = N i [ k =1 [ ξ − k ( t ) , ξ + k ( t )] , and each interval [ ξ − k ( t ) , ξ + k ( t )] is aconnected component ofone of the sets Ξ l ( ϕ ( · , t )) , l = 0 , , , .There exist functions τ k : [ t i , t i +1 ) → R , i = 0 , . . . t M , k = 1 , . . . N i , and t M +1 = ∞ . Theyare such that the unique solution to (2.1) with initial data ϕ ( x, 0) = ϕ ( x ) is given by the followingformula for t ∈ [ t i , t i +1 ) , i = 0 , . . . , Mw ( x, t ) = (cid:26) w ( x ) if x ∈ [0 , π ) \ S N i k =1 [ ξ − k ( t ) , ξ + k ( t )] l k ( x, ξ + k ( t i ) , τ k ( t )) + w ( ξ + k ( t i ) , t i ) if x ∈ [ ξ − k ( t ) , ξ + k ( t )] , k = 1 , . . . , N i (5.3)Moreover, w x ( · , t ) is well-defined a.e., ∂w defined by(4.5)belongs to J - R and k ∂w ( · , t ) k J - R ≤ k ∂w k J - R . 21n addition, ateach timeinstant t i , i = 0 , . . . , M ,one of the following happens:(i) Oneormore zero-curvature facets disappear, i.e. ifone facet disappears at t i ,then ξ + k − ( t ) ≤ ξ − k ( t ) < ξ + k ( t ) ≤ ξ − k +1 ( t ) , for t i < t < t i +1 and lim t → t − i +1 ξ − k − ( t ) = ξ − l ( t i +1 ) , lim t → t − i +1 ξ − k +1 ( t ) = ξ + l ( t i +1 ) , where [ ξ − l ( t i +1 ) , ξ + l ( t i +1 )] is a subset of a connected component of Ξ l ( ϕ ( t i +1 )) , as a result N i +1 Let us suppose that w s = ϕ ∈ J - R and [ ξ − k , ξ + k ] is a connected component of Ξ l ( ϕ ) and s k isits member. Weassume that F k ( ξ − k , ξ + k ) isnot azero curvature facet.(a) If ξ + k < ξ − k +1 ,then for sufficiently small τ k ofaproper sign, there exist ξ ± k ( τ k ) such that w ( ξ ± k ( τ k )) = l k ( ξ ± k ( τ k ) , ξ ± k , τ k ) + w ( ξ ± k ) and ξ ± k (0) = ξ ± k . (5.5)Moreover, the functions τ k ξ ± k ( τ k ) are Lipschitz continuous, provided that w s ( ξ ± k ) = α k . Other-wise, ξ ± k ( τ k ) are locally Lipschitz continuous. Inaddition, dξ + k dτ k ( τ k ) = 1 w s ( ξ + k ) − α k , dξ − k dτ k ( τ k ) = 1 w s ( ξ − k ) − α k for a.e. | τ k | ∈ [0 , ǫ ) . (5.6)(b) If ξ + l < ξ − l +1 ≤ ξ + l +1 = ξ − l +2 ≤ ξ + l +2 = ξ − l +3 . . . ≤ ξ + l + r < ξ − l + r +1 , (in particular we admit ξ − = ξ + N − π ),then w ( ξ + k − ) + l k − ( ξ + k − ( τ k − , τ k )) , ξ + k − , τ k − ) = w ( ξ + k ) + l k ( ξ + k ( τ k , τ k +1 )) , ξ + k , τ k ) (5.7)for k = l + 1 , . . . , l + r . Moreover,thefunctions ( τ k , τ k +1 ) ξ ± k ( τ k , τ k +1 ) , k = l + 1 , . . . , l + r − are Lipschitz continuous. Proof. Before proceeding to the formal proof we will explain the situation by drawing a picture (wherethe subscript k is suppressed). The graph of w ( · ) and the line containing F ( ξ − , ξ + ) moved verticallyby τ intersect at x = ξ − ( τ ) and at x = ξ + ( τ ) .(a) Since F k is not of zero curvature then by the fact that ϕ ∈ J - R it follows that w in a neigh-borhood of [ ξ − k , ξ + k ] is either convex or concave. Let us consider the case of w being convex on22 a, b ) ⊃ [ ξ − k , ξ + k ] , the other case is similar. By convexity, any chord is above the graph of w . Thus,the line l k ( · , ξ + k , τ k ) + w ( ξ + k ) for sufficiently small τ k > intersects the graph of w at exactly twopoints, i.e. for τ k > equation (5.5) has exactly two solutions. One of them, which is greater than ξ + k is called ξ + k ( τ k ) , the other one, smaller than ξ − k is dubbed ξ − k ( τ k ) . The function x w ( x ) − l k ( x, ξ + k , τ k ) − w ( ξ + k ) =: F + k ( x ) (5.8)is increasing for x ∈ [ ξ + k , b ) and this interval is maximal with this property, while the function x w ( x ) − l k ( x, ξ − k , τ k ) − w ( ξ − k ) =: F − k ( x ) (5.9)and decreasing for x ∈ ( a, ξ − k ] and again this interval is maximal with this property. One can see thisby taking the derivative of (5.8) and (5.9), because we have ddx ( w ( x ) − l k ( x, ξ + k , τ k )) = w ′ ( x ) − α k ≥ w ′ ( ξ + k ) − α k > for a.e. x ∈ [ ξ + k , b ) and ddx ( w ( x ) − l k ( x, ξ − k , τ k )) = w ′ ( x ) − α k ≤ w ′ ( ξ − k ) − α k < for a.e. x ∈ ( a, ξ − k ] . Thus, the function [ ξ + k , b ) ∈ x F + k ( x ) (resp. ( a, ξ − k ] F − k ( x ) has a continuous inverse. As aresult, for any τ k belonging to [0 , δ ) ⊂ F + k ([ ξ + k , b )) ∩ F − k (( a, ξ − k ]) , δ > , we may set ξ + k ( τ k ) =( F + k ) − ( τ k ) and ξ − k ( τ k ) = ( F − k ) − ( τ k ) . Moreover, dξ ± k dτ k ( τ k ) = 1 w s ( ξ ± k ( τ k )) − α k , a.e. This formula combined with monotonicity of w s yields, α k − w − s ( b ) ≤ dξ + k dτ k ( τ k ) ≤ w + s ( ξ + k ( τ k )) − α k , w + s ( a ) − α k ≤ (cid:12)(cid:12)(cid:12)(cid:12) dξ − k dτ k ( τ k ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ w − s ( ξ − k ( τ k )) − α k (5.10)for a.e τ k . If w s ( ξ + k ) = α k , then it follows that ξ + k ( · ) is Lipschitz continuous on [0 , ǫ ] , for some ǫ > .A similar statement is valid for ξ − k ( · ) .(b) Functions ξ + l ( τ l , τ l +1 ) , . . . , ξ + l + r − ( τ l + r − , τ l + r ) are defined as unique solutions to the decou-pled system of linear equations (5.7) for any given τ l , . . . , τ l + r . This is indeed possible because α k = α k +1 . The solution ξ + k depends linearly upon τ k , τ k +1 . Subsequently, we set ξ − k +1 := ξ + k ( τ k , τ k +1 ) , k = l, . . . , l + r − . Remark. In the case (a) the derivatives ddτ k ξ ± k are never zero. They may converge to infinity at t = t i ,as well as at t = t i + t ∗ , if at that time instance w s ( ξ ± k ) = α k .The lemma above expressed the evolution of the pre-images of facets in terms of τ k , i.e. theamount of vertical shift of the line l k ( · , ξ − , w ( ξ − )) . However, in order to render (5.3) meaningful, wehave to figure out the time dependence of τ k . At the same time we have to construct Ω . We begin withan explicit case. 23 emma 5.2. Let us suppose that F k ( ξ − k , ξ + k ) is neither of zero-curvature nor interacting and it maybedegenerate. Thenthere exist Ω − k , Ω + k ∈ ∂J ( α k ) andaunique solution τ k : [ t ∗ , t ∗ + T max ) → R tothe equation dτ k dt = Ω + k − Ω − k ξ + k ( τ k ) − ξ − k ( τ k ) , τ k ( t ∗ ) = 0 . (5.11)Theyare such that the function Ω( x, t ) = Ω + k − Ω − k ξ + k ( t ) − ξ − k ( t ) ( x − ξ − k ( t )) + Ω − k (5.12)and w defined by(5.3)satisfy ∂w∂t ( s, t ) = ∂ Ω ∂s ( s, t ) for t ∈ [ t ∗ , t ∗ + T max ) , s ∈ ( ξ − k ( τ k ( t )) , ξ + k ( τ k ( t )) . Proof. The non-interaction assumption implies that α k − ≡ α k − ∆ α < w − ,s ( ξ − ) ≤ α k ≤ w +0 ,s ( ξ + ) < α k +1 ≡ α k + ∆ α or α k − > w − ,s ( ξ − ) ≥ α k ≥ w +0 ,s ( ξ + ) > α k +1 . Keeping this in mind we set Ω + k = lim x → ( ξ + k ( t )) + ∂J∂ϕ ( w ,x ( x )) Ω − k = lim x → ( ξ − k ( t )) − ∂J∂ϕ ( w ,x ( x )) . (5.13)Of course Ω − k , Ω + k ∈ ∂J ( α k ) . We notice that both quantities are well-defined for regular as well asdegenerate facets.Now, we turn our attention to equation (5.11), we notice that this equation states that the timederivative of τ k equals the slope of the straight line passing trough the points ( ξ − k , Ω( ξ − k )) and ( ξ + k , Ω( ξ + k )) . This line provides a section of ∂J , necessary to construct solutions to (2.1).The numerator of (5.11) is constant and if ξ + k ( · ) , ξ − k ( · ) are Lipschitz continuous and ξ + k ( τ k ) >ξ − k ( τ k ) for all the values of τ k , then (5.11) has a unique solution. If however, ξ + k (0) = ξ − k (0) ,then (5.11) is singular and this equation requires special attention. A similar situation arises when w s ( ξ ± k ) = α k . Fortunately, due to a simple structure of (5.11) we may resolve these issues.The ODE (5.11) governing the behavior of a non-interacting facet F k is obtained by taking thetime derivative of (5.5), ddt ( w ( ξ + k ( t )) − α k ξ + k ( t )) = ddt τ k ( t ) , ddt ( w ( ξ − k ( t )) − α k ξ − k ( t )) = ddt τ k ( t ) . (5.14)In reality, we do not assume that w is differentiable everywhere, but its one-sided derivatives do existat each point. Due to monotonicity of ξ ± k the one-sided derivatives suffice in the formula above.By the definition of ξ ± k we rewrite (5.11) as follows (( F + k ) − ( τ k ) − ( F − k ) − ( τ k )) dτ k dt = ∆Ω k . Here, due to the definition of J and (5.13), we have ∆Ω = ∆Ω k = Ω + − Ω − = π . ξ + k ( τ k ) > ξ − k ( τ k ) as long as τ k = 0 , then we deduce that G , the primitive function of ( F + k ) − ( τ k ) − ( F − k ) − ( τ k ) such that G (0) = 0 , is strictly increasing. Thus (5.11) takes the form ddt ( G ( τ k )) = ∆Ω or G ( τ k ) = ∆Ω t . As a result function τ k is given uniquely by the formula τ k ( t ) = G − (∆Ω t ) and τ k (0) = 0 .If we now set Ω by formula (5.12), then by the convexity of the set ∂J ( α k ) , we conclude that Ω( x, t ) ∈ ∂J ( α k ) . Moreover, for w defined by (5.3), the following equality holds by the definition of Ω and τ k , ∂w∂t ( x, t ) = dτ k dt = Ω + k − Ω − k ξ + k ( t ) − ξ − k ( t ) = ∂ Ω ∂x ( x, t ) for t ∈ [ t ∗ , t ∗ + T max ) , x ∈ ( ξ − k ( t ) , ξ + k ( t )) .We note that Ω , which we so far constructed, belongs to W ([0 , π )) for each t > t ∗ , if howeverthe facet does not degenerate, then Ω( · , t ∗ ) ∈ W ([0 , π )) too.We can infer the following observation from Lemma 5.1 and (5.11). Corollary 5.1. Letussupposethat w s isincreasing(resp. decreasing)inaneighborhood ofthepre-image [ ξ − k , ξ + k ] ofanon-interacting facet. Then,thereexistsapositive δ ,suchthatfor t ∈ [ t k , t k + T ) :(a) if ξ + k < ξ − k +1 ,then ddt ξ + k ( τ k ( t )) ≥ δ > a.e. (resp. ddt ξ + k ( τ k ( t )) ≤ δ < a.e.).(b) if ξ + k − < ξ − k ,then ddt ξ − k ( τ k ( t )) ≤ δ < a.e. (resp. ddt ξ − k ( τ k ( t )) ≥ δ > a.e.). Proof. The chain formula yields ddt ξ + k = dξ + k dτ k dτ k dt a.e. In the case (a), by the geometry of the problem,we deduce that dξ + k dτ k > (see (5.6)) as well as dτ k dt > (see (5.11)). Moreover, formulas (5.6) and(5.11) imply that none of the factors may vanish, in fact they are separated from zero.The remaining cases are handled in the same way.We shall state a result corresponding to Lemma 5.2 for a set of interacting facets. It will besomewhat more tedious. Lemma 5.3. Let us suppose that non-degenerate facets F l , ..., F l + r , r > interact, while ξ + l − <ξ − l and ξ + l + r < ξ − l + r +1 . Then, there exist continuous functions ξ ± k : [ t ∗ , t ∗ + T ) → R , k = l, ..., l + r ,suchthattheyarelocallyLipschitz continuous on ( t ∗ , t ∗ + T ) satisfying (5.19)belowandthereare C functions τ k : [ t ∗ , t ∗ + T ) → R , k = l, ..., l + r ,and Ω( · , t ) ∈ W ( ξ − l ( t ) , ξ + l + r ( t )) . Theyareall such that w defined by(5.3)satisfies ∂w∂t ( s, t ) = ∂ Ω ∂s ( s, t ) for t ∈ [ t ∗ , t ∗ + T max ) , s ∈ ( ξ − l ( t ) , ξ + l + r ( t )) . (5.15) Remark. The above Lemma includes the case when the set S \ Ξ( w ,s ) consists of a single component. Proof. By our assumption the pairs of facets F l − , F l and F l + r , F l + r +1 do not interact. Thus, theevolution of the end points ξ − l and ξ + l + r is determined as for a single non-interacting facet. This remain25pplicable, unless Ξ( w s ( · , t i )) = [0 , π ) . We proceed as in Lemma 5.2, but we have to determine ξ ± l + i , τ l + i , i = 1 , . . . , r and Ω simultaneously. We keep in mind that ξ − l + i = ξ + l + i − , i = 1 , . . . , r . In orderto obtain their time evolution, we differentiate (5.7) with respect to time. This yields, α k ˙ ξ + k + ˙ τ k = α k +1 ˙ ξ + k +1 + ˙ τ k +1 . (5.16)The equation for τ k should be similar to (5.11), if so we have to select Ω ± l + i , i = 1 , . . . , r . We define Ω − l and Ω + l + r as in (5.13), i.e. Ω + l + r = lim x → ( ξ + l + r ( t )) + ∂J∂ϕ ( w ,x ( x )) , Ω − l = lim x → ( ξ − k ( t )) − ∂J∂ϕ ( w ,x ( x )) . (5.17)We have to define the remaining Ω ± k ’s while keeping in mind Ω + k = Ω − k +1 . By the properties of deriva-tive ∂J∂ϕ ( ϕ ) and the subdifferential ∂J ( α k ) the number Ω − l is one endpoint of the interval ∂J ( α k ) , thuswe inductively define Ω + k as follows, Ω + k +1 = (cid:26) Ω + k , if the facet F k has zero curvaturethe other endpoint of the interval ∂J ( α k ) , otherwise.We have to check that Ω + l + r defined in this way agrees with (5.17) . We prove this by induction withrespect to r , the number of interacting facets. If r = 1 , then the claim follows from the precedingconsiderations. Let us suppose validity of the claim for some r ≥ , we will show it for r + 1 . Letus suppose that w corresponds to a group of r + 1 interacting facets satisfying the assumptions ofthe Lemma. We consider such a mollification w ǫ of w in a neighborhood of ξ + l + r = ξ − l + r +1 that w ǫ = w for x satisfying | x − ξ + l + r | ≥ ǫ and w ǫ is smooth. Moreover, we require that w ,s and w ǫ ,s are simultaneously increasing or decreasing. Thus the facets corresponding to w ǫ are F l , . . . , ˜ F l + r , ˜ F l + r +1 . We notice that facet ˜ F l + r + i is of zero curvature iff facet F l + r + i is of zero curvature, i = 0 , .Moreover, facets ˜ F l + r , ˜ F l + r +1 do not interact. By the inductive assumption ˜Ω + r + l = Ω + r + l is equalto Ω − r + l +1 . At the same time ˜Ω + r + l = ˜Ω − r + l +1 is determined from ˜Ω + r + l +1 = Ω + r + l +1 and w ,s as inLemma 5.2. The two ways of course coincide, due to formulae (5.13). Our claim follows.We now write equations for τ k , k = l , . . . , l + r , they are as (5.11), dτ k dt = Ω + k − Ω − k ξ + k ( τ k ) − ξ − k ( τ k ) , τ k ( t ∗ ) = 0 for k = l, . . . , l + r. (5.18)Since we do not admit degenerate facets, these equations are not singular. We combine them with(5.13) and after writing η = ( ξ + l , . . . , ξ + l + r − ) , we arrive at A ˙ η = B ( η ) , (5.19)where A = α l − α l +1 . . . α l +1 − α l +2 . . . . . . ... − α l + r − . . . α l + r − ,B ( η ) k = − Ω + k − Ω − k η + k − η + k − + Ω + k +1 − Ω − k +1 η + k +1 − η + k , k = l + 1 , . . . , l + r − , ( η ) l + r − = α l + r ddt ξ + l + r − Ω + l + r − − Ω − l + r − η + l + r − − η + l + r − + Ω + l + r − Ω − l + r ξ + l + r − η + l + r − . Under our assumptions, there is a separate equation for ddt ξ + l + r i.e. (5.11). Due to the assumption ofabsence of degenerate interacting facets, this system is uniquely solvable on [ t ∗ , t ∗ + T ) .We have to define Ω , it will be a continuous piece-wise linear function, Ω( x, t ) = Ω + l + i − Ω − l + i ξ + l + i ( t ) − ξ − l + i ( t ) ( x − ξ − l + i ( t )) + Ω − l + i . (5.20)Moreover, w and Ω satisfy (5.15).We claim in theorem 5.1that the number of facets decreases in time. The result below explainsthat certain phenomena are forbidden. Namely, no facet with non-zero curvature may degenerate. Proposition 5.3. In anygroup of interacting facets F k , k = l, . . . , l + r , r > only afacet with zerocurvature maydegenerate. Proof. Let us suppose that F l , . . . , F l + r , r > is a maximal group of interacting facets with non-zero curvature. For the sake of definiteness, we will proceed while assuming that w s is increasing on ( a, b ) ⊃ [ ξ − l , ξ + l + r ] . Step 1. Let us observe that for a facet F k to disappear, it is necessary, (but not sufficient) that one ofneighboring facets moves upward faster than F k , i.e. either V k +1 = dτ k +1 dt > dτ k dt = V k or V k − = dτ k − dt > dτ k dt = V k . Indeed, the position of F k is defined by the intersection of the lines containing F k , F k +1 moved vertically by τ k and respectively by τ k +1 and the intersection of lines containing F k , F k − moved vertically by τ k and respectively by τ k − . Thus, if the lines containing F k +1 and F k − are moved up so much that their intersection is above the line containing F k moved vertically by τ k ,then facet F k is going to disappear. This situation may occur only if V k +1 > V k or V k − > V k . Step 2. Let us suppose that facets F k , F k − interact, hence by (5.19) α k ˙ ξ k − α k − ˙ ξ k − = ˙ τ k − − ˙ τ k . (5.21)By the monotonicity assumption on w s we notice that ˙ τ k − and ˙ τ k are positive. If the length of F k ,which is equal to ξ k − ξ k − , stays bounded on [ t ∗ , t ∗ + T ) while the length of F k − vanishes at t = t ∗ + T , then in a neighborhood of t ∗ + T we have ˙ τ k − − ˙ τ k < . Thus, by (5.21) we can see that α k − ( ˙ ξ k − ˙ ξ k − ) + ( α k − α k − ) ˙ ξ k < and by (5.18) the left-hand-side (LHS) converges to −∞ when t tends to t ∗ + T . Since ˙ ξ k − ˙ ξ k − must be bounded from above, we deduce that ˙ ξ k < for t close to t ∗ + T . Step 3. Since always ˙ ξ − l − < and ˙ ξ + l + r +1 > (unless Ξ( ϕ ) = [0 , π ) ), we conclude that not allof the facets vanish simultaneously at t = t ∗ + T . As a result we may assume the length ℓ ( F l − ) of F l − is greater than d > on [ t i , T ) . Thus, we conclude by step 1, that for t close to t ∗ + T we have V l > V l − . By induction we obtain that V k +1 > V k , k = l, . . . , j + r − . (5.22)We notice that we have the following possibilities for facet F l + r : (a) there is an adjacent zero-curvature facet F l + r +1 ; (b) ξ + l + r +1 is defined as ( F + l + r +1 ) − ( τ l + r +1 ) (see the proof of Lemma 5.1). Incase (a) we can see that τ l + r +1 = 0 while in (b) τ l + r +1 > .27he condition (5.22) combined with (5.18) implies that ξ + l − ξ − l > . . . > ξ + r + l +1 − ξ − r + l +1 . Hence, the endpoints of F k , k = l, . . . , j + r − converge to a common limit p . But by step 2 ξ + r + l +1 ( t ) > ξ + r + l +1 ( t ∗ ) > ξ + l +1 ( t ∗ ) > ξ + l +1 ( t ) . This is a contradiction, our claim follows.This observation shows that the initial time t = 0 is special. If the data are poor from the view-point of dynamics, but still acceptable, then they get immediately regularized. That is all non-zerocurvature degenerate facet become regular.We are now ready for the proof of the main result. Proof of Theorem 5.1. P ART A. We start with data free from degenerate interacting facets. We set t = 0 , we have to define time instance t i , i = 1 , . . . , M postulated by the theorem. We shall proceediteratively.It follows from Proposition 5.3, that degenerate, non-zero curvature facets are possible only at t = 0 , i.e. at the initial time instance.Let us suppose that [ ξ − k ( t i ) , ξ + k ( t i )] is a connected component of Ξ l ( w s ( t i )) . We have six possi-bilities for F k = F k ( ξ − k ( t i ) , ξ + k ( t i )) :(a) F k is regular, does not have zero curvature, is non-interacting;(b) F k is regular, does not have zero curvature, is interacting;(c) F k is regular, has zero curvature, is non-interacting;(d) F k is regular, has zero curvature, is interacting;(e) F k is degenerate, does not have zero curvature, is non-interacting;(f) F k is degenerate, has zero curvature, is non-interacting.Cases (a) and (e) are solved in Lemma 5.2, where corresponding ξ ± k are constructed.The construction of ξ ± k corresponding to (b), (d) is performed in Lemma 5.3. We stress that in allthese cases τ k , is given by (5.11).The definition of ξ ± k is simple if (c) or (f) holds, we just set ξ − k ( t ) = ξ − k , ξ + k ( t ) = ξ + k , τ k ( t ) = 0 . (5.23)We have to define Ω . By the very definition of zero-curvature facets the intersection ∂J ( ξ + k + ǫ ) ∩ ∂J ( ξ − k − ǫ ) is a singleton { α } for any positive ǫ < min { ξ − k +1 − ξ + k , ξ − k − ξ + k − } . Moreover, α ∈ A ,hence we set Ω( x, t ) = α, for x ∈ [ ξ − k ( t ) , ξ + k ( t )] . (5.24)Thus, we have specified evolution of ξ ± k for every configuration. In all these cases the functions ξ ± k , k = 1 , . . . , N i are defined on maximal intervals [ t i , t i + T ± k ] . The numbers T ± k are defined asfollows.In (a) and (e) the positive number T + k (resp. T − k ) is such that ξ + k ( t ) < ξ − k +1 ( t ) (resp. ξ + k − ( t ) <ξ − k ( t ) ) for t < t i + T + k (resp. t < t i + T − k ), while equality occurs at t = T + k (resp. t = T − k ), i.e. thefacet begins to interact with its neighbor. By Corollary 5.1 T ± k are finite.If a group of interacting facets F l , . . . F l + r does not contain any zero-curvature facet, then byProposition 5.3 it may not vanish and its maximal existence time is defined as in (a) for ξ l + r . Thus,at T + r + l the group begins to interact with another facet. On the other hand, if this group of interactingfacets F l , . . . F l + r contains a zero-curvature facet, say F p , then T + p is defined as the extinction time of28 p , i.e. ξ − p ( t ) < ξ + ( t ) for t ∈ [ t i , t i + T + p ) , while ξ − p ( t i + T + p ) = ξ + ( t i + T + p ) . Thus, the number offacets drops by one.Cases (c) and (f) do not contribute to the definition of t i +1 , because (5.23) is valid for all t ≥ t i .We have to define also Ω( x, t ) . An attempt to do so reveals another difficulty related to con-struction of ξ ± k starting from t = 0 . Let us consider two interacting facets F k ( ξ − k , ξ + k ) , F k ′ ( ξ − k ′ , ξ + k ′ ) ,where [ ξ − k , ξ + k ] ⊂ Ξ l ( w ,s ) , [ ξ − k ′ , ξ + k ′ ] ⊂ Ξ r ( w ,s ) . (5.25)It is obvious that for any s ∈ [ ξ − k , ξ + k ] and s ′ ∈ [ ξ − k ′ , ξ + k ′ ] the intersection ∂J ( w ,s ( s )) ∩ ∂J ( w ,s ( s ′ )) is non-empty if and only if | l − r | = 1 . If the above intersection is non-empty, we can construct thedesired Ω( x, t ) . On the other hand, if this intersection is void, then we have no chance to construct a W section of ∂J ( w s ) .Let us suppose then that (5.25) holds and | l − r | = p + 1 , p > . Let us suppose for simplicitythat l < r . Thus, a single point ξ is a connected component of Ξ j ( w ,s ) , j = l, l + 1 , . . . , r , i.e. ξ = ξ − j = ξ + j , j = l, l + 1 , . . . , r. In other words, we have a number of degenerate, interacting facets at ξ . The system of ODE’s (5.19)is singular. The problem of evolution of interacting degenerate facets shall be dealt with below in PartB of the proof. It occurs only at t = 0 .Finally, we check that w ( x, t ) and Ω( x, t ) fulfill the conditions postulated in the definition of theweak solution. They satisfy the equation w t ( x, t ) = Ω x ( x, t ) (5.26)and the initial and boundary conditions are satisfied. Integral identity in Definition 2.1 follows.P ART B. After finishing part A, i.e. the case of data satisfying (5.4), we consider the interactionof facets during creation, i.e. (5.4) is no longer valid. We have two cases to consider:(g) F k is degenerate, with nonzero curvature and interacting;(h) F k is degenerate, with zero curvature and interacting.We begin with (g). Let us suppose that w violates (5.4) at some ξ . Thus, we are dealing with thesituation when one sided derivatives of w differ at ξ , i.e., w − ,s ( ξ ) < α k < w +0 ,s ( ξ ) for some a k ∈ A . It may as well happen that the reverse inequalities occur, however for the sake ofdefiniteness we shall stick to the above choice.We shall construct two functions w ǫ , w ǫ such that their derivatives belong to J - R , w ǫ ( x ) C. We have to deal with the points outside of Ξ( w s ( · , t )) ≡ S N k i =1 [ ξ − k , ξ + k ] . By the definitionof Ξ( w s ( · , t )) , its complement is open [0 , π ] \ Ξ( w s ( · , t )) = N i [ l =1 ( ξ + k , ξ − k +1 ) . where ( ξ + N i , ξ − N i +1 ) should be understood as ( ξ + N i , π ] ∪ [0 , ξ − ) , (with the understanding that ≤ ξ ± k ≤ π , k = 1 , . . . , N i ). Using again the definition of Ξ , we come to the conclusion that, if x belongs toany of the intervals ( ξ + k , ξ − k +1 ) , then either w s ( x, t ) exists or w + s ( x, t ) = w − s ( x, t ) . In either case, theset ∂w ( x, t ) (see (4.5)) does not intersect A . Since ∂w ( x, t ) is an interval, we deduce that there exists α k ∈ A such that ∂w ( x, t ) ⊂ ( α k , α k +1 ) . (5.28)We have to make sure that the choice of α k , in the formula above, depends only on the interval ( ξ + k , ξ − k +1 ) , but it is independent from a specific point x ∈ ( ξ + k , ξ − k +1 ) . Indeed, by the definition ofthe J - R class ∂w = M − f or ∂w = f − M , where f is a continuous increasing function and M amaximal monotone operator. Thus, the images f ( ξ + k , ξ − k +1 ) and M ( ξ + k , ξ − k +1 ) are connected intervals,so is the image ∂w ( ξ + k , ξ − k +1 ) , which is disjoint from A . Our claim follows.As a result, our definition of w ( x, t ) for x Ξ( w s ( · , t )) is as follows, w ( x, t ) = w ( x, t k ) and Ω( x, t ) = dJdϕ ( w s ( y, t k )) for x ∈ ( ξ + i , ξ − i +1 ) . where y ∈ ( ξ + i , ξ − i +1 ) is any differentiability point of w ( · , t k ) .P ART D. We have to define t k +1 . We do this inductively. Once t k is given, we set t k +1 = t k + min { min i T + i , min i T − i } . Thus at t k +1 two facets begin to interact, due to the shrinkage of [ ξ + i , ξ − i +1 ] to a point or due to thedisappearance of a facet. By Proposition 5.3, we know that only zero-curvature facets may disappear.We set N i +1 = N i − m, m is the number of removed degenerate, interacting, zero-curvature facets at t = t i +1 .The last thing to show is the estimate k w s ( · , t ) k J - R ≤ k w s ( · , s )) k J - R , whenever t > s . Bythe construction above, the number of connected components of Ξ( w s ( · , t )) drops at time instances t k , k = 1 , . . . , M N , hence K ( w s ( · , t )) ≤ K ( w s ( · , s )) , whenever s ≤ t . It remains to show that k w s ( · , t ) k T V ( S ) ≤ k w s ( · , s ) k T V ( S ) , where we denoted by k f k T V ( E ) the total variation of function f over set E .We first consider the case t > s such that Ξ( w s ( · , t )) = S , we know that we always have Ξ( w s ( · , t )) ⊃ Ξ( w s ( · , s )) for s < t . By the general properties of the total variation, we noticethat k w s ( · , t ) k T V ( S ) = k w s ( · , t ) k T V (Ξ( t )) + k w s ( · , t ) k T V ( S \ Ξ( t )) , where we wrote Ξ( σ ) for Ξ( w s ( · , σ ) . Now, by the definition of w ( x, t ) , we notice that k w s ( · , t ) k T V ( S \ Ξ( t )) = k w s ( · , s ) k T V ( S \ Ξ( t )) ≤ k w s ( · , s ) k T V ( S \ Ξ( s )) . We turn our attention to k w s ( · , t ) k T V (Ξ( t )) . On the intervals forming Ξ( t ) function w s ( · , t ) is piece-wise constant. The jumps occur at the endpoint of these intervals. They are no bigger and no morenumerous than the jumps of w s ( · , s ) . Thus our claim follows in the considered case of t . In fact, thecase of t such that Ξ( t ) = S is not much different. Finally, we can see that w s is a difference of twomonotone functions and one of them is continuous, the other one a maximal monotone operator.Our theorem is proved.We close this subsection with a formula, which might be called “morphing a circle into a square”. Example. Let us suppose that φ ( s ) = s or w ( s ) = s . Due to the high symmetry of the problem,it is sufficient to consider just formation of one facet. Then, w ( x, t ) , the unique solution to (2.1), isgiven by the formula, w ( x, t ) = (cid:26) s s ∈ [0 , ξ − ( t )] ∪ [ ξ +1 ( t ) , π ] , π s − π + τ ( t ) s ∈ [ ξ − ( t ) , ξ +1 ( t )] . Here, ξ ± = π ± √ τ and τ = (cid:16) √ π t (cid:17) / . Let us note that at T = π / we have Ω + − Ω − = ξ + − ξ − , so for later times ˙ τ = 1 .We can make this observation more general. Proposition 5.4. Let us suppose that the assumptions of Theorem 5.1 are satisfied. Then, thereexist T fa , such that if t > T fa , then w ( · , t ) is fully faceted, i.e. w s ( · , t ) is piece-wise linear. Moreprecisely, for Ξ( w s ( · , t )) ⊂ [0 , π ) for t < T fa and Ξ( w s ( · , t )) = [0 , π ) for t ≥ T fa . Proof. Let us consider w . It is fully faceted or not. If not, then by the proof of Theorem 5.1, wededuce that after at some t i we have Ξ( t i ) = [0 , π ) and our claim follows. We show that after some depending upon the initial data, the solution becomes such that w s = ϕ ismonotone decreasing or increasing. We shall call this process by convexification. Proposition 5.5. Let us suppose that the assumptions of Theorem 5.1 are satisfied. Then, thereexist T cx ,such that if t ≥ T cx ,then w s ( · , t ) ismonotone, while this is nottrue for t < T cx . roof. If w ,s is monotone, then we are done. Otherwise, let us suppose that t j is the largest time suchthat at t j a zero curvature facet disappears. Since the zero-curvature facets cannot persist because theirendpoints necessarily move, it follows that T cx = t j has the desired properties. Remark. All possibilities can be realized T cx > T fa as well as T cx < T fa . Here, we consider the last stage of evolution, when t ≥ t M and N M = 4 . In this case, it is sufficient tospecify only ξ + k , k = 1 , , , . Furthermore, the system for interacting facets, (5.19) takes the form, α ˙ ξ − α ˙ ξ = ˙ τ − ˙ τ α ˙ ξ − α ˙ ξ = ˙ τ − ˙ τ α ˙ ξ − α ˙ ξ = ˙ τ − ˙ τ (5.29) α ˙ ξ − α ˙ ξ = ˙ τ − ˙ τ ξ k ( t M ) = ξ k , k = 1 , , , . We notice that the stationary points of (5.29) are such that ˙ τ = . . . = ˙ τ . This occurs if and onlyif Ω + k − Ω − k = ξ k − ξ k − , k = 1 , , , , where by ξ we understand ξ . Moreover, due to ourassumptions on J we have Ω + k − Ω − k = ∆Ω , k = 1 , , , .Additionally, system (5.29) possesses a Liapunov functional. Namely, let us write F ( ~ξ ) = X k =1 ln( ξ k − ξ k − ) ∆Ω , with the understanding of ξ as above. By direct calculation, we check that ddt F ( ~ξ ) = ∇ ξ F · ddt ~ξ < . This derivative vanishes if and only if ξ is the only equilibrium point. Thus, we have a completepicture of the asymptotic behavior of Λ . Theorem 5.2. Let us assume that ϕ ∈ J - R and w is the corresponding unique solution to (2.1).Then,there exists T , max { T cx , T fa } ≤ T ≤ ∞ withthe following property:(a)If T < ∞ ,then ξ l ( t ) = − π + π l + α, forsome α ≥ , l = 0 , ...,3,and t ≥ T ,inotherwords, w is theminimal solution for t > T ;(b) If T = ∞ ,then lim t →∞ ξ l ( t ) = − π + π l + α, l = 0 , ...,3 forsome α ≥ . In the course of proof of Theorem 5.1, we exhibited a quite explicit construction of the weak solutionwith such initial data that ϕ ∈ J - R . Now, we have to show that is has all the postulated propertiesof the almost classical solution. We have already noticed that w s = Λ s + s belongs to the J-R class,furthermore k w s ( · , t ) k J - R ≤ k w s ( · , k J - R . The key point, however, is to realize that Ω = ∂J ¯ ◦ ∂w, (5.30)32here ∂w is the multivalued map whose section is w s . We defined ∂w in (4.5). Checking that (5.30)indeed holds requires recalling the steps of construction of Ω , we will do this below. Finally, after weset N = { , t , . . . , t M } we see that Λ t = ∂∂s ∂J ¯ ◦ (Λ s + s ) , holds for all t ∈ (0 , + ∞ ) \ N in the L sense, more precisely it holds pointwise except x ∈ [0 , π ) \ { ξ ± i : i = 1 , . . . , N k } . Indeed, the definitions (5.11), (5.18), (5.23) of τ k ( t ) were suchthat ddt τ k ( t ) = ∂ Ω ∂s . Moreover, ∂ Λ ∂t = ddt τ k ( t ) , see Lemma 5.2, Lemma 5.3 and eq. (5.26). We recallthat by definition functions τ k ( · ) are continuous on [ t i , t i +1 ] and differentiable in ( t i , t i +1 ) . Moreover,the right derivative of τ k ( t ) is well-defined for all t , except possibly t = t . Hence, ∂ Λ ∂t is definedeverywhere, except the points t i , i = 0 , . . . , M , but the right time derivative ∂ Λ + ∂t is defined for all t > .We will check below that Ω , constructed in the course of proof of Theorem 5.1, coincides with ∂J ¯ ◦ ∂w , — see (5.12), (5.20), (5.24), where w s ( s, t ) = Λ s ( s, t ) + s . In order to see that we examinethe steps of the construction of Ω and compare it with the definition of the composition ¯ ◦ . Let us fix t ∈ ( t k , t k +1 ) , at the end we will consider t = t k +1 , then we compose ∂w ( · , t ) : [0 , π ] → [ a, b ] with ∂J : R → R . We have to identify the sets D s , D f and D r appearing in the Definition 2.3. For ourchoice of t we have D s ( t ) = { s ∈ [0 , π ] : w + x ( s, t ) = w − x ( s, t ) } . In particular, D s ( t ) contains all points ξ ± i ( t ) , i = 1 , . . . , N k . We can see that D f ( t ) = N k [ i =1 ( ξ − i ( t ) , ξ + i ( t )) , i.e., it is the sum of interiors of intervals contained in Ξ( w s ( · , t )) . Finally, by the definition D r ( t ) = [0 , π ] \ ( D s ( t ) ∪ D f ( t )) . We shall consider these cases separately. o case D r . If s ∈ D r ( t ) , then w is differentiable at s and w s ( s, t ) 6∈ A . Thus, by (2.4) ∂J ¯ ◦ ∂w ( s, t ) = dJdϕ ( w s ( s, t )) . We notice that D r ( t ) ⊂ [0 , π ] \ Ξ( w s ( · , t )) , hence by Part C ofthe proof of Theorem 5.1 we immediately see that ∂J ¯ ◦ ∂w ( s, t ) equals Ω( s, t ) on D r . o case D f . By its definition D f ( t ) is the sum of interiors of pre-images of facets, as noticedabove. Moreover, on each interval ( ξ − i ( t ) , ξ + i ( t )) , the set ∂w ( x, t ) is a singleton equal to { α k } ⊂ A .Then, the cases of the Definition 2.3, see formulas (2.5)–(2.8) have their counterparts in the formulas(5.12), (5.20) and (5.24). o case D s . We notice that, if t > , then the set Ξ( w s ( · , t )) has no component, which is asingleton. Thus, if s ∈ D s ( t ) , then the set ∂w ( s, t ) does not intersect A . As a result, formula (2.9)for the composition yields a singleton, because on the RHS of (2.9) the limit of constant functions aretaken. This in agreement with the discussion of Part C.Finally we have to deal with the case t = t k +1 . On one hand Ω( · , t k +1 ) is defined by the left timecontinuity of Ω , on the other hand we have to check that Ω = ∂J ¯ ◦ ∂w .By the very definition of t k +1 (see Part D of the proof of Theorem 5.1), at this time instant azero-curvature curvature facet disappears or two facets begin to interact or merge, i.e., lim t → t − k +1 ξ + i ( t ) = a = lim t → t − k +1 ξ − i +1 ( t ) . 33e have then two possibilities, either a = ξ + j ( t k +1 ) = ξ − j +1 ( t k +1 ) or a ∈ ( ξ − j ( t k +1 ) , ξ + j ( t k +1 )) where this interval is a connected component of Ξ( w s ( · , t k +1 )) . Once we realize this, it is clear that Ω( · , t k +1 ) = ∂J ¯ ◦ ∂w ( · , t k +1 ) . Here, we consider closed curves, we view them as graphs over a smooth, convex reference closedcurve M . We do not make here any attempt to consider non-smooth reference curves, which isreasonable because this would add up difficulties while not giving advantages.Let us suppose that x ( s ) is an arc-length parameterization of M and e t ( s ) , e n ( s ) are unit tangentand normal vectors, respectively, such that ( e t ( s ) , e t ( s )) is positively oriented. Then all points in aneighborhood of M can be uniquely written as x = x ( s ) + e n Λ , as a result we can parameterize ourcurve Γ( t ) as x ( s, t ) = x ( s ) + e n ( s )Λ( s, t ) . Since M is convex we may write e n uniquely as e n ( ϕ ( s )) = (cos ϕ ( s ) , sin ϕ ( s )) , where ϕ is themeasure of the angle between the x axis and e n . Moreover, dds e n ( ϕ ( s )) = − e t ( ϕ ( s )) dϕds = − κ e t ( ϕ ( s )) . We note ∂x∂s ( s, t ) = e t (1 − κ Λ) + e n Λ s , because | ˙ x ( s ) | = 1 . With this formula at hand, we can write the expression for the tangent andnormal to Γ( t ) , they are τ = W ( e t (1 − κ Λ) + e n Λ s ) , n = W ( − Λ s e t + (1 − κ Λ) e n ) , where W = (1 − κ Λ) + Λ s . Hence, the LHS of (1.1) takes the form βV = β dxdt · n = 1 W (1 − κ Λ)Λ t . The RHS of (1.1) is div S ∇ ξ γ ( ξ ) | ξ = n . In our paper [MRy], we have shown that it is equal to κ = dds (cid:18) ∂∂ϕ I θ ( ϕ ) (cid:19) . We defined I θ ( ϕ ) as follows, I ϑ ( ϕ ) = ¯ γ ( n ( ϕ )) + R ϕϑ dψ R ψϑ ¯ γ ( n ( t )) dt. We noted that this functionis convex iff the stored energy function ¯ γ is convex. However, in general I ϑ does not enjoy higherregularity properties. It is not differentiable at angles corresponding to the normals to the Wulff shape.Finally, equation (1.1) takes the form β n · e n Λ t = dds (cid:18) ∂∂α I θ ( α ) (cid:19) , (6.1)where α is the measure of the angle between the x axis and n .One may study evolution of convex curves defined by their angle parameterization. We notice α = ϕ + ψ, where ψ is the measure of the angle between τ and e t . We notice that τ · e n = sin ψ = Λ s W ,34 · e n = cos ψ = − Λ W . Thus, we see that ψ = Arg ( τ · e n + iτ · e n ) , in fact we have ψ = arctan (cid:16) Λ s − Λ (cid:17) . Thus, (1.1) takes the form β n · e n Λ t = dds (cid:18) ∂∂φ I θ (cid:18) ϕ + arctan (cid:18) Λ s − Λ (cid:19)(cid:19)(cid:19) . This equation is rather involved, we prefer to simplify it by dropping the terms which at this stage wedeem not important, thus we come to (1.2). J We may also consider any properly chosen piecewise linear, convex J , J l ( ϕ ) = N X i =1 b i | ϕ − α i | . (6.2)We require that N ≥ , b i > and α < α < . . . < α N < α +2 π , we will write S = [ α , α +2 π ) .In order to stick to geometrically relevant data, we also impose the condition that P Ni =1 b i = π , whichguarantees that ∂I ( S ) is an interval of length π . In addition, we assume that the following functionyields an angle parameterization of closed curve, which encompasses a convex region. Namely, weset Ω j = j X i =1 b i − N X i = j +1 b i , j = 0 , . . . , N, (6.3)with the convention that the summation over an empty set of parameters yields zero. Then we define Φ : [ α , α + 2 π ) → R by the formula Φ( s ) = N X i =0 Ω i χ [ α i ,α i +1 ) , (6.4)(with the convention α N +1 = α + 2 π ) is an angle parameterization of closed curve. We notice thatour assumptions imply that Ω + 2 π = Ω N .The analysis of behavior of solutions presented in Section 5 is valid also for J given by (1.4) and J l , however the actual calculations for J l are more lengthy. In addition we may show existence ofweak solution for a general, piecewise smooth, convex J , but in this case we cannot offer detailedanalysis of solutions, yet. Acknowledgment. The present work has been partially supported by Polish KBN grant No. 1 P03A 037 28.Both authors thank Université de Paris-Sud XI, where a part of the research for this paper was per-formed, for its hospitality. References [AW] F.Almgren and L.Wang, Mathematical existence of crystal growth with Gibbs–Thomson cur-vature effects, J. Geom. Anal. , , (2000), 1-100.[ABC1] F.Andreu, C.Ballester, V.Caselles, J.M.Mazón, The Dirichlet problem for the total variationflow, J. Funct. Anal. , , (2001), 347–403.35ABC2] F.Andreu, C.Ballester, V.Caselles, J.M.Mazón, Minimizing total variation flow, DifferentialIntegral Equations , , (2001), 321–360.[ACM] F.Andreu, V.Caselles, J.M.Mazón, “Parabolic Quasilinear Equations Minimizing LinearGrowth Functionals”, Progress in Mathematics, 223. Birkhäuser Verlag, Basel, 2004.[AG] S. B. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure 2.Evolution of an isothermal interface, Arch. Rational Mech. Anal. , (1989), 323-391.[BCN] G.Bellettini, V.Caselles, M.Novaga, The total variation flow in R N , J.Differential Equations, (2002), 475–525.[BNP1] G.Bellettini, M.Novaga, M.Paolini, Characterization of facet breaking for nonsmooth meancurvature flow in the convex case, Interfaces and Free Boundaries , , (2001), 415–446.[BNP2] G.Bellettini, M.Novaga, M.Paolini, On a crystalline variational problem, part I: First varia-tion and global L ∞ regularity, Arch. Rational Mech. Anal. , , (2001), 165–191.[BNP3] G.Bellettini, M.Novaga, M.Paolini, On a crystalline variational problem, part II: BV regular-ity and structure of minimizers on facets, Arch. Rational Mech. Anal., , (2001), 193–217[Ch] A.Chambolle, An algorithm for mean curvature motion Interfaces Free Bound. , , (2004),195–218.[CR] X.Chen and F.Reitich, Local existence and uniqueness of solutions to the Stefan problemwith surface tension and kinetic undercooling, J. Math. Anal. Appl. , , (1992), 350–362.[EPS] J. Escher, J. Prss, G. Simonett, Analytic solutions for a Stefan problem with Gibbs-Thomsoncorrection, J. Reine. Angew. Math. , (2003), 1–52.[ES] J. Escher, G. Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J.Math. Anal. , , (1997), 1028–1047.[FP] X.Feng, A.Prohl, Analysis of total variation flow and its finite element approximations. M2AN Math. Model. Numer. Anal. , (2003), 533–556.[FG] T. Fukui, Y.Giga, Motion of a graph by nonsmooth weighted curvature, in “World congressof nonlinear analysts ’92”, vol I, ed. V.Lakshmikantham, Walter de Gruyter, Berlin, 1996,47-56.[GG] M.-H.Giga, Y.Giga, A subdifferential interpretation of crystalline motion under nonuniformdriving force. Dynamical systems and differential equations, vol. I (Springfield, MO, 1996). Discrete Contin. Dynam. Systems, Added Volume I , 276–287 (1998).[GK] Y.Giga, R.Kobayashi, Equations with singular diffusivity, J.Stat.Physics, , (1999), 1187-1220.[GGK] M.-H.Giga, Y.Giga, R.Kobayashi, Very singular diffusion equation, Advanced Studies inPure Mathematics , (2001), 93-125.[GR1] Y.Giga, P.Rybka, Facet bending in the driven crystalline curvature flow in the plane, TheJournal of Geometric Analysis , (2008), to appear36GR2] Y.Giga, P.Rybka, Facet bending driven by the planar crystalline curvature with a genericnonuniform forcing term, preprint. [L] S.Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs–Thomson law forthe melting temperature, Euro. J. Appl. Math., , (1990), 101–111.[Mo] J.S.Moll, The anisotropic total variation flow. Math. Ann. , , (2005), 177–218.[Mu] P.B.Mucha, On the Stefan problem with surface tension in the L p framework, Adv. Differen-tial Equations , (2005), 861–900.[MRa] P.B.Mucha, R.Rautmann, Convergence of Rothe’s scheme for the Navier-Stokes equationswith slip conditions in 2D domains. Z. Angew. Math. Mech. , , (2006), 691–701.[MRy] P.B.Mucha, P.Rybka, A new look at equilibria in Stefan type problems in the plane, SIAM J.Math. Analysis , , No.4, 1120-1134.[LSU] O.A.Ladyženskaja, V.A.Solonnikov, N.N.Ural’ceva, “Linear and Quasilinear Equations ofParabolic Type,” Am. Math. Soc., Providence, R. I., 1968.[Ra] E.V.Radkevich, The Gibbs–Thompson correction and condition for the existence of classicalsolution of the modified Stefan problem, Soviet Math. Dokl., , (1991), 274–278.[RO] L. Rudin, S. Osher, Total variation based image restoration with free local constraints, in “Proc. of the IEEE ICIP-94 Austin, TX,” Vol. 1, pp. 31-35, 1994.[ROF] L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal, Physica D , , (1992), 259-268.[Ry] P.Rybka, The crystalline version of the modified Stefan problem in the plane and its proper-ties, SIAM J.Math. Anal., (1999), 736–786.[S] J.Smoller, “Shock waves and reaction-diffusion equations. Second edition”. Grundlehren derMathematischen Wissenschaften, 258. Springer-Verlag, New York, 1994.[T] J.E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, in : Dif-ferential Geometry (eds. B. Lawson and K. Tanenblat), Proceedings of the Conference onDifferential Geometry, Rio de Janeiro, Pitman Monographs Surveys in Pure and AppliedMath.52