Curve shortening flow in heterogeneous media

Transcription

1 Curve shortening flow in heterogeneous media Annalisa Cesaroni Matteo Novaga Enrico Valdinoci Abstract In recent years, there has been a growing interest in geometric evolutions in heterogeneous media. Here we consider curvature driven flows of planar curves with an additional space-dependent forcing term, and we look for estimates which depend only on the L -norm of the forcing term. Our motivation comes from a homogenization problem, which we can rigorously solve in the special case when the initial curve is a graph and the forcing term does not depend on the vertical direction. In such case, we are also able to define a soluton of the evolution even if the forcing term is just a bounded function, not necessarily continuous. 1 Introduction In this paper we consider the curvature shortening flow of planar curves in a heterogeneous medium, which is modeled by a spatially-dependent additive forcing term. The evolution law reads: v = κ + gν, 1 where ν is inward normal vector to the curve, κ is the curvature of the curve, v is the normal velocity vector, and g L R represents the forcing term. The original motivation for our analysis comes from a homogenization problem related to the averaged behaviour of an interface moving by curvature plus a rapidly oscillating forcing term. More precisely, the evolution law is given by v = κ + g ε, y ε ν, where g is a 1-periodic Lipschitz continuous function. When the forcing term is periodic, Equation 1 was recently considered in [6], where the authors prove eistence and uniqueness of planar pulsating waves in every direction of propagation. This result leads to the homogenization of for plane-like initial data see Section 3. Related results on the homogenization of interfaces moving with normal velocity given by v = εκ + g ε, y ν, ε Dipartimento di Matematica Pura e Applicata, Università di Padova, via Trieste 63, 3511 Padova, Italy, acesar@math.unipd.it, novaga@math.unipd.it Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 133 Roma, Italy, valdinoc@mat.uniroma.it 1

2 have been obtained in [5] and [1], under suitable assumptions on the forcing term including the fact that it does not change sign, and in [4] under more general assumptions. In particular, the authors show that the homogenized evolution law, when it eists, is a first order anisotropic geometric law of the form v = cν ν. Coming back to our problem, as a first step we look for geometric estimates for solutions to 1, which depend only on the L -norm of g. In particular, reasoning as in the case of the unperturbed curvature flow [9,, 11], in Section we classify all possible singularities which can arise during the evolution. As a consequence, in Section.5 we can show that, when g is smooth and the initial curve is embedded, the eistence time of a regular solution to 1 is bounded below by a quantity depending only on g and on the initial curve. Unfortunately, since we have no estimates on the curvature in terms of g, we are not able to obtain a general eistence result for 1 in the nonsmooth case, i.e. when g L. However, in Section.6 we overcome this difficulty by assuming that the initial curve is the graph of a function u, for instance in the vertical direction. In this case Equation 1 becomes u t = u 1 + u + g, u 1 + u. 3 In Lemma.18 we establish an L p -estimate on u which depends only on g. In Proposition.1 we consider a sequence of smooth forcings g n weakly converging to g L. Using the estimate on u and the results of the Section.5, and letting u n be the solution corresponding to g n, we can pass to the limit as n and obtain that u n u H 1 [, T ], L [, 1] L [, T ], H 1 [, 1], for some time T > depending only on g and on the initial datum. When g does not depend on u, we obtain a stronger estimate on u t, which allows us to show that u W 1, [, T ], L [, 1] L [, T ], W, [, 1]. As a first application, this leads to an eistence and uniqueness result for solutions to 3, when g is a L -function which is independent of u see Theorems.3 and.5. The second application of our result is to the homogenization problem. In section 3, under the assumptions of Theorem.3, that is, when the curve is a graph and g is independent of the vertical direction, we can pass to the limit in as ε, and show that the limit curve moves according to the evolution law v = κ + g, yddy ν. [,1] Local eistence of solutions In this section we are concerned with the local eistence for 1, under the assumption that the forcing term g is smooth and bounded, i.e. g C R L R. If we parametrize counterclockwise the evolving curve with a function = 1, : [, 1] [, T ] R, with, = 1,, problem 1 becomes t = κ + gν = +, 1 g, 4 where ξ denotes the component of the vector ξ orthogonal to. As usual we let τ, ν, κ be respectively the unit tangent vector, the unit normal vector and the curvature of the

3 evolving curve. Denoting by s the arclength paramter of the curve, so that s = /, by the classical Frenet Serret formulas we have s = τ, ss = τ s = κ ν, ν s = κ τ. 5 We recall the following local in time eistence result for 4, proved in [7, Thm 4.1]. Theorem.1. Let : [, 1] R be a smooth map such that = 1 and > for all [, 1]. Then there eist T > and a smooth solution to 4, defined on [, 1] [, T ], such that, = for all [, 1]..1 Estimates on the curvature and its derivatives Lemma.. The following commutation rule holds: Moreover, Proof. By definition of arclength, we have Therefore, from 4 and 5, t s = s t + κκ + g s. 6 τ t = κ + g s ν, 7 ν t = κ + g s τ 8 κ t = κ + g ss + κ κ + g. 9 s =. t s s t = 3 t = τ t = τ κ + gν s = τ κ + gν s = κκ + g s, that is 6. Now, applying 6 to 4 and 5, we obtain which is 7. Also, since ν = 1, τ t = s t = t s + κκ + g s = κ + g s ν + κ + gν s + κκ + gτ = κ + g s ν, and so, from 7, = ν ν t = ν ν t ν t = ν t νν + ν t ττ = ν t ττ = ν τ t ν τ t τ = ν τt τ = κ + g s τ, 3

4 that is 8, and that is 9. κ t = κν t ν = τ s t ν = τ t s ν + κκ + gτ s ν = κ + g s ν s ν + κ κ + g = κ + g ss + κ κ + g, Let us compute the evolution for the spatial derivaties of the curvature. We denote by p j,k l sκ, m s g a generic polynomial depending on the derivatives up to order j of κ and the derivatives up to order k of g. Lemma.3. For all j N, j 1, it holds t j sκ = j sκ ss + j + 3κ + j + κg j sκ + p j 1,j+ l sκ, m s g. 1 Proof. The proof is by induction on j. When j = 1 from 6 and 9 we easily get t κ s = κ s ss + 4κ + 3κgκ s + κ g s + g sss. Assume now 1 for some j N. Using 6, we compute recursively t s j+1 κ = s t sκ j + κκ + g s j+1 κ = s j+1 κ ss + j + 3κ + j + κg + κκ + g s j+1 κ + p j,j+3 sκ, l s m g, which gives 1 for all j. We recall that in [7] similar equations as 1 for the evolution of the second fondamental form and its derivatives are obtained for forced mean curvature flow of hypersurfaces in any dimension. We now compute the evolution equation of w := log. Lemma.4. There holds w t = κκ + g. 11 Proof. A direct computation using 4 gives w t = t = τ s t = κκ + g. Lemma.5. Assume that 4 admits a smooth solution on [, t], with t >. Then ma [,1] [, t] j sκ C j for all j N, where the constants C j depend only on the initial curve, on t, on ma [,1] [, t] κ and on g C j+. Proof. Following [8], we let K j, t := j sκ, t M j t := ma [,1] K j, t. 4

5 For all such that K j, t = M j t we have and so Recalling 1, for a.e. t [, t] we have s K j = ss K j = sκ j s + sκ j sκ j ss j sκ, t j sκ ss, t. 1 Ṁ j t = ma : K j,t=m jt tk j = ma : K j,t=m jt j sκ j sκ t = ma : K j,t=m jt j sκ j sκ ss + A j j sκ + B j where the constants A j, B j depend on M l and g C k, with l < j and k j +. Hence, using 1 we get Ṁ j A j M j + B j. By Gronwall s Lemma it then follows that the quantities M j are uniformly bounded on [, t]. Since the eistence result in Theorem.1 is first established in the usual Hölder parabolic spaces C k+α,k+α [, 1] [, T ] see [14], if we still denote by T the maimal eistence time of the evolution, we have that, if T < +, either 1 or j sκ blow up as t T, for some j N. Proposition.6. Let T be the maimal eistence time of the evolution 4, and assume T < +. Then lim t T κ L = Proof. Assume by contradiction that κ is uniformly bounded for all t [, T and [, 1]. Equation 11 implies that and 1/ are also uniformly bounded on [, 1] [, T. But Lemma.5 implies that also the quantities j κ, t are uniformly bounded on [, 1] [, T for all j N, thus reaching a contradiction. We then proved lim sup κ L = +. t T Notice that the lim sup is indeed a full limit due to 9. The following Lemma provides a lower bound to 13. Lemma.7. Let T as above and assume T < +. The following curvature lower bound holds: lim inf T t κ L t T 5

6 Proof. Notice that 9 can be written as κ + g t = κ + g ss + κ + gκ + κ + g g ν. 15 Let w := κ + g. Observe that, for ε >, κ 1 + εw ε g. So, from 15 it follows w t = w ss κ + g s + wκ + w g ν w ss + w 1 + εw g + g L w ε w ss εw g W 1, w. ε Letting M := ma [,1] κ + g, from 16 we get d dt M + C = Ṁ 1 + εm g W 1, M 1 + εm + C, 16 ε where C = [1 + 1/ε/1 + ε] g W 1,, so that d dt 1 M + C Integrating on [t, s] [, T we thus obtain 1 Mt + C 1 Ms + C 1 + ε εs t. Letting now s T and recalling that Ms + by Proposition.6, we get that is which gives the thesis. 1 Mt + C Mt 1 + εt t, εt t C, From 17 and Proposition.6 we obtain the following estimate on the maimal eistence time of the evolution. Proposition.8. Let T be the maimal eistence time of 4, then T c κ L, g W 1,. Remark.9. Note that, differently from the case of curve shortening flow, in our case, due to the presence of the forcing term g, self-intersections may arise. Nevertheless if the initial curve is embedded then, thanks to Proposition.8, it remains embedded in a time interval [, T ], with T > depending only on the initial datum and on g W 1,. We think it is an interesting problem to determine whether or not the constant c in Proposition.8 depends only on the initial set and on the L -norm of g see for instance Section.5 below for a special case. 6

7 . Huisken s monotonicity formula In the following we derive a monotonicity formula for curvature flow with a forcing term, and apply it to the analysis of singularities. By a standard computation, using the fact that solves 4, we get the following formula. Lemma.1. Let τ > and let f : R [, τ R be a smooth function. Then d fs, t, tds = [f t κκ + gf + κ + g f ν] ds. 18 dt We denote by L t the length of the curve [, 1], t, that is 1 L t := d = ds. When no confusion can arise, we write L instead of L t. Corollary.11. Let : [, 1] [, T ] R be a solution to 4. Then L t L e g t t [, T ]. 19 Proof. Taking f 1 in 18 we have t L t = which gives 19 by Gronwall s Lemma. κκ + gds κ + g ds, We now apply Lemma.1 with fp, t = e p p /4T t, p, p R, and we get 4πT t d dt = = + e s,t p /4T t ds 4πT t e p /4T t 4πT t [ p 4T t 1 e p /4T t 4πT t [ e p /4T t 4πT t [ p T t + κ + g T t + κκ + g + κ + g p ν T t ] ν ds + 1 e p /4T t g ds 4 4πT t 1 T t + κ p ν T t Following [9, Theorem 3.1], the last term can be actually written as = [ e p /4T t 4πT t e p /4T t 4πT t ] ds ] ds. 1 1 T t + κ p ν T t p τ T t ds. 7 ] ds

8 Substituing this in 1 we obtain an analog of Huisken s monotonicity formula [9] d dt = e s,t p /4T t 4πT t ds [ e s,t p /4T t κ + s, t p ν + g ] + 1 4πT t T t 4 g ds. In the net paragraph we will apply this formula in the analysis of type I singularities..3 Type I singularities We assume that at time T the flow is developing a singularity of type I, i.e. there eists a constant C > 1 such that ma κ, t C. 3 [,1] T t Observe that for every and t r < T, r, t C T t C T r + g r t. 4 This implies that the functions, t converge uniformly to a function T as t T. Now we fi [, 1] such that, t T =: p and κ, t becomes unbounded as t T. We rescale the curve around the point p as follows:, z :=, tz p T tz zt := log T t. From 4, we deduce, tz p, z = C + 1 g e z C + 1 g, T tz so in particular, z remains bounded as z +. The evolution law satisfied by the rescaled curve is z = κ + e z g ν + z [ log T, +. 5 We also have the rescaled version of the monotonicity formula : letting y = compute d dz = e s,z / d s = T t d dt e s,z / e s,t p /4T t T t ds [ κ + s, z ν + e z g ] + e z g p, we T t d s. 6 8

9 Letting F z := e s,z / d s, Equation 6 gives Integrating 7 we obtain d dz F z g e z F z g e z. 7 F z e g T/4 F log T z log T. In particular, we deduce that for every R > there eists a uniform bound on H 1 [, 1], z B, R, where H 1 denotes the 1 dimensional Hausdorff measure. Indeed H 1 [, 1], z B, R = χ B,R d s χ B,R e R,z d s 8 for some positive constant K. e R F z K, Proposition.1. Under Assumption 3, for each sequence z j + there eists a subsequence z jk such that the curve, z jk, rescaled around p, locally smoothly converges to some smooth, nonflat limit curve, such that κ + ν =. 9 Proof. The proof follows the same argument as in [9, Proposition 3.4]. Indeed, the limit curve is smooth thanks to 8, Proposition.6 and the fact that the rescaled curve has uniformly bounded curvature. Moreover, it is nonflat by 14. Finally, the limit curve satisfies 9 thanks to 6 and 7. Remark.13. Proposition.1 implies that the type I singularities of 1 are modeled by homothetic solutions of the flow, as for the spatially homogeneous case [9]. We recall that, among such solutions, the circle is the only embedded one [1], hence, under Assumption 3, T is actually the etinction time for the evolution. From this we can conclude that.4 Type II singularities T c κ L, g L. We consider now the case that at time T the flow is developing a singularity of type II, i.e. lim sup t T ma κ, t T t = +. 3 [,1] Proposition.14. Under condition 3, there eists a sequence of points and times n, t n on which the curvature blows up such that the rescaled curve along this sequence converges in C to a planar, conve limiting solution, which moves by translation. 9

10 Proof. By means of 9 and 19, an easy calculation implies that d κ κ ds = κ s + dt κ g ssds g L e g t + g : κ,t= κ ds. 31 From 31, using Gronwall lemma, we obtain that t κ ds is uniformly bounded in [, T ] and admits a bounded limit as t T. Following [] we choose a sequence n, t n such that t n [, T 1 n and t n < t n+1 ; k n = κ n, t n + and k n T 1 n t n = ma t [,T 1/n] κ T 1n t + as n +. 3 We define the new parameter u as follows u = k nt t n, u [ k nt n, k nt t n ] and the rescaled curve along the sequence n, t n as n, u = k n, tu n, t n, for [, 1]. Observe that n n, =, and κ n n, = 1; moreover v n = d du κ + gν n = = κ n + g n ν n 33 k n where g n y = gy/k n + n, t n /k n. In the sequel, we shall write for simplicity κ n instead of κ n. Note that for every ε, ω >, there eists n such that κ n 1 + ε for u [ k nt n, ω]. Indeed, using 3, we get κ n, u = κ, tu k n T 1/n t n T 1/n tu = T 1/n t n T 1/n t n u/kn. This implies that, on every bounded interval of time, the curvatures of the rescaled curves are uniformly bounded. Moreover, from this, we deduce uniform bounds also on the derivatives of the curvature, using Lemma.5 and recalling that n satisfies 33 and the fact that. By the same argument of [, Theorem 7.3], this implies that there eists a subsequence along which the rescaled curves converge smoothly to a smooth, non trivial limit defined in, +. Moreover evolves by mean curvature flow, L t = + and κ = 1 = κ,. We prove now that is conve. j g n = j g /k j+1 n Recall that from 31 we get that t κ is uniformly bounded and admits a limit as t T. The same also holds for t n κ n. Moreover, from 31 we also obtain d κ n κ n ds = κ n s + du n κ n g n ss ds. So M Mκ n= κ n,u= κ n s du = κ n, M κ n, M ds n = κ, t n + M kn κ, t n M kn ds 1 n M M M M n n κ n κ n g n ss dsdu κ n κ n g n ss dsdu.

11 Letting n + along the subsequence on which n, we get κ, t n + M kn ds κ, t n M kn ds. We argue as in 31, using the definition of g n and the fact that, by 19, L t n k n K for some constant K just depending on g and T, M κ M n M κ n g g n ss dsdu M kn 3 L u n + g kn κ n ds du C n kn n as n +. In particular, this gives M M,κ n,u= κ n s du as n +, and we can conclude as in [, Theorem 7.7] that is a conve eternal solution to the curvature flow, that is, is the so-called Grim Reaper..5 The embedded case In this section we strengthen Proposition.8 in the case of embedded planar curves. Following [1] we define where ηt := inf <y y L,y t := min σ, t dσ,, t y, t, 34 L,y t σ, t dσ + 1 y σ, t dσ. Notice that the infimum in 34 is in fact a minimum, moreover η is a continuous function in [, T, where T is the first singularity time. Since the initial curve is embedded, we also have η >. Let now Et := {, y : < y and ηt = }, t y, t. L,y t In the following we assume for simplicity that L,y t = y σ, t dσ, the other case being analogous. Notice that, if ηt < /, we have the estimate y κ σ, t dσ ma z,w [,y] τz, t τw, t > π 35 for all, y Et. Indeed, if τz, t τw, t π for all z, w [, y], then [, y] is the graph of a 1-Lipschitz function, which in turn implies ηt /. Letting c := π/, from 11

12 35 we get y κκ + g σ, t dσ L,y c 1 y y κ σ, t dσ g κ σ, t dσ c g 36 L,y whenever L,y c/ g. Moreover, reasoning as in [1], from the minimality condition it follows that κν κyνy y, 37 for all, y Et. When ηt < /, using 36, 37 and the so called Hamilton s trick see [15] we compute [ 1 y y ] ηt = min [κ + gν κy + gyνy],y Et L,y y + η κκ + g dσ [ min g + cη ] c g 38,y Et L,y L,y L,y [ ] min + π g + π,y Et 4 L,y 4L η.,y Theorem.15. Let be an embedding and let T be the maimal eistence time of 4. Then T c, g. 39 Proof. Remark.13 assures that the statement is true if the evolution develops a type I singularity at t = T. Now, we can assume that the evolution develops a type II singularity at t = T. In particular it follows that ηt =. Let τ := sup{t [, T ] : ηs > on [, t]}. Notice that τ > due to the fact that is an embedding. The thesis will follow if we show that τ is bounded below by a constant depending only on and g. Since ητ =, we can find t 1 < t τ such that ηt 1 = η := min η, ηt = η ηt η, η for all t t 1, t. In particular, letting a := + π /4 g and b := π /4, from 38 we have η a + L,y b L,y η a + L,y b η L,y a b η, which implies τ t t 1 b η a = η g π 1

13 .6 The graph case We assume now that the curve can be parametrized as, t =, u, t, with [, 1] and u C [, 1], with the following periodic-type boundary conditions: u, t u, = u1, t u1, u, t = u 1, t for all t [, T ]. Notice that is not a closed curve, however it can be etended to a periodic infinite curve, that is, it is a closed curve on a suitable flat torus, and all the results of the previous sections also apply to this case. In this parametrization, Equation 4 becomes u 4 u t = 1 + u + g, u 1 + u. 41 We say that solves 41 if, t =, u, t, where the function u solves 41. Let us recall the following interpolation inequalities [16]. Proposition.16. Let u H 1 [, 1] L p [, 1], with p [, + ]. We have u L p where the constants C p, B p depend only on p. p p+ p p C p u L u L + B p u L. 4 The following Proposition can be easily derived from Proposition.16 see [3]. We recall that L p is the intrinsic L p space on the curve, see [3]. Proposition.17. Let z be a smooth function defined on the support of, where is a C 1 curve, and let p [, + ]. We have p z L p C p p z s p+ L z p where the constants C p, B p depend on p but are independent of. L + B p z L. 43 In particular, choosing p = 4, 43 becomes z 4 ds = z 4 L 4 z C s L z 3 L + z 4 L. 44 Lemma.18. Let u be a smooth solution of 4-41, and let We have F := arctant dt = arctan log t 1 + u d C 1 + u d 45 1 t F u d C u d t 1 + u d C + C 1 + u d, 47 where the constants C > depend only on g L. 13

14 Proof. Inequality 45 can be obtained eactly as. In order to show 46, we compute 1 t F u d = = u t arctan u d u t + gu t 1 + u d 48 g u d which leads to 46. We now prove 47. Letting e 1 = 1, R and z := 1/τ e 1 = 1 + u, from 7 we get z t = κ + g s z ν e We compute t z ds = = = zz t κκ + gz ds = κ + g 3κz + 6z z s ν e 1 ds = 3 z 3 κ + g s ν e 1 κκ + gz ds z 1 z 1 z 1 z s + gz s ds ν e 1 z 1 z 1 z s + gz z z 1 1 z 1 z s κ + gz s 1 + z ν e 1 ν e 1 zs + g z z 1 ds 3 zs + g z 4 ds 3 z s L + C g z s L z 3 L + z 4 L C g 4 where we used 44 to estimate z L 4. 3 z ds + C g z ds, Proposition.19. Let g, y C [, 1], and let u C [, 1], with u = u 1. Then, there eists T > depending only on u W 1, and g L such that Equations 4-41 admit a smooth solution u C [, 1] [, T ]. Moreover ut, H1 [,1] K for every t [, T ], where K depends only on u W 1, and g L. Proof. By standard parabolic regularity theory [14], it is enough to show that the gradient u remains bounded for a time T as above. From 47 we obtain that u L 3 [, 1] [, T ], with norm bounded by a constant depending only on T and g. Theorem.15 gives that κ = u 1 + u 3/ L [, 1] [, T ], for T depending only on u W 1, and g L. Therefore, we also get u L [, 1] [, T ]. ds ds 14

15 Lemma.. We have the continuous embedding H 1 [, T ], L [, 1] L [, T ], H 1 [, 1] C 1, 1 4 [, 1] [, T ]. Proof. Let u H 1 [, T ], L [, 1] L [, T ], H 1 [, 1], and let, t, y, s [, 1] [, T ], with < y, t < s. Since u L [, T ], H 1 [, 1], we have u, t uy, t y Moreover, being also u H 1 [, T ], L [, 1], we have 1 u, t u, s L = u, t u, s d s t u dσ C y. 5 [,1] [,T ] u t ddt Cs t. By 4 with p =, this implies u, t u, s L C u, t u, s 1 L u, t u, s 1 L + u, t u, s L The thesis follows from 5 and 51. C s t Proposition.1. Let u H 1 [, 1], let g n C [, 1] L [, 1], with g n L C for every n, and let u n C [, 1] [, T ] be the solutions of 4-41 given by Proposition.19, with g = g n and with initial data u n converging to u in H 1 [, 1]. Then there eists u H 1 [, T ], L [, 1] L [, T ], H 1 [, 1] such that u n u, up to a subsequence, uniformly on [, 1] [, T ]. Proof. By Proposition.19 there eist T >, depending only on u H 1 and C, such that the solutions u n are uniformly bounded in L [, T ], H 1 [, 1]. Moreover, using the equality for 48, we obtain 1 u n t d g n un d t F u n d and integrating it in time we also get a uniform bound of u n in H 1 [, T ], L [, 1]. It then follows that the sequence u n converges, up to a subsequence as n +, to a limit function u in the weak topology of H 1 [, T ], L [, 1] L [, T ], H 1 [, 1]. By Lemma., u n are uniformly Hölder continuous and then we conclude by Arzelà-Ascoli theorem that, along a subsequence, u n u uniformly. We are interested in studying solutions of 41 when g is only a L -function. We consider the simpler case in which g is independent of u, i.e. g, y = g. In this case we define the following notion of weak solution. Definition.. We say that a function u H 1 [, T ], L [, 1] L [, T ], H 1 [, 1] is a weak solution of 41 if u t ϕ + arctanu ϕ g 1 + u ϕ ddt = 5 [,1] [,T ] for all test functions ϕ C 1 c [, 1], T, with periodic boundary conditions. 15

16 We have the following eistence theorem for weak solutions to 41. Theorem.3. Let g, y = g, with g L [, 1], and let u W, [, 1] satisfying u = u 1. Then, there eists T > depending only on u and g such that Equation 41 admits a weak solution u W 1, [, T ], L [, 1] L [, T ], W, [, 1] with initial datum u. Proof. Let g n C [, 1] be a sequence of smooth functions which converge to g weakly* in L [, 1]. By Propositions.19 and.1 there eist T >, depending only on u H 1 and g L, and smooth solutions u n of 41 which converge, up to a subsequence, to a limit function u in uniformly and in the weak topology of H 1 [, T ], L [, 1] L [, T ], H 1 [, 1]. Let us prove that u is a weak solution of 41. The main point is showing that u n converge to u almost everywhere, so that we can pass to the limit in 5. We compute t u t u = u t u tt = u t 1 + u + g 1 + u = u t u t 1 + u t u u u u t u 1 + u + g t 1 + u 1 u t u 1 + u + g u u 1 + u 1 + u u t In particular, applying the same computation as 53 to u n, we obtain that u nt is decreasing in time. Indeed if M n t = sup [,1] u n t /, 53 gives that M nt. Therefore u W 1, [, T ], L [, 1]. Moreover, since g depends only on we have t 1 u t d = u t u tt d = u t 1 + u u t 1 + u 1. arctanu t u t + g 1 + u u t d t + gu u t u t 1 + u d + g u u t d 1 g u t u L = C where the constant C > depends only on u and g. We then get arctanu = u t g 1 + u d C t [, T ] [,1] [,1] [,T ] [,1] arctanu t d dt = [,1] [,T ] u t u d dt C. 54 As a consequence, the function arctanu n is uniformly bounded in H 1 [, T ], L [, T ] L [, T ], H 1 [, 1]. Therefore, the sequence arctanu n converges, up to a subsequence, to arctanu uniformly on [, 1] [, T ]. Since arctan is injective this implies that the 16

17 sequence u n converges to u a.e. on [, 1] [, T ], and we can pass to the limit in 5, obtaining that u is a weak solution of 41. Finally, being arctanu continuous, possibly reducing T we have that u is also continuous hence bounded on [, 1] [, T ]. In particular, recalling 41 the uniform bound on u t implies an analogous bound on u, that is u L [, T ], W, [, 1]. Remark.4. If u is only in H 1 [, 1], since the sequence u n is uniformly bounded in H 1 [, T ], L [, T ], reasoning as in Theorem.3 we get u W 1, loc, T ], L [, 1] L loc, T ], W, [, 1]. We conclude the section with a comparison and uniqueness result for solutions to 41. Theorem.5. Let g, y = g, with g L [, 1], and let u 1, u be two solutions to 41 such that u 1, u, for all [, 1]. Then u 1 u on [, 1] [, T ]. In particular, there is a unique solution to 41, given an initial datum u W, [, 1]. Proof. Let dt := min [,1] u, t u 1, t. Possibly replacing u 1, with u 1, δ, we can assume that d = δ >. The thesis now follows if we can show that dt δ for all t [, T ]. Let w = u u 1, so that From 54 it follows w t = u 1 + u u u 1 + g 1 + u 1 + u w t L [, T ], H 1 [, 1] L [, T ], C α [, 1] for all α < 1/. Choose now t [, T ] such that w t, t C α [, 1] and notice that, for all [, 1] such that dt = w, t, we have w = u u 1 =. In particular, recalling 55, w is twice differentiable at and we have w t = u 1 + u u u 1 For almost every t [, T ] we then get w = 1 + u 1. dt = min w t, t, : dt=w,t which gives the thesis. 17

18 Remark.6. We point out that in general we have T < + in Theorem.3, since the derivative u may blow up in finite time. This is related to the so-called fingering phenomenon, and we shall give an eplicit eample of such behavior. Let u =, g = M for [, 1/, and g = M for [1/, 1], where the constant M is greater than 4. For all t, we set u, t := u +, t := 1 + M 4 t M 4 t 4, 1 1, 1. A direct computation shows that u, u + are respectively a subsolution and a supersolution of 41 on their intervals of definition. By Theorem.5, for all t [, T ] it follows that u, t u, t on, 1/, and u, t u +, t on 1/, 1. Since u is continuous, this necessarily implies T 1/4M 4. More precisely, if we etend the solution u on a maimal time interval [, T ma, we have T ma 1/4M 4 and the derivative u 1/, t blows up in absolute value as t T ma. 3 A homogenization problem Given a smooth function g which is periodic on [, 1], we consider the following homogenization problem u t = u 1 + u + g ε, u 1 + u ε 56 with initial data u, = u, satisfying 4. Notice that, after the parabolic rescaling s = t/ε, y = /ε and v = u/ε, problem 56 becomes v s = v yy 1 + vy + εg y, v 1 + vy 57 which has been studied in [6], where eistence of traveling wave solutions for 57 has been established. Moreover in [13] see also [5] the authors discuss the uniqueness of traveling fronts and characterize the asymptotic speed in some particular case. A straightforward application of the results in Subsection.6 gives the first result about the convergence of the solutions to the perturbed problem 56. Proposition 3.1. Let u ε be the solutions to 56 with initial data u H 1 [, 1] satisfying 4. Then, up to a subsequence, u ε u H 1 [, T ], L [, 1] L [, T ], H 1 [, 1] as ε, uniformly on [, 1] [, T ]. Proof. By Proposition.19 there eists T > independent of ε and a family of smooth solution u ε of 56, which are uniformly bounded in H 1 [, T ], L [, 1] L [, T ], H 1 [, 1]. In particular, as in Propositon.1, we can pass to the limit, up to a subsequence as ε, and obtain that u ε u H 1 [, T ], L [, 1] L [, T ], H 1 [, 1] uniformly on [, 1] [, T ]. 18

19 There are two main open problems related to the homogenization of equation 56: - the characterization of u as the solution of an appropriate homogenized equation - the convergence on large time intervals. About the second question, we epect the following result. Conjecture 3.. If the initial datum u is Lipschitz continuous, then the convergence in Proposition 3.1 is on [, 1] [, +. Let L > be the Lipschitz constant of u. Due to the comparison principle and the periodicity of g, for all N N we have the estimate u ε, t u ε + Nε, t [L] + 1Nε, 58 where [L] denotes the integer part of L. Passing to the limit in 58 as ε, we get u, t uy, t L y, that is the norm u, t W 1, is non increasing in t. We epect this bound to be true also for the approimating sequence u ε, which would imply that we can take T = +. About the first question, we have some partial results about the characterization of the limit equation. We state a result when g depends only on. Theorem 3.3. Let g, y = g L [, 1], and let u W, [, 1] with u = u 1. Then, there eists T > depending only on u and g such that the solutions u ε to 56 converge in W 1, [, T ], L [, 1] L [, T ], W, [, 1], as ε, to the unique solution u of u t = u u + g d + u 59 with initial datum u and boundary conditions 4. In particular, u C [, 1], T ]. Proof. The proof is a straightforward adaptation of Theorem.3. Remark 3.4. When g, y = g W 1, [, 1] and periodic, an analogous result can be obtained in general dimensions and for all times, i.e. the solutions of u ε to 56 converge, as ε, to the unique solution u of 59 locally uniformly in [, 1] [, +. This can be proved by the so-called perturbed test function method cf. [4], which is by now a standard method in viscosity solutions theory applied to homogenization problems. More precisely, one considers the formal asymptotic epansion u ε, t = u, t + ε χ ε,, t 6 where the corrector χξ,, t = ψξ, Du, t is periodic in ξ and solves, for every fied p = Du, t, the cell problem [ tr I p p ] p Dξξχ = gd gξ + p in R n. [,1] n 19

20 Plugging the epansion 6 in 56, and using the comparison principle for viscosity solutions to 59, one obtains that u solves 59. In the general case, we can determine the limit equation satisfied by u only in a very specific case, that is when the initial data are plane-like. Proposition 3.5. For every ε > let u ε H 1 [, 1] satisfying 4 and such that u ε α in H 1 [, 1] as ε, for some α R. Then, letting u ε be the solution to 56, 4 with initial datum u ε, we have uniformly in [, 1] [, T ], with cα := lim u ε, t = u, t := α + cα 1 + α t, ε { if Gs = for some s [, 1] Gs ds otherwise, where 1 Gs := lim L L L g, α + s d. Proof. By [6, Thm. 4.1] see also [13] for all α R there eist global smooth solutions û α,ε of 56, with average slope α, which are pulsating waves, that is, there eist τ > and a vector v 1, v Z, depending on α, ε and such that We let û α,ε, t + τ = û α,ε εv 1, t + εv, t R. cα, ε = ε v 1, v ν α τ α where ν α =, α 1 + α be the velocity of the wave in the normal direction ν α and we set cα, ε = if û α,ε is a standing wave. In particular, in [6, Thm. 4.1] it is shown that û ε can be represented as û α,ε, t = α + cα, ε 1 + α t + Oε, t R, 61 where Oε Cε, for a constant C depending only on the C -norm of g. Moreover, by [6, Cor..5] the derivatives û α,ε, t are uniformly bounded for all, t R and for all ε small enough. Notice that, by [6, Prop. 4.4], for all, t R we have cα, ε = û α,ε t = cα, ε > û α,ε t > cα, ε < û α,ε t <. In particular, without loss of generality we can assume that û α,ε t, t R.

21 One can now argue as in [5, 4.9 and 4.1] and conclude by maimum principle that û α,ε α Cε, t R, 6 for a constant C depends only on g. Integrating 56 on [, 1] and using 6, a direct computation as in [5, Prop. 6] gives lim ε cα, ε = cα. Finally, by comparison principle for solutions to 56, we can use the functions û α,ε as barriers for u ε, and obtain that uniformly in [, 1] [, T ]. lim u ε, t = u, t = α + cα 1 + α t, ε Remark 3.6. Notice that cα = [,1] g for all α Q, so that the function α cα is not necessarily continuous. This suggests that the homogenization limit of should be a geometric evolution of the form v = cν, κ ν where the function c is in general discontinuous. References [1] U. Abresch, J. Langer. The normalized curve shortening flow and homothetic solutions. J. Differential Geom , no., [] S. Altschuler. Singularities of the curve shrinking flow for space curves. J. Differential Geom , [3] T. Aubin. Some Nonlinear Problems in Riemannian Geometry. Springer, New York [4] P. Cardaliaguet, P.-L. Lions, P. E. Souganidis. A discussion about the homogenization of moving interfaces. J. Math. Pures Appl. 91 9, [5] B. Craciun, K. Bhattacharya. Effective motion of a curvature-sensitive interface through a heterogeneous medium. Interfaces Free Bound. 6 4, [6] N. Dirr, G. Karali, N. K. Yip. Pulsating wave for mean curvature flow in inhomogeneous medium. European J. Appl. Math. 19 8, [7] K. Ecker, G. Huisken. Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. Comm. Math. Phys , no. 3, [8] M. Gage, R. S. Hamilton. The heat equation shrinking conve plane curves. J. Differential Geom , [9] G. Huisken. Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom , no. 1,

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