A variational approach to a quasi-static droplet model

Transcription

1 A variational approac to a quasi-static droplet model Natalie Grunewald and Inwon Kim Abstract We consider a quasi-static droplet motion based on contact angle dynamics on a planar surface. We derive a natural time discretization and prove te existence of a weak global-in-time solution in te continuum limit. Te time discrete interface motion is described in comparison wit barrier functions, wic are classical sub- and super-solutions in a local neigborood. Tis barrier property is different from standard viscosity solutions since tere is no comparison principle for our problem. In te continuum limit te barrier properties still old in a modified sense. Contents 1 Introduction 1 2 Construction of a time discrete solution 4 3 Te barrier properties for time discrete solutions 8 4 Te continuum limit and existence of weak solutions 15 1 Introduction Te motion of liquid drops on a planar surface is a widely studied topic. We consider a quasi stationary free boundary model, derived in [8], [10] and [12]. Te model is contact angle driven, i.e. te motion of te boundary of te wetted region is due to a deviation of te contact angle from te ideal contact angle. It is also quasi stationary in te sense tat te actual profile of te drop adjusts itself to te wetted region by minimizing a surface energy under a volume constraint. We derive a natural time discretization by exploiting a formal gradient flow structure of te model. Te time-discrete solutions satisfy barrier properties similar to standard viscosity solutions. Tese barrier properties stay valid in a modified sense as te time step size goes to zero. Let us begin by a formal introduction of te model. Te profile of te droplet is given by te eigt function u : R N (0, T ) R wit N = 2, te positive pase {u > 0} denotes te wetted region and te Inst. für Angewandte Matematik, Universität Bonn, Germany, grunewald@iam.uni-bonn.de Dept. of Matematics, UCLA, Los Angeles CA 90095, USA, ikim@mat.ucla.edu 1

2 free boundary {u > 0} denotes te contact line between drop, air and surface. It sould be pointed out tat our analysis is performed in general space dimension N. Trougout te paper we denote te spatial derivative of u by Du. Te motion of te droplet is described by contact angle dynamics - te free boundary {u > 0} evolves by a relationsip between te outward normal velocity V and te contact angle Du of te droplet wit te surface. In tis paper te normal velocity is given by V = Du 2 1 on {u > 0}. Te square of te contact angle in te velocity law seems natural, as it is te only power for wic we directly ave a gradient flow structure like te one considered in tis paper. For discussion of te contact angle dynamics in form of more general free boundary velocities we refer to [18]. On te oter and te sape of te drop adjusts to te wetted region by obeying two constraints: First te volume in eac component i of te drop is kept constant over time. Secondly te liquid/vapor interface is minimal in te sense tat it minimizes te Diriclet integral, leading to te Euler Lagrange equation u(, t) λ in eac connected components of {u > 0}. Tis equation, a simplification of minimal surface equation, defines te sape of a quasi-static droplet. By coosing a suitable Lagrange multiplier λ = λ I (x, t), te volume of droplets in eac component can be preserved. Summarizing above discussion we arrive at te following free boundary problem: u(, t) = λ i (t) in i (t); (P ) V = Du 2 1 on i (t); i(t) u(, t) dx c i, were, as mentioned above, V is te outward normal velocity of te connected component of te support of te drop i (t), so for Du 0 one as V = tu Du. As te overall volume is conserved we ave i c i 1. Several serious callenges arise in developing a global notion of solutions for te model described above: Most importantly, (P ) does not satisfy te comparison principle between solutions, even in te case of single components. For example consider two sets D 1 D 2 IR n wit te droplet profile u i (x, 0) supported in D i for i = 1, 2. Suppose we ave te same volume constraint, i.e., u i = 1. Since we assume a quasi-stationary profile for u i, tey satisfy te first equation in (P ): u i (x, 0) = λ i in D i, Due to te volume constraint and te fact tat D 1 D 2, it is clear tat λ 1 > λ 2. Terefore it may be te case tat u 1 > u 2 in some parts of D 1. Also te fact tat λ 1 > λ 2 and te second equation in (P ) 2

3 suggests te possibility tat te free boundary velocity of {u 1 > 0} is bigger tan in {u 2 > 0} in some parts, and terefore te evolution of D 1 and D 2 by (P ) may reverse te inclusion order between te sets. Due to te failure of comparison principle, te viscosity solutions approac applied to mean curvature flow (see [7] and [4] for example) does not apply ere, even if we assume tat tere is no topology cange. Observe tat if λ is independent of time ten standard viscosity solution teory as in [14] applies. Based on tis observation a discrete-time approximation wit fixed λ in eac time step was carried out in [9]. Tis way a unique weak solution is obtained for star-saped initial data, for sort times (as long as te wet region stays star-saped). However approximating (P ) wit fixed λ in small time intervals (apparently) does not work well wit topology canges. On te oter and topology canges seem unavoidable. Splitting of droplets into multiple components is generic for non-convex droplets, even if we start te evolution wit a simply connected droplet. Merging of different parts of te droplet also naturally occurs. (Recall tat our model is quasi-stationary. Tis means tat te dynamics inside te liquid pase is not modeled. In some sense wen a topology cange occurs we fast forward te time so tat te droplet becomes quasi-stationary again.) In addition to te topological canges, we expect corner or cusp formation on te interface, due to merging, splitting, and also srinking of droplets (see [9]). Lastly, tere is a bifurcation (non-uniqueness) of solutions in te event of merging. More precisely, two stationary drops toucing eac oter at exactly one point can eiter decide to stay as tey are, or see eac oter and develop into one big drop. A similar bifurcation was also observed, for solutions of a flame propagation model ([17]). Our goal is to introduce a global-time notion of weak solution wic describes (P ) past topological canges and singularities. We take a variational approac, based on te following observation. Formally speaking te droplet evolution (P ) is a gradient flow for te energy E() := Du 2 dx +, (1.1) were denotes te (Lebesgue) measure of. Te gradient flow takes place on te manifold of possible supports of te droplet. Te droplet eigt u itself is ten part of a tangent bundle above te manifold. We refer to Section 2 for detailed discussion of tis structure. In Section 2 we approximate te solution (P ) by a time-discrete gradient flow (JKO) sceme, originated by [13]. Tis sceme defines te solution in te next time step as a minimizer of a composited functional. Tis functional consists partly of te energy and partly of te distance to te previous time step. See Section 2 for details. Suc approac was taken before by Almgren,Taylor and Wang [1] and Luckaus and Sturzenecker [15] for mean curvature motion. In [4] it was sown tat a particular selection of te discrete sceme in [1] converges to viscosity solution of te mean curvature flow in te sense of [7]. As mentioned above our problem lacks te comparison property even in simple settings, wic prevents us to develop any connection to standard viscosity solutions approac. However it is still possible to describe te evolution of solutions by barrier properties (Proposition 3.1 and 3.3) of te time-discrete weak solutions. Rougly speaking tis means tat te time-discrete solutions evolve wit te free boundary velocity given by (P ), at regular points of te interface. 3

4 In te continuum limit we sow tat a global-in-time weak solution (see Definition 4.4) exists. At te moment, we are only able to describe te limiting free boundary beavior in terms of te liminf and limsup of te time-discrete solutions. We refer to Section 4 for definition of weak solutions (Definition 4.4) and precise statements (Teorem 4.6). 2 Construction of a time discrete solution We consider a generalized version of (P ) wit curvature: (P ɛ ) u(, t) = λ i (t) in i (t); V = Du 2 1 ɛκ on i (t); i(t) u(, t)dx u(x, 0)dx. Here ɛ 0 and κ = ( Du Du ) denotes te mean curvature of te interface, positive if te positive pase {u(, t) > 0} is convex. Te curvature term in (P ) ɛ is introduced to use te structure of Caccioppoli sets in te variational arguments in Section 3. However te regularized problem (P ) ɛ and teir properties are also of independent interest. Let us start wit te definitions: Definition 2.1 Let B := {x IR n : x < R} wit R a sufficiently large constant. (a) Let us define te set of Caccioppoli sets Cacc := { B ; is a Borel set wit finite perimeter}. (b) For any Cacc and any volume c { u,c := argmin Du 2 dx : u H 1 (), supp u, } u dx = c. Remark 2.2 Note tat te minimizer u,c exists for any c > 0 and any set Cacc tat admits one H 1 function u wit supp u. Definition 2.3 For a nonnegative function u H 1 (IR n ) and for x Ω(u) we define λ(u)(x) := Du 2 dx udx (2.1) were is te connected component of Ω(u) wic contains x. If udx = 0 we set λ(u)(x) = 0. Note tat u,c = λ(u,c )(x) λ(u,c ) in its positive set, if as a single component wit smoot boundary. 4

5 Problem (P ɛ ) is a formal gradient flow on Cacc for te energy E ɛ () := Du,1 2 dx + + ɛ per(), (2.2) were and per() respectively denote te Lebesgue measure and te perimeter of. To see tis we calculate te differential of E ɛ for some normal velocity field ṽ applied to and δu te cange of u introduced by ṽ: diff E ɛ ().ṽ = 2Du Dδu dx + (1 + Du 2 ) ṽ ds + ɛκ ṽ ds = 2 u δu dx + 2 Du δu + (1 + Du 2 + ɛκ) ṽ ds = λ δu dx + 2 Du 2 ṽ + (1 + Du 2 + ɛκ) ṽ ds = (1 Du 2 + ɛκ) ṽ ds. Tis gives (P ɛ ) for te Riemanian structure g (v, ṽ) := v ṽ ds v, ṽ T Cacc, (2.3) on Cacc, by te volume conservation and Definition 2.1. As te distance connected to (2.3) are difficult to model, we introduce a modified distance, wic was originally introduced in [1] and [15] (also see e.g. [5] and [4].) dist 2 ( 0, 1 ) := dist(x, 0 ) dx. 0 1 Here dist is te distance function, and 0 1 denotes te symmetric difference between te two sets. Note tat dist 2 is not a (squared) distance function (it lacks e. g. symmetry), but an approximation of te distance connected to (2.3). Following [15] and te JKO sceme [13], i+1 is determined from te previous set i by } i+1 { 1 = argmin Cacc dist 2 ( i, ) + E ɛ (). Lemma 2.4 For fixed > 0, fixed volume c and any 0 Cacc tere exists at least one minimizer Cacc of min c F() := 1 dist 2 ( 0, ) + E ɛ (). Note tat we do not sow uniqueness. We also do not expect uniqueness for (P ) or (P ɛ ), see Section 1. Te dependence on c is suppressed in te notation of E ɛ. 5

6 Proof. Tere exist sets B suc tat F() < (e.g. speres around 0 ) and F() 0. Terefore tere exists a minimizing sequence { k } Cacc suc tat F( k ) k inf{f() : B}. By te definition of E ɛ () we ave k + ɛ per( k ) < C and terefore te indicator functions χ k are uniformly bounded in BV norm. Tus (see e.g. [6], p.176) tere exists a subsequence and a function χ BV (B) suc tat χ k χ in L 1 (B) Since χ k take values in {0, 1} so does χ and tere exists a set c min B suc tat χ = χ min. It c remains to sow tat F(c min ) inf F( k ). Tis is direct for te part of te energy k + ɛ per( k ), by te lower semi continuity of te perimeter and te L 1 convergence of χ k. For te remaining part of te energy we ave to take into account te convergence of te corresponding droplet wit volume c, u k,c. By te boundedness of te H 1 norm of u k,c u k,c ũ in L 2 (B). Were ũ = c and u k,c = u k,c χ k ũ χ min c a.e. in B. Terefore by te lower semi-continuity of H 1 norm and Definition 2.1 inf Du k,c 2 Dũ 2 Du min,c 2. k,c min c On te oter and dist 2 is continuous wit respect to te L 1 topology of te indicator functions: dist 2 ( 0, ) dist 2 ( 0, ) = dist(x, 0 )dx dist(x, 0 )dx 0 0 = dist(x, 0 )dx. ( ) 0 Tis vanises as χ χ L1 (IR N ) = 0, by te boundedness of te distance function in B. min c Definition of te time discrete evolution. We define a time discrete evolution of (P ). Rougly speaking we do te minimization in Lemma 2.4 for eac component of te drop separately. If two components merge at te next time step, we go back and do te same minimization step but for te two components togeter. Splitting of a component is already taken care of in te minimization in Lemma 2.4, as min migt ave several components. To be more precise: for fixed > 0 and i N take te previous state i Cacc wit possibly infinitely many connected components i,k Cacc, k N. For eac connected component (in te classical sense) we ave some droplet u i,k,c by Remark 2.2 and k Lemma 2.4. Ten i+1 is given by i+1 := k c min k if for any l m : c min l c min m =, (2.4) 6

7 were c min k is a minimizer in Lemma 2.4 for te connected component i,k. If c min l c min m l, m), ten we define for only one pair (l, m) and if it does not intersect wit oter components ( min c k i+1 := ( k l,m c min k ) c min l +c k. were c min l +c k is a minimizer in Lemma 2.4 for initial set i,l i,m. In general te process of sorting out merging components is non-unique: we will prescribe te following process to proceed witout ambiguity. Let us first consider te maximal index set I 1 suc tat eac element c min k wit k I 1 intersects wit c min 1. Next take te first element c min k wit k / I 1 and repeat te process to create te second index set I 2. If I 2 intersects wit I 1, ten we replace I 1 wit I 1 I 2. If not, ceck weter intersects wit min I 2 index set I 3, and ceck against min I 1 sets I 1, I 2,... suc tat min I k Now define min I 1 := min Σc k, k I 1, k. If yes ten still replace I 1 wit I 1 I 2. If no, ten proceed to create te tird and I min 2. Tis way we end up wit a sequence of (disjoint) index are all disjoint. Ten i+1 := k min I k. and u (, t) := k u Ik := u min I,Σ j Ik c j k u Ik for t [ i, (i + 1) ). (2.5) Tis way u is a H 1 function in B at any time t [0, T ]. Tus u L 2 loc(h 1 (B)). Te total volume of u at time t is u (, t)dx = k c k = 1. As te JKO sceme is constructed to describe a time discrete gradient flow, we ave te energy decrease for free: Suppressing in te notation te dependence of te energy on te volumes in eac component, we ave: Lemma 2.5 Te time evolution defined in (2.4) and (2.5) satisfies E ɛ () i 1 dist 2 ( i, i+1 ) + E ɛ( i+1 ). (2.6) Proof. Equation (2.6) is obvious for any components k I j i,k and te corresponding minimizer I min j = i+1, as Lemma 2.4 can be tested wit te set k I j i,k. Furtermore I j E ɛ ( i ) = j E ɛ ( k I j i,k ) 1 dist 2 ( j i,k k I j, min I j ) + j 1 dist 2 ( i, i+1 ) + E ɛ( i+1 ). E ɛ ( min I j ) 7

8 3 Te barrier properties for time discrete solutions In tis section we sow, tat for fixed time step > 0 te discrete time solution constructed above satisfies te free boundary motion law in time scale, in te sense tat it is comparable to smoot sub and super solutions of (P ɛ ) in local neigboroods. A more precise statement will follow in Propositions 3.1 and 3.3 for wic we need te following notation: Let us denote te positive pase of a function u(x, t) : IR N [0, ) IR + and its boundary by: and te positive pase in space time by: Ω t (u) := {u(, t) > 0} and Γ t (u) := {u(, t) > 0}, Ω(u) := {u > 0} {B [0, )} and Γ(u) := Ω(u). Next we sow te barrier properties for te time discrete solutions. We begin wit te barrier property for u being a super solution. Tat is, u can be compared to a barrier function φ tat is below. If φ is not fast enoug at te boundary and not curved enoug in te interior, ten te ordering will persist: Proposition 3.1 (Super solution barrier property) Let u be defined by (2.5). Given a ball B r (x 0 ) in B let λ := inf {λ(u (0, ))(x), λ(u (, ))(x)}, x B r(x 0) were λ(u)(x) is as defined by (2.1). Suppose tere exists a smoot function φ wit Dφ 0 in B r (x 0 ) [0, ]. Furter suppose tat for some small δ > 0 φ(, t) < λ δ in B r (x 0 ) [0, ], φ t Dφ ( Dφ 2 1 ɛκ φ ) < δ on Γ(φ) (B r (x 0 ) [0, ]), (3.1) were κ φ := ( Dφ Dφ ) is te mean curvature of te corresponding level set of φ. Ten for sufficiently small > 0 depending on δ, r, te minimum of Dφ and te C 2 -norm of φ in B r (x 0 ) [0, ] te following olds: If φ u on te parabolic boundary of B r (x 0 ) [0, ], ten φ(, ) u (, ) in B r (x 0 ). φ Note tat t Dφ = V, were V is te outward normal velocity of {φ > 0} wit respect to te positive set of φ. Terefore, Proposition 3.1 sows tat a function φ wic is a sub-solution of (P ɛ ) can not cross te discrete time solution u. Tus, u is a super solution. We also mention tat a local barrier function like te ones in Proposition 3.1 can always be extended to a global barrier function satisfying (3.1), wic is not restricted to a ball B r. We begin by a lemma wic states tat te support of φ cannot cross too muc. 8

9 Lemma 3.2 Under te assumptions of Proposition 3.1 and for sufficiently small > 0, tere exists a constant C > 0 independent of > 0 suc tat (φ(x, t) C 1/2 ) + u (x, t) in B r (x 0 ) [0, ]. (3.2) In particular, if φ crosses u on B r (x 0 ) [0, ], we can coose 2 < τ < C 1/2 suc tat ϕ(x, t) := (φ(x, t) τ) + crosses from below by o( 2 ), i.e., 0 < (Ω (ϕ) ) B r (x 0 ) = o( 2 ). (3.3) Proof. Once (3.2) is proved, our second claim follows from te fact tat Ω (φ) does cross and φ is smoot wit Dφ > 0 near Ω(φ). To prove (3.2), first observe tat were Terefore, we ave for ϕ wit Ω (φ) { Ω 0 (φ) {x B : d(x, Γ 0 (φ)) C 1 } }, C 1 = sup φ t / Dφ. B r(x 0) [0,] Ω (ϕ) {x Ω 0 (φ) : d(x, Γ 0 (φ)) C 2 τ C 1 } C 2 = inf B Dφ 1 (x, t). r(x 0) [0,] Tus, (3.2) follows by te comparison principle if we sow To prove (3.4), let us define S := {x (Ω 0 (φ) B r (x 0 )) : d(x, Γ 0 (φ)) C 3 1/2 }. (3.4) ˆ := S and Σ := ˆ = S. Suppose tat Σ 0, oterwise we are done. Since Ω 0 (φ) 0, we ave by te smootness of φ dist 2 ( 0, ˆ ) dist 2 ( 0, ) C 4 Σ. were C 4 is proportional to te size of C 3. On te oter and, since te Diriclet energy decreases wen te domain increases, E ɛ (ˆ ) E ɛ ( ) Σ + ɛ per(ˆ ) ɛ per( ) Σ + ɛ S ɛ S C 5 Σ, were C 5 depends on φ. Te last inequality follows from S S Σ Dφ (x, ) η ds Dφ = Σ ( Dφ )(x, ) dx Dφ = Σ κ φ dx. 9

10 Here η is te outward normal vector at x Σ and κ φ is te mean curvature of te level set of φ. We conclude tat if C 3 is cosen sufficiently large, ten Tis contradicts te minimizing property of. dist 2 ( 0, ˆ ) + E ɛ (ˆ ) < dist 2 ( 0, ) + E ɛ ( ). Proof of Proposition 3.1: 1) Suppose te proposition is not true. Ten φ(x 0, ) > u (x 0, ) at some point x 0 Ω (u ). Due to te maximum principle for armonic functions, tis implies tat Ω (φ) (B\ ) for one of te components in Ω (u ). Let 0 be a corresponding component in Ω 0 (u ) wic gives rise to. For notational simplicity, we prove te proposition assuming tat is indeed te only component generated by 0, i.e. 0 as not splitted into multiple components and is generated by only one component: te proof for te general case is parallel. 2) Let us define ϕ(x, t) := (φ(x, t) τ) + were 2 < τ < C 1/2 is as given in Lemma 3.2. In te proof we use ϕ instead of φ, wic is possible witout violating te assumptions according to Lemma 3.2. Next set We claim tat := ( Ω (ϕ) B r (x 0 ) ). dist 2 ( 0, ) + E ɛ ( ) < dist 2 ( 0, ) + E ɛ ( ), (3.5) wic yields a contradiction to te minimizing property of. 3) To prove (3.5) first observe tat dist 2 ( 0, ) dist 2 ( 0, ) = signdist(x, 0 )dx signdist(x, Γ 0 (ϕ))dx, were signdist is te signed distance function, tat is negative inside te set. Here te first equality is due to straigtforward computation, and te inequality is due to te fact tat Ω 0 (ϕ) is a subset of 0. By construction of ϕ, for eac point x tere exists a time t wit 0 t + o( 2 ) suc tat x Γ t (ϕ). Terefore, as ϕ t Dϕ Next we consider te energy difference (0, ) denotes te outward normal velocity of Γ(ϕ), signdist(x, Γ 0 (ϕ)) ϕ t Dϕ E ɛ ( ) E ɛ ( ) = I + II + III + o(). (3.6) 10

11 were I = Du 2 (, ) Dũ 2, II = 1 dx, III = ɛ per( ) ɛ per( ). Here ũ (x) := u, R u. In te next step we will sow tat I Dϕ 2 (, ) dx and III ɛκ ϕ dx Tis proves our claim by (3.1) and (3.6). Note tat (3.1) is strict and terefore extends to a small region inside. 4) Let us estimate III. Note tat, as before, per( ) per( ) \ Dϕ Dϕ (, ) η ds \ Dϕ (, ) η ds Dϕ = ( Dϕ )(, )dx Dϕ = κ ϕ dx were η = Dϕ/ Dϕ (x, ) is te outward normal vector at x, η is te outward normal vector at x, and κ ϕ is te mean curvature of te level sets of ϕ. It remains to estimate I. To tis end let us define two auxiliary functions, ū and v: ū = λ in wit supp (ū) =, v = 0 in wit v = ϕ(, ) on. (3.7) We remark tat ū is defined by approximation from outside and v is defined by approximation from inside, i.e. ū(x) := inf{f(x) : f = λ in {f > 0} wit w {f > 0}}, and v(x) := sup{f(x) : f = 0 in wit f < φ on B\ }. Let us define c := u (, ), c := ū and ũ := c cū. Ten te following olds: Du (, ) 2 Dũ 2 = λ u (, ) λ(ũ) ũ = c c λ ( ū u). (3.8) Furtermore, ū max ( (u + v), ϕ ) (, ) since ( ( = Ω max (u + v), ϕ )) and max ( (u + v), ϕ)(, ) λ. For te same reason, on te reduced boundary of we ave, for te inward normal η, η (u (, ) + v) η ϕ (, ). (3.9) 11

12 Tus, ( ) λ ū u (, ) by (3.9) and (3.7). Tus togeter wit (3.8) we ave I c c Dϕ 2 (, ). λ v + λ ϕ(, ) (u (, ) + v) v ( ϕ ϕ)(, ) (D(u (, ) + v))dv + η (u (, ) + v) v + Dϕ 2 (, ) ( η ϕϕ)(, ) Dv 2 + Dϕ 2 (, ), Lastly, note tat as τ τ 0, λ(ũ) converges to λ(u ), due to te minimizing property of. Hence c c and we can coose τ τ 0 given in (3.3) small enoug tat c c(1 + o()) to conclude. By a parallel argument in te proof of Proposition 3.1, u can also be compared wit barriers wic are super-solutions of (P ɛ ): Proposition 3.3 (Sub-solution barrier property) B, let Let u be defined by (2.5). Given a ball B r (x 0 ) in λ := inf {λ(u (0, ))(x), λ(u (, ))(x)}, x B r(x 0) were λ(u)(x) is as defined in (2.1). Suppose tere exists a smoot function φ wit Dφ 0 in B r (x 0 ) [0, ]. Furter suppose tat for some small δ > 0 φ(, t) > λ + δ and φ t Dφ ( Dφ 2 1 ɛκ φ ) > δ in B r (x 0 ) [0, ]. (3.10) Ten for sufficiently small > 0 depending on δ, r, te minimum of Dφ and te C 2 -norm of φ in B r (x 0 ) [0, ] te following olds: If u φ + := max(φ, 0) on te parabolic boundary of B r (x 0 ) [0, ], ten u (, ) φ(, ) + in B r (x 0 ). Proof. Te proof is analogous to te proof of Proposition 3.1. We still present it, as te estimation of te Diriclet integral as a non-trivial difference from te previous proof. Suppose te above proposition is not true. Ten φ(, ) crosses u (, ) from above at some point in B r (x 0 ). As before, te maximum principle for armonic functions states tat ten Ω (φ) is 12

13 nonempty for a component of Ω (u ). Set 0 be te component of Ω 0 (u ) wic generates. Again we construct a contradiction to te minimizing property of and u. Wit a parallel argument to Lemma 3.2 one can cange φ to ϕ := (φ + τ) +, 2 τ C 1/2, suc tat u (x, ) (φ(x, ) + τ) + and 0 < ( Ω (ϕ)) B r (x 0 ) = o( 2 ). Tis time we denote: We claim tat = (( Ω (ϕ)) B r (x 0 )) ( (B\B r (x 0 )). dist 2 ( 0, ) + E ɛ ( ) < dist 2 ( 0, ) + E ɛ ( ). First observe tat tis time dist 2 ( 0, ) dist 2 ( 0, ) = signdist(x, 0 )dx signdist(x, Γ 0 (ϕ))dx. By integration of te velocity of Γ t (ϕ) we ave Next we consider te energy difference signdist(x, Γ 0 (ϕ)) ϕ t Dϕ + o(). (3.11) E ɛ ( ) E ɛ ( ) = I + II + III (3.12) were I = Du 2 (, ) Dũ 2, II = 1 dx, III = ɛ per( ) ɛ per( ). Here ũ(x) solves ũ = λ wit support, were λ is cosen suc tat ũ = u (, ). We will sow tat I Dϕ 2 (, ) dx and III ɛκ ϕ dx Tis proves our claim by (3.10), (3.12) and (3.11). First let us estimate III: per( ) per( ) Dϕ \ Dϕ (, ) η ds Dϕ \ (, ) η ds Dϕ = ( Dϕ )(, )dx Dϕ = κ ϕ dx were η = Dϕ/ Dϕ (x, ) is te outward normal vector at x, η is te outward normal vector at x, and κ ϕ is te mean curvature of te level sets of ϕ. 13

14 It remains to estimate I. We again consider te two auxiliary functions, ū and v defined by (3.7). As before we ave for c := u (, ) and c := ū: Du 2 (, ) Dũ 2 = c c λ ( ū u (, )). (3.13) But tis time te inequality (min[ū, ϕ(, )] v) + u (, ) olds on, as = supp (min[ū, ϕ(, )] v) + and (min[ū, ϕ(, )] v) λ. For te same reason we ave for te outward normal η of η (u (, ) + v) η ϕ (, ). (3.14) Tus, as min(ū, ϕ) = ū in and min(ū, ϕ) = ϕ in, using te smootness of ϕ it follows tat λ ( ū u (, )) λ v λ ϕ(, ) (u (, ) + v) v + ( ϕϕ)(, ) + ( ϕ λ) ϕ(, ) (D(u (, ) + v)) Dv η (u (, ) + v) v Dϕ 2 (, ) + ( η ϕϕ)(, ) + o() Dv 2 Dϕ 2 (, ) + o() by (3.14) and (3.7). Tus togeter wit (3.13) we ave I c c Dϕ 2. Lastly we need to sow tat c c as τ τ 0 0. To see tis, first note tat u (x, ) (φ(x, ) + τ 0 ) +. In particular u (, ) C τ τ 0 on w w {x : φ(x, ) + τ 0} were C depends on te C 2 -norm of φ. It follows tat u (, ) c c + O(τ τ 0 ). Hence we conclude. ū + C τ τ 0, and terefore 14

15 4 Te continuum limit and existence of weak solutions In tis section we sow tat in te limit 0 and ɛ = te time discrete solution u converges to a weak solution u(, t) H 1 (B) of (P ) in te sense tat te liminf and limsup -envelopes satisfy te barrier property at infinitesimal time scale (see Definition 4.4). We begin by defining viscosity sub- and supersolutions for a given multiplier function λ(x, t) : B [0, ) [0, ). Definition 4.1 A lower semi-continuous function u : B [0, ) IR is a viscosity super-solution on [t 1, t 2 ] wit respect to λ(x, t) if following olds: For a given function φ C 2,1 ({φ > 0}) wit Dφ 0 in B r (x 0 ) [t 1, t 2 ], suppose tat φ u on te parabolic boundary of B r (x 0 ) [t 1, t 2 ] wit Ten φ u in B r (x 0 ) [t 1, t 2 ]. φ(, t) < λ(, t) in B r (x 0 ) [t 1, t 2 ], φ t Dφ ( Dφ 2 1) < 0 in Γ(φ) {B r (x 0 ) [t 1, t 2 ]}. For te subsolution part, in te context of our limit as 0, we ave to take into account te possibility tat {u > 0} leave tin segments or isolated points in te limit, wic are not traceable from te limit of u. We get around tis difficulty by including a set Σ in te definition: Definition 4.2 Let u : B [0, ) IR + be upper semi-continuous, and let Σ be a closed subset of B [0, ) containing Ω(u). Ten te pair (u, Σ) is a viscosity sub-solutionon [t 1, t 2 ] wit respect to λ(x, t) if te following olds: For a given function φ C 2,1 ({φ > 0}) wit Dφ 0 in B r (x 0 ) [t 1, t 2 ], suppose tat φ(, t) > λ(, t) in B r (x 0 ) [t 1, t 2 ], φ t Dφ ( Dφ 2 1) > 0 in Γ(φ) {B r (x 0 ) [t 1, t 2 ]}. If u φ and Σ Ω(φ) on te parabolic boundary of B r (x 0 ) [t 1, t 2 ], ten u φ and Σ Ω(φ) in B r (x 0 ) [t 1, t 2 ]. Let us go back to te time discrete solutions u. Define G := {k2 n : k, n N} and = (n) = 2 n, n N. Ten u is defined on grid times t G by (2.5), wit te coice of ɛ =. Due to te Diriclet energy bound, along a subsequence u (, t) u(, t) weakly in H 1 (IR N ) for eac t G. (4.1) We ten coose a common subsequence of (n) suc tat (4.1) olds along te same sequence for eac time. We obtain a weak form of convergence in te continuum limit 0 along a subsequence. 15

16 Unfortunately a stronger, point-wise convergence of u cannot be obtained witout extra regularity of u suc as equicontinuity in time. Instead we consider te limit infimum and supremum: u (x, t) := lim r 0 inf u (y, s) (4.2) { x y r, s t r, r} and u (x, t) := lim r 0 sup u (y, s). (4.3) { x y r, s t r, r} Let us also define Σ := {(x, t) a sequence (x, t ) (x, t) suc tat x Ω(u (, t ))}. (4.4) Note tat Σ contains Ω(u ). Σ is a closed set, including traces of supports of u (, t) wic may degenerate into zero set of u in te limit 0. Let us denote Σ(s) := Σ {t = s}. Next we define appropriate limits for te multipliers to be used for u and u. Definition 4.3 For B, let us define and λ in (, c) := lim δ 0 λ(u δ,c) λ out (, c) := lim δ 0 λ(u δ,c), wit δ := {x : d(x, ) δ} and δ := {x : B δ (x) }. Clearly λ in λ out, as δ δ. Now we are ready to define our weak solution: Definition 4.4 For functions u 1, u 2 : B [0, ) and a closed set Σ B [0, ), te triple (u 1, u 2, Σ) is a weak solution of (P ) if te following olds: (a) u 1 u 2 and {u 2 > 0} Σ; (b) u 1 is a viscosity supersolution wit respect to λ 1 (x, t) := λ out (, c 1 ), were is te connected component of Ω t (u 2 ) containing x and c 1 := u 1 (, t)dx. (c) (u 2, Σ) is a viscosity subsolution wit respect to λ 2 (x, t) := λ in (, c 2 ), were is te connected component of Ω t (u 1 ) containing x and c 2 := u 2 (, t)dx, were D is te connected component of Ω t (u 2 ) containing. D 16

17 Rougly speaking, λ 1 and λ 2 are respectively te smallest and te largest possible value of te multiplier one can obtain by te lim sup and lim inf operation at a given point (x, t). Tese definitions are tailored for u 1 = u and u 2 = u. Remark 4.5 Classical solutions (i.e. u C 2,1 ( Ω(u)) wit smoot Γ(u) satysfying (P ) in te classical sense), if tey exist, would be weak solutions of (P ) in our definition wit u 1 = u 2 = u and Σ = Ω(u). In te rest of te section we will sow te following: Teorem 4.6 Te triple (u, u, Σ) defined in (4.2) - (4.4) is a weak solution of (P ). Remark In [9] it was proven tat starting from a star-saped intial data, tere is a unique star-saped weak solution (u, u, Ω(u)) of (P ) for a sort time, and for global time wit additional symmetries in te initial data. Sort-time existence of any nature for general smoot initial data is an open problem. 2. For free boundary problems wic satisfy a comparison principle (suc as te mean-curvature flow or (P ) wit fixed λ), te sub-solution would stay below te super-solution, wic would ten yield tat u u. Tis in turn yields u = u and in particular te uniform convergence of u to a weak solution readily follows. Unfortunately for us tis line of argument cannot be applied since (P ) does not satisfy a comparison principle. Proposition 4.8 Let us define λ 1 (x, t) and λ 2 (x, t) as in Definition 4.4 wit u 1 = u and u 2 = u. (a) Suppose (x, t) Σ. Tere is x Ω t (u ) suc tat (x, t ) (x, t). Let w be te connected component of Σ(t) containing x and let w be te corresponding connected component containing x. Ten λ 1 (x, t) lim inf λ(u (, t))(x) 0 (b) Suppose (x, t) Ω(u ). Ten tere is x Ω t (u ) suc tat (x, t ) (x, t). Let w be te connected component of Ω(u ) containing x, and let w be te corresponding connected component containing x. Ten lim sup λ(u (, t))(x) λ 2 (x, t). 0 Proof. To prove (a), first note tat for fixed δ we ave tat u (, t ) converges uniformly to zero outside of,δ := {x : d(x, ) δ}. Terefore, we can lower u to its essential part: tere exists ε 0 suc tat ũ := (u ε ) + satisfies Ω t (ũ ),δ. Moreover we ave, by definition of c 1, lim inf u dx c 1. 0,δ 17

18 Terefore, λ 1 (x, t) λ(u,δ,c 1 (, t))(x) Dũ lim inf,δ 2 (, t)dx 0 c 1 = lim inf 0 λ(u (, t))(x). To prove (b), note tat for any δ > 0, tere exists 0 < δ, suc tat w δ := {x : B δ (x) w } is contained in w 0 : Suppose δ for some δ > 0 for any small > 0. Ten tere exists a sequence of points x converging to a point x in δ suc tat u (t, x ) = 0. Tis is a contradiction to te fact δ Ω t (u ). Terefore δ at least for a sequence of converging to zero. Furtermore, by definition of c 2, lim sup u (, t)dx c 2. 0 δ And lim sup λ(u (, t))(x) lim λ(u δ,c 2 )(x) λ 2 (x, t). 0 δ 0 Proof of Teorem 4.6 Te proof carries over te barrier properties of te time discrete solutions. We will only sow tat u is a viscosity supersolution of (P ) wit respect to λ 1 (x, t). Te subsolution part can be sown via parallel arguments. Suppose tere exists a smoot function φ as in Definition 4.1 in S := B r (x 0 ) [t 1, t 0 ] suc tat φ crosses u from below at (x 0, t 0 ).: i.e. u φ as a minimum zero at (x 0, t 0 ). By using φ(x, t) := (φ(x, t) σ(x x 0 ) 2 + σ(t t 0 )) + wit small σ > 0 if necessary, one may assume tat te minimum is strict in S. Ten for small > 0 te function u φ also as a minimum at (x, t ) in S wit (x, t ) (x 0, t 0 ) as 0. Since by Definition 4.1 φ(x, t) < λ 1 (x, t), we ave tat by Proposition 4.8(a) tere exists δ > 0 suc tat φ < λ(u )(, t 0 ) in B δ (x 0 ) for 0 < < δ. Te above inequality as well as te second inequality in Definition 4.1 yield tat φ satisfies (3.1) for and r sufficiently small. Hence Proposition 3.1, applied to φ and u at (x, t ) in S, yields a contradiction. Acknowledgements Natalie Grunewald was supported by te German Science Foundation, DFG, troug fellowsip GR 3391/1-1. Inwon Kim is supported by NSF DMS Te autors tank Karl Glasner for elpful discussions. Natalie Grunewald tanks UCLA for its ospitality during er stay in te last year, during wic most of tis paper was written. 18

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