Harmonic Maps in Connection of Phase Transitions with Higher Dimensional Potential Wells*

Fanghua LIN Changyou WANG

(Dedicated to Professor Andrew J.Majda with deep admiration)

Abstract This is in the sequel of authors’paper [Lin,F.H.,Pan,X.B.and Wang,C.Y.,Phase transition for potentials of high dimensional wells,Comm.Pure Appl.Math.,65(6),2012,833-888] in which the authors had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells.The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition.The authors also present a proof,under simplified geometric assumptions,of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature.In a forthcoming paper,a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg’s work (in 1989)on the fast reaction,slow diffusion and motion by the mean curvature would be addressed.

Keywords Partially free and partially constrained boundary,Boundary partial regularity,Boundary monotonicity inequality

1 Introduction

This is a continuation of our previous work Lin-Pan-Wang [12] in which we had set up a program to verify various phenomena associated with multiple components phase transitions with higher dimensional wells.One of the goals here is to show rigorously the formal asymptotic arguments for the description of fast reaction,slow diffusion and sharp interface dynamics using the Ginzburg-Landau approximation as in the celebrated papers [17-18] by Keller-Rubinstein-Sternberg.For the leading term of the energy functional in the static energy minimization,we showed in [12] that the sharp interfaces for these general phase transition problem must be area minimizing hypersurfaces with weights.For the energy minimization,each of weights must be a constant giving by the length of a so-called minimal connection between a pair of potential wells.Therefore for the gradient flow,the dynamic of these sharp interfaces would simply be the motion by mean curvature provided that this weight function remains to be a constant that equals the length of a minimal connection.The latter leads to a challenging issue of studying energy minimizing maps(phases)and its gradient flows that lie in multiple potential wells (submanifolds)of high dimensions and,that each patch of such maps (phases)possesses a specific and non-standard boundary condition at corresponding sharp interfaces.The phases and their dynamics within each of the potential wells would be derived from the“slow diffusion”part as in [17-18],and it is hence in the next term of formal asymptotic for the energy of the system.This gives a nonlinear coupling between terms of different orders(in formal expansions)of the energy through boundary conditions,and it leads us to the study of harmonic maps with these unusual boundary conditions.In this paper,we show a boundary regularity theory of minimizing harmonic maps in the above described problems.We also establish a theorem on the short time existence of classical solutions to the corresponding heat flows.In a forthcoming work,we will address these dynamical issues in a more general context.

Let us first recall the Cahn-Hilliard energy functional that models the phase transition described by a scalar function v:

where Ω ?Rnis assumed to be a bounded,smooth domain in Rnthroughout this paper,v:ΩR is the density function,andis a double-well potential function that has two minima (zeros)at ±1.The term ∈|?v|2is the interfacial energy that penalizes the formation of interface.The asymptotic behavior of minimizers v∈of E∈(·)under the constraintwas first studied by Modica-Mortola [16],Modica [15],and Luckhaus-Modica [13]:They have showed that the separation region between the two stable phases has O(∈)thickness and the phase transition converges to a minimal hypersurface within the frame work of De Giorgi’s Γ-convergence theory.There are many important contributions to this problem (see for examples [5,10,13-15,21-22]).

Rubinstein-Sternberg-Keller [17-18] introduced the vector-valued system of fast reaction and slow diffusion:

Next we recall the main results of [12].For k ＞1,let

be the union of two disjoint,compact,connected,smooth Riemannian manifolds N±?Rkwithout boundaries.For δ ＞0,let denote the δ-neighborhood of N.It is well known that there exists δN＞0 such that d2(p,N)∈C∞(NδN).Consider the class of double-well potential functions depending only on the distance function from N,namely,

where f ∈C∞(R+,R+)satisfies the property that there exist c1,c2,c3＞0 such that

Consider the family of Cahn-Hiliard functional

that are singular perturbations of the functional of phase transitions of high dimensional wells:

For the boundary conditions,we let Σ±??Ω be two disjoint,connected,open subsets of ?Ω such that

(1)?Σ+=?Σ-=Σ is a connected (n-2)-dimensional smooth manifold;

(2)?Ω=Σ+∪Σ-∪Σ.

For any small η ＞0,let Ση= {x ∈Rn:d(x,Σ)＜η} be the η-neighborhood of Σ,and denoteAssume that for some β ＞0,R ＞0,L ＞0,and C ＞0,g∈:satisfy:

(2)for any p±∈N±,?extension maps

such that

where ?τdenotes the tangential derivative on hypersurfaces in Rn.

Set

In [12],we proved the following theorem.

Theorem AAssume that F ∈C∞(Rk)satisfies (1.1),Γ ?Ω is an area-minimizing hypersurface with ?Γ=Σ and g∈:?ΩRksatisfies conditions (1.2)and (1.3).Then

where cF0is the energy of the minimal connecting orbits between N+and N-defined by

and

Let

be the euclidean distance between N+and N-,and

be the pair of minimal sets in N±.

Assume that g∈is almost optimal near Σ in the sense that its limitgives the minimal connecting orbits between N+and N-(see [12,pp.804-841] for more details).Then we also proved in [12] the following result.

Theorem BAssume F(p)= f(d2(p,N))satisfies (1.1),Γ is a unique area minimizing hypersurface with ?Γ=Σ,which is smooth and strictly stable.Assume also that

Then

where

Furthermore,if {u∈} is a sequence of minimizers of E(∈),then there exists u ∈A attaining the value D such that after taking a possible subsequence,u∈converges to u in L1(Ω,Rk).

The first aim of this paper is to study the boundary regularity of a minimizing harmonic map v ∈A that attains D near the sharp interface Γ.In order to achieve it,we make some further assumptions on the minimal sets M±.More precisely,let M+?N+and M-?Nbe such that

· M+and M-are connected,C1-manifolds without boundaries,equipped with induced metric from N+and N-respectively;

·there exists a C1diffeomorphism,whose inverse map is

Let Γ ?Ω be a smooth hypersurface with boundary Σ,i.e.,?Γ=Σ.Denote the two connected components of Ω separated by Γ by Ω±,i.e.,ΩΓ=Ω+∪Ω-,so that

Let g:?Ω →N be a given map such that g ∈H1(Σ±,N±),and the two one-side trace values of g on Σ satisfy:

The minimization problem seeks

where

It is readily seen that if the configuration space

is non-empty,then there exists at least one energy minimizing map u ∈A,i.e.,

Note that for n ≥3 if,up to a diffeomorphism,Ω=B1?Rn,the unit ball,Σ=?B1∩{xn=0},Σ±=?B1∩Rn±,Γ=B1∩{xn=0},and g ∈H1(Σ±,N±)satisfies (1.11),thenIn fact,it is not hard to verify that the homogeneous of degree zero extensionbelongs to A.In general,we have the following lemma.

Lemma 1.1Assume that Π1(N+)= Π1(N-)= {0},g:?ΩN satisfies g|Σ±∈H1(Σ±,N±),and the condition (1.11)holds.Then A is non-empty.

ProofDenote the two one side trace of g on Σ by g±(x)for x ∈Σ.Then by (1.11)g±∈(Σ,M±).First,we want to extend g±:ΣM±to maps G±:ΓM±.By (1.11),it suffices to construct an extension map G+of g+,since G-(x)= Φ+(G+(x))for x ∈Γ will provide an extension of g-.Since M+is connected,i.e.,Π0(M+)={0},Theorem 6.2 of Hardt-Lin [7-8] implies that for any 1 ＜p ＜2,there exists an extension map G+∈W1,p(Γ,M+)such thatin the trace sense.Now we let u+∈H1(Ω+,Rk)solve

Since Π1(N+)=0,by applying the extension Lemma 6.1 of [8] as in the proof of Theorem 6.2 of [8] we conclude that there exists a mapsuch thatand

Similarly,we can find an extension map∈H1(Ω-,N-)such that= g on Σ-and= G-on Γ.Now if we set:ΩN by lettingfor x ∈Ω±,then∈A.This completes the proof.

For a minimizing harmonic map u ∈A,denote the set of discontinuous points of u in Ω±∪Γ by S±(u)?Ω±∪Γ and define

as the set of discontinuous points of u in Ω.

It follows from the interior regularity theory of minimizing harmonic maps by Schoen-Uhlenbeck [19] that S(u)∩(Ω{Γ})has Hausdorffdimension at most n-3.

Our first main result concerns the boundary partial regularity at Γ for a minimizing harmonic map u in A,which is stated as follows.

Theorem 1.1Assume that the boundary value g ∈H1(Σ±,N±)satisfies the condition(1.11).If u ∈A is an energy minimizing harmonic map,then

(i)S(u)∩Γ is discrete for n=3;

(ii)S(u)∩Γ is of Hausdorffdimension at most (n-3)for n ≥4.

The paper is organized as follows.In §2,we will give a proof of Theorem 1.1.In §3,we will discuss the corresponding problem on the heat flow and establish the existence of short time regular solutions.In§4,we will provide boundary monotonicity inequalities for both stationary harmonic maps and their corresponding heat flows under the same boundary condition in Theorem 1.1,which may have its own interest and are useful to future studies.

2 Proof of Theorem 1.1

2.1 Euler-Lagrange equation

In this subsection,we will derive the Euler-Lagrange equation for energy minimizing maps in A.

Assume that u ∈A is an energy minimizing map.For a sufficiently small δ ＞0,let u(t,·)∈A,t ∈(-δ,δ),be a family of comparison maps for u,i.e.,u(0,·)=u(·).For t ∈(-δ,δ),let u±(t,x)denote the two one-sided trace value of u(t,x)for x ∈Γ.Then for t ∈(-δ,δ),we have

and

For the test function φ,if we denote by φ±(x)the two one-sided trace value of φ on Γ from Ω±,then

and

Let A±denote the second fundamental form of N±in Rkand denoteThen by integration by parts u satisfies

Here (·)T(x):denotes the orthogonal projection map for x ∈Γ,and

denotes the adjoint of the linear map

It is not hard to see that the 5th equation of (2.1)can also be written as

2.2 Boundary monotonicity inequality

In order to establish the partial boundary regularity for energy minimizing maps in A,we need a version of boundary monotonicity inequality.

For R ＞0,denote by BR?Rnthe ball of radius R and center 0,B±R= BR∩Rn±.Since Γ is smooth,there exists r0= r0(Γ)＞0 such that for any x0∈Γ,0 ＜r ≤r1:=min{r0,dist(x0,?Ω)},there exist C ＞0 and C1-diffeomorphism Ψ:Br(x0)=Br(x0)∩ΩBrso that

Here Inis the identity matrix of order n.By Fubini’s theorem,u ∈H1(?Br(x0)∩Ω±,N±)for almost all r ∈(0,r1)so that if we define

Utilizing (2.2)and direct calculations,we have that

Therefore,for any x0∈Γ and r ∈(0,r1),we have that

holds,provided u ∈A is an energy minimizing map.In particular,by integrating (2.3)with respect to r,we obtain that for any x0∈Γ and 0 ＜R1≤R2＜r1,

holds for any energy minimizing map u ∈A.

2.3 Boundary extension lemma

A crucial ingredient to prove Theorem 1.1 is the following boundary extension lemma,similar to [9,Lemma 3.1].

Lemma 2.1There exist positive constants δ,q,and C such that,if 0 ＜∈＜1,x0∈Γ,and 0 ＜r0＜dist(x0,?Ω),if η±∈H1(?Br0(x0)∩Ω±,N±)satisfies

for some p±∈Rk,and if η±:?Br0(x0)∩ΓM±satisfies

then there exist maps ω±∈H1(Br0(x0)∩Ω±,N±)such that ω±=η±on ?Br0(x0)∩Ω±,and ω±:Br0(x0)∩ΓM±satisfies

Furthermore,it holds that

Here ?tandenotes the tangential gradient on ?Br0(x0).

ProofThe proof can be done by suitable modifications of the arguments from [8-9] and[19].It is based on an induction of the dimension n.There are two crucial ingredients of the construction:

(i)Construction in dimension n=2;

(ii)Homogeneous of degree zero extension for n ≥3.

For simplicity,we will only indicate how to implement these two ingredients in our situation.The interested readers can consult with [8-9,19] for more details.

Case 1n = 2 (linear interpolation).Since the problem is invariant under bi-Lipschitz transformations,we may assume that x0= 0,r0= 1,Ω = B1,and Γ = Γ1(= B21∩{x2= 0}).Denote by S±1??B21the half unit circles.Choose θ±0∈S±1so that

Then it is easy to see that

By Sobolev’s embedding inequalitywe have that

Set

Then we have

Recall that there exists δ0= δ0(M±)＞0 such that for any 0 ＜δ ＜δ0,the nearest point projection maps ΠM±:(M±)δM±and ΠN±:(N±)δN±are smooth,where (M±)δ(or (N±)δrespectively)denotes the δ-neighborhood of M±(or N±respectively)in Rk.Let v+:B+1Rksolve

Thus we can define

To construct ω-,first let

so that w-(Γ1)?M-.Let v-:solve

Then we also have

so that we can define

It follows directly from the above construction that ω-(x)= Φ+(ω+(x))for x ∈Γ1,and (2.6)follows from the standard estimate on harmonic functions.

Case 2n ≥3(homogeneous of degree zero extension).For 0 ＜δ ＜1,letbe(n-1)-dimensional half balls of radius δ ＞0,andbe the n-dimensional half cylinders of size δ.Letbe the (n -2)-dimensional half spheres of radius δ so that

Lemma 2.2Forif u±1(x)=u±(x,-δ)and u±2(x)= u±(x,δ),x ∈,satisfiesif u±(x,t)= u±0(x)forwithand if

Then there exist extension mapssuch that

and

Here

and

for some fixed p*∈RL.

ProofBy scaling,we may assume δ = 1.There exists a bi-Lipschitz homeomorphismsuch thatis also a bi-Lipschitz homeomorphism.Letbe the radial projection map.Define the projection mapThen define

It is easy to see that (2.7)implies thatsatisfies the trace condition (2.8)on Γ1.It is also easy to see that

where K is a constant depending on the Lipschitz constants of f±andThis implies(2.9).Similar argument for W also yields (2.10).

Corollary 2.1There is a constant c ＞0 such that under the same assumptions of Lemma 2.1,if u ∈H1(Ω±,N±)∩A is energy minimizing among all maps in A,and for any x0∈Γ and 0 ＜r0＜dist(x0,?Ω),

then

where u±=u|Ω±denotes the restriction of u on Ω±,and

is the average of the one-side trace of u±in Br0(x0)∩Γ.

ProofFor simplicity,we assume r0= 1.Since u±:Ω±→N±and N±is compact,it follows

From the Poincaré inequality,we have that

From the trace estimate and the Poincaré inequality,we also have that

Applying Fubini’s theorem,we can choosesuch that

and

By choosing a sufficiently small c ＞0,we can apply Lemma 2.1 with η±=u±|?Br(x0)∩Ω±andto obtain an extension map ω±∈H1(Br(x0)∩Ω±,N±)such that ω±= u±onhas image in M±that satisfies

and the estimate (2.6).If we define:Ω →N by

which,combined with (2.6),then implies (2.11).This completes the proof.

2.4 Small energy regularity

Another crucial step to prove Theorem 1.1 is the following energy improvement property.

Lemma 2.3There exist positive constants ∈,C,and θ ＜1 such that if u ∈A is an energy minimizing map that satisfies,for x0∈Γ and some 0 ＜r0＜dist(x0,?Ω),

then

The proof of Lemma 2.3 is based on a blowing up argument,similar to [9,Theorem 3.3].Before presenting it,we need the following regularity estimate on the linear equation,resulting from the blow-up process of the nonlinear harmonic map equation (2.2).

Denote by B+1and B-1the upper half and lower half unit ball,and set Γ1=B1∩{xn=0}.For a+∈M+,let a-= Φ+(a+)∈M-.Let Tan(a±,M±)denote the tangent space of M±at a±,and Nor(a±,M±)denote the normal space of M±?N±at a±,i.e.,

For any vector v±∈Tan(a±,N±),we decompose it as

where vt±denotes the orthogonal projection of v±into Tan(a±,M±),and vn±denotes the orthogonal projection of v±into Nor(a±,M±).

Lemma 2.4Suppose that v±∈H1(B±1,Tan(a±,N±))are two harmonic functions,with tracesTan(a±,M±)),satisfying

ProofSince a±∈M±,we can decompose v±=vt±+vn±so that

and

Since v±(x)∈Tan(a±,M±)for Hn-1a.e.x ∈Γ1,we have that

It is readily seen that by (2.17)and (2.18),and for any l ≥1,

To show regularity of vt±,we denote P =DΦ+(a+)and proceed as follows.DefineTa+N+be an even extension v-,i.e.,

Then it is easy to see that

and

From the standard theory of harmonic functions,we see that (2.20)and (2.21)imply

and it holds that,for any l ≥1,

If PPt=Ik,i.e.,P ∈O(k)is an orthogonal matrix,then we have

This and (2.22)easily yield (2.15).

If PPtIk,then P-1Ptand we can also see easily that(2.15)follows from(2.22).This completes the proof.

Proof of Lemma 2.4The proof follows from a blow-up argument,Lemma 2.4,and the boundary extension Lemma 2.2.Here we only sketch the argument.

For simplicity,assume that x0= 0,r0= 1,Ω = B1,and Γ = Γ1so that Lip(Γ)= 0.Suppose that the conclusion were false.Then for any θ ∈(0,1),there would exist ∈i→0 and a sequence of minimizing harmonic maps ui∈A that satisfy

and

Therefore for i sufficiently large there is a unique nearest pointsuch that

Now we define the corresponding blow-up sequence vi:B1→Rkby letting

It is easy to see that

and

By (2.25)and the H1-trace theory,we have

Hence,after taking a subsequence,there exists v:B1→Rk,withsuch that vi±converge to v±weakly in H1(B1±,Rk).In particular,by (2.25),we have

Again passing to a subsequence,we assume that

It is not hard to verify that v+(x)∈Ta+N+for a.e.x ∈B+1,and v-(x)∈Ta-N-for a.e.x ∈B1-.Since(x)∈M±for Hn-1a.e.x ∈Γ1,it is also not hard to see that

and

we have,after taking i to infinity,that

Since vi±also satisfies the trace condition

we obtain,after taking i to infinity,that

This implies

so that

By (2.28)-(2.30),we can apply Lemma 2.4 to conclude thatMoreover,by(2.27)and (2.31)we have that for any 0 ＜θ ＜1,

Combining(2.11)with (2.34).we can repeat the argument of[8] to get a desired contradiction.

Proof of Theorem 1.1It is well-known that iterations of Lemma 2.3,combined with the interior ∈-regularity,implies that there exist ∈0＞0 and α0∈(0,1)such that if for x0∈Γ,there exists r0＞0 such that

3 On the Local Existence of Regular Solutions to Heat Flow

In this section,we will consider the gradient flow associated with the minimization problem(1.12),or,equivalently,the parabolic version of the harmonic map equation (2.1).Under some further assumptions on M±and Γ,to be specified below,we will establish the local existence of regular solutions of the heat flow under the initial and corresponding boundary conditions.For the harmonic map heat flow,the reader can refer to the articles [2-3,24-25].

Before describing the corresponding heat flow problem,we first need to introduce some notations.For a given T ＞0,let{Γ(t):t ∈[0,T]}be a smooth family of smooth hypersurfaces,with Γ(0)=Γ,such that

For t ∈[0,T],decompose ΩΓ(t)into the disjoint union of two simply connected components Ω+(t)and Ω-(t),i.e.,

Denote Ω±=Ω±(0),and write

so that ?Ω±=Γ ∪Σ±.Set

and

The harmonic heat flow problem corresponding to (2.1)can be formulated as follows.We are looking for u±:Q±TN±,with u±(x,t)∈M±for (x,t)∈ΓT,that solves

Here u±0:Ω±N±,with u±0(x)∈M±satisfying u-0(x)= Φ+(u+0(x))for x ∈Γ,andare given initial and boundary values.

In order to establish the short time existence of regular solutions to (3.1),we need to set up the problem appropriately by specifying the assumptions (A),(B),and (C)on N±and M±:

(A)The target Riemannian manifolds(N±,h±)have the same dimension dim(N±)=k+m.For,otherwise,if k1=dim(N+)＜k2=dim(N-),then we can replace (N+,h+)by

where hcandenotes the standard metric on Sk2-k1.Notice thatMoreover,for any map u:Ω+(t)×[0,T] →N+,if we define(x,t)= (u(x,t),e):Ω+(t)×[0,T] →where e ∈Sk2-k1,then we can show that if u is a solution to the heat flow of harmonic maps to N+,thenis also a solution to the heat flow of harmonic maps to.This follows from the chain rule and the fact that (N+,h+)is a totally geodesic sub-manifold of

(B)The manifolds M±?N±are two k-dimensional compact smooth sub-manifolds,with?M±= ?,such that there exists a smooth diffeomorphism Φ+:M+M-,whose inverse is denoted by Φ-:M-M+.Moreover,there exists r0= r0(M+)＞0 such that for any p+∈M+,Φ+can be extended into a smooth diffeomorphism,still denoted as itself,

whose inverse is also denoted by Φ-.

(C)There exists a 0 ＜r1= r1(N+)≤r0(M+)such that for any p+∈N+,there exists a local parametrization ofi.e.,

provides a local representation ofvia the diffeomorphism

We may assume that U(p+)=(0,0),and if p+∈M+then

and the Riemannian metric h+oncan be expressed by

and the induced metric of h+onis given by

It is readily seen that for p+∈M+and p-= Φ+(p+),through the diffeomorphism Φ+:provides a local parametrization ofthrough the diffeomorphismIn particular,U(p-)=(0,0),

and the Riemannian metric h-oncan be expressed by

and the induced metric of h-onis given by

We may assume henceforth that r1(N+)=r0(M+)in the assumptions (B)and (C).

Remark 3.1Under the assumptions(A),(B),and(C),it is not hard to see that by choosing a sufficiently small r0= r0(M+)＞0,under the above local parametrization ofthe local representations of the Riemannian metrics h±enjoy the following properties:

such that

for some C ＞0 depending only on M±and N±.

Now we are ready to state a theorem on the local existence of regular solutions to (3.1),whose full proof will be given in another future work.

Theorem 3.1Under the assumptions (A),(B),and (C)on N±and M±,for 0 ＜α ＜1,letbe given initial and boundary data such thatsatisfies u-0(x)=Φ+(u+0(x))andfor x ∈Γ.Then there exist T0＞0,depending on ‖u±0‖C1,α(Ω±),and a unique solution u±∈of the initial and boundary value problem (3.1).

The proof of Theorem 3.1 is more delicate than the usual proofs of short time smooth solutions to the heat flow of harmonic maps under the Dirichlet boundary condition (see [1,6])or the free boundary condition (see [25]).It involves to first show the local existence of regular solutions over small balls,and then patch these local solutions by extending the Schwarz alternating method on linear parabolic equations to the quasilinear harmonic map heat flows into small neighborhoods of points in N±.For this,we have to overcome major difficulties that arise near the interface Γ.A detailed proof will be addressed in a forthcoming work.The approach that we will utilize is based on the Schwartz reflection method adapted to the parabolic settings,see [4] and [7] for some backgrounds on this method.

In this part,we will indicate a proof of Theorem 3.1 when the images of u±is contained in a single coordinate chart of N±.Before doing it,we want to rewrite the system (3.1)in an intrinsic form near a small neighborhood of a point (x0,t0)∈ΓTand also derive a generalized energy inequality.

3.1 Local representation of (3.1)

For t0∈(0,T)and x0∈Γ(t0),choose a small δ0＞0,depending on ‖u±‖C0(Q±T),such that

where Pδ0(x0,t0)= Bδ0(x0)×(t0-δ20,t0+δ20).Then,by employing the local representations given by the assumptions (B)and (C)on M±,N±,we can rewrite the harmonic heat flow equation (3.1)as

where U = (U1,U2):Q+T∩Pδ0(x0,t0)Bk1×Bm1is the local representation of u = u±:Q+T∩Pδ0(x0,t0)N,and ?！?·)(·,·)is the Christoffel symbol of N±.

Observe that within this local coordinate system,the boundary condition the 4th equation of (3.1)on the free interface ΓTgives rise to

and by (3.2)the boundary condition the 5th equation of (3.1)on the free interface ΓTreduces to

3.2 Parametrization of domains

Since Ω±(t)is t-dependent over[0,T],in this subsection we will re-parametrize the domains and rewrite(3.1)so that it can be viewed as the heat flow of harmonic maps over fixed domain but with time-dependent metrics on the domain.

Given that u±:Q±TN±satisfies (3.1),we want to derive the equation fornow.To do it,first set

and

Then direct calculations imply that

and

Hence the 1st and 2nd equation of (3.1)becomes

where

Observe that the boundary condition the 4th equation of(3.1)on the free interface ΓTgives rise to

while the boundary condition the 5th equation of (3.1)on the free interface ΓTgives rise to

where ν(=ν(t))is the unit outer normal of Γ with respect to the metric

First we observe that a sufficiently regular solution of (3.1)enjoys a generalized energy inequality.For 1 ＜p ＜∞,T ＞0,and an open set E ?Rn,denote

Lemma 3.1For T ＞0,and g ∈C1(Σ±,N±),if u±∈withL2(Q±T),is a strong solution of (3.1),then there exists constant C ＞0 depending on ΓTsuch that

for all 0 ≤s ＜t ∈[0,T].

ProofLet Ψ(·,t):Ω×[0,T]Ω be a smooth family of diffeomorphism given by (3.6).Defineby

Within this time dependent parametrization,we can write

It is easy to see that

While,applying the integration by parts,(3.8),the boundary conditions (3.9),(3.10)and(3.12),and the fact thatfor (x,t)∈ΓT,andon Σ±×[0,T],we can show that the boundary contributions on both Γ and ?Ω are zeroes.Hence we can estimate I by

It is easy to see that

Hence

On the other hand,it follows from the chain rule (3.7)that

Putting all these estimate together,we obtain

which,combined with Gronwall’s inequality,implies (3.11).

We will sketch a proof of Theorem 3.1 by employing the fixed point argument,under two extra assumptions that

(i)the images of u±0is contained in a single coordinate chart,i.e.,

for a pair of points p±0∈M±that satisfies p-0=Φ+(p+0); and

(ii)

First we will give some heuristic arguments to indicate that the appropriate function spaces for the local existence of regular solutions are

which is equipped with the norm

To see this,assume that Γ(t)≡Γ for 0 ≤t ≤T.Letbe given,andbe a local representation ofConsider V =(V1,V2):QTBk1×Bm1that is a weak solution of

under the initial and boundary condition:

It follows from the regularity of linear parabolic equations thatMoreover,since

for any 1 ＜p ＜∞.

By the Sobolev’s embedding theorem (see [11,Lemma II.3.3]),we conclude that V ∈and

Proof of Theorem 3.1 under the assumptions (3.13)and (3.14)For a pair of initial and boundary data(u0,g)given by Theorem 3.1,letbe a local representation of u0.It follows from the assumptions(3.13)and(3.14)thatif and only if its local representation U belongs to the space

From the condition on U0,we know that there exists ∈0＞0 such that

Hence by the maximum principle,we have that

As a consequence,for any 0 ＜∈≤∈0,we can see that

Now we define the solution mapby letting V =be the solution of

subject to the initial and boundary condition (3.16).

Now we need the following lemma.

Lemma 3.2There exist ∈＞0 and T ＞0 such thatis a contractive map,i.e.,for any θ ∈(0,1),we can find ∈＞0 and T ＞0 such that

Therefore there exists a uniquesuch that U =T(U).In particular,ifQ±TN±has U as its local representation,then u is a unique regular solution of (3.1)in QT.

ProofFor U ∈B(U0,∈),since V -U0satisfies

and

Hence,similar to the earlier discussion,we have that for some p=p(α)＞n+2,

provided we choose a sufficiently small T = T0＞0,depending only on U0and α.Hence

and

Hence we can conclude that for any θ ∈(0,1)such that for p=p(α)＞n+2,

provided T =T0＞0 is chosen so that

This completes the proof of both Lemma 3.2 and Theorem 3.1 under the assumptions (3.13)and (3.14).

4 Boundary Monotonicity Inequality of (3.1)

In this section,we will derive a boundary monotonicity inequality on (3.1),analogous to Struwe’s monotonicity formula,which may have its own interest.

To simplify the presentation,we assume that

Let u±:Rn±×[0,+∞)→N±,with u±(x,t)∈M±for (x,t)∈?Rn+×(0,∞),satisfy

For (x0,t0)∈Rn×(0,+∞)andlet

denote the backward heat kernel on Rn.Set

Lemma 4.1Suppose that(x0,t0)=(0,0)∈?Rn+×(-∞,0]andis a solution to the system (4.1).Then

ProofWrite G(x,t)for G(0,0)(x,t)and define

It is easy to see that

For simplicity,we only verify (4.2)at R=1.Since

we have

Since

and

we have

Since x=(x′,0)for x ∈?Rn±,and

we have

so that

Since

we have

on ?Rn±×(-∞,0)and hence

Therefore we have

where we have used the boundary condition the 5th equation of (4.1)in the last step.Putting all these calculations together,we obtain

This completes the proof.