Zhenghuan Gao,Xinan Ma,Peihe Wang and Liangjun Weng,3

1School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China;

2 School of Mathematical Sciences,Qufu Normal University,Qufu 273165,China;

3 Mathematisches Institut,Albert-Ludwigs-Universit¨at Freiburg,Freiburg im Breisgau 79104,Germany.

Abstract.For any bounded strictly convex domain ? in Rnwith smooth boundary,we find the prescribed contact angle which is nearly perpendicular such that nonparametric mean curvature flow with contact angle boundary condition converge to ones which move by translation.Subsequently,the existence and uniqueness of smooth solutions to the capillary problem without gravity on strictly convex domain are also discussed.

Key words:Mean curvature flow,prescribed contact angle,asymptotic behavior,capillary problem.

1 Introduction

In this paper,we are interested in the study of the evolution of graphs defined over bounded strict convex domains ??Rnby the nonparametric mean curvature flow,whose speed in the direction of their normal is equal to their mean curvature and with a prescribed contact angle to ??.

Various results have been obtained for mean curvature flow of hypersurfaces with Dirichlet boundary conditions[26],zero-Neumann boundary condition[15],[18]and general Neumann boundary condition[25].We study the evolution of graphs for u=u(x,t)with the speed depending on the mean curvature of the surface{(x,u(x,t)):x∈?}and with the prescribed contact angle boundary condition,that is,

Guan[14]proved the global existence of solutions to(1.2)with prescribed contact angle condition for general bounded domain ?.Recently,Zhou generalized Guan’s results to the domain ? on Riemannian manifold in[30].

As for studying the asymptotic behavior of u(x,t)in(1.2),Guan[14]or Zhou[30]only obtained the convergence results for F(x,u,Du)with specific form,say F:=φ(x,u)·which excluded F ≡0.In[15],Huisken studied the fixed vertical contact angle case of(1.1),i.e.,so uν=0 on ??.By using the Sobolevtype inequalities and an iteration method,Huisken proved that the solution u(·,t)of(1.1)converges to a constant function as t→+∞.For the non-perpendicular case,Altschuler and Wu in[1]firstly considered the problem(1.1)with fixed contact angle boundary condition in one dimension,and showed that u(x,t)converges to translating solitons.Subsequently,they studied in[2]for two dimension case and proved that the solutions of(1.1)converge to one which moves only by translation,under the condition that ??R2is strictly convex and kDθkC0

From above discussions,we rewrite(1.1)into the following equivalent form,

The crucial part of the proof is to derive an a priori estimate for the spatial gradient of u(x,t),which is time-independent.This will be achieved by choosing an appropriate auxiliary function and combining with the maximum principle.Our auxiliary function and approach are motivated by methods used in[10,24,25,27].

where k is usually referred to as the capillarity constant(see[8],Chapter 1).Results about the positive gravity k>0 case are extensively studied and quite well-known.Ural’tseva[29],Simon-Spruck[28]and Gerhardt[10]had obtained the existence results of(1.6)for any dimensions.More results related to positive gravity capillary problem also could be seen in[12],the wonderful exposition book by Finn in[8]and references therein.We only discuss and focus on k=0(gravity free)in(1.6)in the rest part of this paper,i.e.

If there exists a solution to(1.7)with constant angle θ≡ θ0,integrating by parts on ? yields that

As pointed out by Concus and Finn in[6],Eq.(1.7)may not have any solution,even for constant angle.In[12],Giusti proved the following results.

Theorem 1.2([12]).Let ? ?Rn(n≥2)be a Lipschitz bounded domain.and there exists ε0>0 such that

holds for all proper subdomains ?0??.Then there exists a solution u∈ BV(?)solving(1.7)in weak sense,with θ≡θ0be a fixed constant.

Remark 1.2.The convexity condition of the domain is necessary in the sense that Finn-Giusti[9]gave an example of nonexistence for the equation(1.7)when the domain is non-convex.And if the domain is nonsmooth,there are already many works related to the generalized solution using the variational methods,see e.g.[8](particularly Chapters 6,7),[9],[12],and references therein.

The main difference and difficulty between the positive gravity(k>0 in(1.6))and free gravity(1.7)is that there is no C0estimate for the solutions to(1.7),since a solution plus any constant is still a solution to(1.7).Thus one can not use the continuity method to get the existence.In order to overcome this difficulty,we use an approximation argument and obtain the uniform gradient estimate of the approximation equation,which is independent of kukC0.This approach has been used previously in several different settings,see,e.g.,[16,17,25,27].Those results also motivate our work here.Additionally,we want to point out that this approach is different with many former methods about the capillary problem,say e.g.[8,12,13],where they usually proved the existence of generalized solutions firstly,hereafter to show that the generalized solutions possess some regularity.Here our methods are able to get the existence of smooth solution directly.

This article is structured as follows.In Section 2,the uniform gradient estimate is established for(1.3).In Section 3,the asymptotic behavior of solution to(1.3),i.e.Theorem 1.1 is demonstrated and followed as the same as the approach used in[2],once we get the uniform gradient estimate.The last section is devoted to prove Theorem 1.3,after obtaining the uniform gradient estimate for the solutions to approximation equations.

2 Uniform gradient estimate for mean curvature flow

In this section,in order to study the asymptotic behavior of the nonparametric mean curvature flow with prescribed contact angle boundary condition,we establish the uniform gradient estimate for the solution to(1.3)under the condition(1.4).

We have the following facts when ? is a strictly convex smooth domain.By the classical result(see for example Caffarelli-Nirenberg-Spruck[4]Section 2,and we can take g0(0)=1 in their definition of u in page 275),there exists a smooth defining function h for ? such that h<0 in ? and h=0 on ??,{hij}≥k1{δij}for some constant k1>0 and sup?|Dh|≤1,hν=?1 and|Dh|=1 on ??.Because of the strict convexity of the domain,we may assume that the curvature matrix of?? satisfies

for some constant κ0>0.For convenience,we denote by

and define the big O notation O(s),which means that there exists a constant C>0,such that|O(s)|≤Cs for s large enough.In particular,we have the positive constant C only depending on M1,M2and n in the rest setting of this paper.

Using the maximum principle,the same as in[2],we have a priori bound on|ut|2.

Next we obtain the uniform gradient estimate for(1.3),which turns the quasilinear evolution equation(1.3)into a uniformly parabolic equation and the infinite time existence of smooth solutions follows by standard regularity theory.

3 Elliptic interlude and asymptotic behavior

As the approach in the two dimension case in the paper[2],our gradient estimate also can be used to solve the elliptic version of the problem.The elliptic version of equation(1.3)is

where τ∈R is a uniquely determined constant.In fact,by using the integration by parts,one can see that

We can obtain the following existence result for(3.1)in high space dimension case under the condition(1.4).For 2 dimension,this result was proved by Altschuler-Wu(see Theorem 2.6 in[2])under more generally condition on ? and θ.

The uniform estimate of|Du|also implies that|D(εu)|→ 0 as ε→ 0.Thus one can conclude that εu→τ as ε→0 for some τ∈R(One can see the detail proof on the existence part for the similar limited equation in Theorem 1.3).

To show the uniqueness,as in[2](see the proof of Theorem 2.6 there)one assumes that if(u1,τ1)and(u2,τ2)are two solutions to(3.1).Without loss of generality,we assume τ1≤τ2,denote u:=u1?u2.By direct computation,we obtain that u is the super-solution of the following elliptic operator

From the maximum principle,u attains the maximum value on the boundary,say at x0∈??.Thus combining with Hopf lemma,we have?0u(x0)=0 and Dνu(x0)<0,that is,|?0u1|=|?0u2|=q and Dνu1

According to the claim,we have

Proof of Theorem 1.1.From Lemma 2.1 and Theorem 2.1 and Schauder estimate,we obtain uniform estimates in any Ck?norm for the derivatives of u,and locally(in time)uniform bounds for the C0norm.So we get longtime existence with uniform bounds on all higher derivatives of u.From Corollary 3 and Lemma 3.1,the limit of any solution to Eq.(1.1)is ?w=w+λt up to a constant,where(λ,w)is the solution to Eq.(3.1)by Theorem 3.1.

4 Constant mean curvature equation with prescribed contact angle boundary value condition

In this section,we consider the capillary problem with prescribed contact angle boundary condition.We first obtain the following uniform gradient estimate of u for equations(4.1).Note that,for any fixed ε>0,the existence of solutions to(4.1)is well-known,see[8]for example.Now we can prove the following lemma.

Acknowledgments

The second and the third author are supported by the National Natural Science Foundation of China(Grant No.11471188).The second author is also supported by the National Natural Science Foundation of China(Grant No.11721101)and the National Natural Science Foundation of China(Grant No.11871255).