TRANSLATING SOLUTIONS OF NON-PARAMETRIC MEAN CURVATURE FLOWS WITH CAPILLARY-TYPE BOUNDARY VALUE PROBLEMS

. In this note, we study the mean curvature ﬂow and the prescribed mean curvature type equation with general capillary-type boundary condition, which is u ν = − φ ( x )(1 + | Du | 2 ) 1 − q 2 for any parameter q > 0. Using the maximum principle, we prove the gradient estimates for the solutions of such a class of boundary value problems. As a consequence, we obtain the corre- sponding existence theorem for a class of mean curvature equations. In addi-tion, we study the related additive eigenvalue problem for general boundary value problems and describe the asymptotic behavior of the solution at inﬁnity time. The originality of the paper lies in the range 0 < q < 1, since there are no any related results before. For parabolic case, we generalize the result of Ma-Wang-Wei [25] to any q > 0. And in elliptic case, we generalize the results in [32] to any q ≥ 0 and to any bounded smooth domain.


1.
Introduction. In this article, we are interested in the large time behavior of solutions of the mean curvature flow on a strictly convex bounded domain with general capillary-type boundary conditions. The main originality of this paper is twofold: on the one hand, we obtain existence results for these nonlinear capillarytype problems, and on the other hand we apply the classical approaches to prove asymptotic problems.
First, we introduce the following mean curvature flow and mean curvature equation with general capillary-type boundary conditions in Ω × (0, ∞), and in Ω , where Ω ⊂ R n (n ≥ 2) is a bounded domain with smooth boundary ∂Ω, ν is an inward unit normal vector to ∂Ω, q ≥ 0, v = 1 + |Du| 2 , u i = u xi , u ij = u xixj , and φ(x) is smooth. Also u 0 (x) and φ(x) are smooth functions satisfying on ∂Ω.
In (1) and (2), when q = 0, it is the capillary problem and when q = 1, it is the Neumann boundary value problem.
The capillary problem has been studied for more than forty years. Ural'tseva [30] firstly got the boundary gradient estimates and the corresponding existence theorem on the positive gravity case. In 1976, Simon-Spruck [28] and Gerhardt [9] also obtained the existence theorem on the positive gravity case respectively. For more general quasi-linear divergence structure equation with conormal derivative boundary value problem, Lieberman [20] got the gradient estimate. They all obtained these estimates via test function technique. In 1975, Spruck [29] first used the maximum principle for positive gravity case in R 2 . Later, Korevaar [18] and Lieberman [19] obtained the gradient estimates respectively based on the maximum principle.
Lieberman [20] also proved the gradient estimates for a more general class of quasi-linear elliptic equations with the boundary condition for q = 0 and q > 1 in zero gravity case (f u = 0). In the physical interpretation of the capillary problem, f u measures gravitational effects. Recently, the third author in this article and Ma [26] got the boundary gradient estimates of mean curvature equations with Neumann problem via the maximum principle and derived an existence result in positive gravity case (f u ≥ C > 0) for some positive constant C. In [32], the third author gave a new maximum principle proof of boundary gradient estimates for q > 1 and q = 0 in zero gravity case. But so far no further progress has been made for 0 < q < 1. In this paper, we mainly want to study the case 0 < q < 1 on a strictly convex domain but our method still holds for arbitrary q > 0.
For the mean curvature flow, it has also been studied by many mathematicians. In [5,13,14], Brakke and Huisken studied the motion of the parametric surfaces by their mean curvature. Their work suggests that it's geometrically natural to consider the surfaces whose speed in the direction of their unit normal is equal to the mean curvature. In the graphical setting, it's natural to consider the parabolic equation. In [15], Huisken studied the vertical capillary problem (1) with φ(x) = 0 and proved that the solution asymptotically converges to a constant function. In his paper, he used integral methods to prove a time-independent gradient bound by the Sobolev inequality and iteration method. In R 2 , Altschuler and Wu [1] considered the prescribed contact angle problem where the boundary value condition is They proved that if Ω is strictly convex and |Dϕ| < min ∂Ω κ, where κ is the curvature of ∂Ω, the solution of mean curvature flow converges to a surface which moves at a constant speed up to a translation. For n ≥ 2, Guan [12] studied the more generalized mean curvature type equation with the prescribed contact angle problem as below in Ω × (0, ∞), where ν is an inward unit normal vector to ∂Ω. He considered the asymptotic behavior of the solutions as t → ∞ for two special cases: Guan proved that in both cases the solution asymptotically approaches the solution of the corresponding stationary equation. But in [12], ϕ(u, Du) must have crucial monotonicity with respect to u. Andrews and Clutterbuck [2] studied the mean curvature equation on a convex domain with the homogeneous Neumann boundary and initial data u 0 ∈ C(Ω) for q = 1.
Recently, the third author in this note deduced the existence theorem for the mean curvature flow of graphs with general Neumann boundary condition in [33] by applying the technique in [26]. In [25], Ma, Wang and the second author adjust the auxiliary function to give a uniform gradient estimate for the mean curvature type equation and the mean curvature equation with Neumann boundary on a strictly convex domain and describe the asymptotic behavior of the corresponding mean curvature flow.
But for the general capillary-type boundary with 0 < q < 1, there are no any related results. Therefore, in this article our goal is to solve this type of problems on a strictly convex bounded domain for mean curvature equation and mean curvature flow.
Assume f is a C 1 function, and for some positive constants L 1 , L 2 , Now we state the first existence and uniqueness of solution to the equation (1) as below.
Before stating the asymptotic behavior of the mean curvature flow with the capillary-type boundary, we need to introduce the additive eigenvalue problem, which will play a vital role in the whole paper.
The additive eigenvalue problem is to be considered the stationary equation with the following boundary condition in Ω, This problem is to seek for a pair (λ, u) such that u is a viscosity solution to the stationary problem (7). If (λ, u) is such a pair, then λ and u are called an additive eigenvalue and an additive eigenfunction of equation (7) respectively. The additive eigenvalue problem appears in ergodic optimal control and the homogenization of Hamilton-Jacobi equations. In ergodic optimal control the additive eigenvalue λ corresponds to averaged long-run optimal costs while λ determines the effective Hamiltonian in the homogenization of Hamilton-Jacobi equations. This problem is often called the ergodic problem in the viewpoint of ergodic optimal control. The additive eigenvalue problem has been studied by so many mathematicians, such as Lions, Ishii, Barles, Giga et al. They applied the additive eigenvalue problem to the large time behavior of the Cauchy problem of Hamilton-Jacobi equations. We also refer to the literatures [3,4,7,23,24] for the asymptotic problems which treat Hamiltion-Jacobi equations under various boundary conditions including four types of boundary conditions: dynamical boundary condition, state constraint boundary condition, Dirichlet boundary condition and Neumann boundary condition. For Hamiltion-Jacobi equation, the regularity of the solutions is generally no more than Lipschitz continuity and the strong maximum principle can not be expected there. More details can be found in [3,4,17,22] and their references therein.
Next we give our second result.
Let Ω be a strictly convex bounded domain in R n (n ≥ 2) with smooth boundary and q > 0 be a constant. For φ(x), ϕ(x) ∈ C ∞ (Ω), there exists a unique λ ∈ R and u ∈ C ∞ (Ω) solving in Ω, where ν is an inward unit normal vector to ∂Ω. Moreover, the solution u is unique up to a constant.
Then, following the argument in [25,27], we have the third result as follows.
Next we will state another main existence and uniqueness theorem of capillarytype boundary value problem.
Let Ω be a strictly convex bounded domain in R n (n ≥ 2) with smooth boundary and q > 0 be a constant. For any φ ∈ C ∞ (Ω) and k > 0, there exists a unique function u ∈ C ∞ (Ω) solving where ν is an inward unit normal vector to ∂Ω.

Remark 2.
In [26], the third author in this note and Ma proved that for any bounded domain Ω with smooth boundary, has a unique solution. To get the existence, they need to get C 0 estimate first and then obtain C 1 estimate with C 0 estimate. Here we will directly compute the C 1 estimate without the C 0 estimate but depending on the convexity of the domain. Actually, we can generalize Theorem 1.4 to any q ≥ 0 and any bounded smooth domain Ω by adjusting the auxiliary function in [32] , which will be given a remark in the last section of the paper .
In addition, we give the existence of the additive eigenvalue problem for some mean curvature equations.
Let Ω be a strictly convex bounded domain in R n (n ≥ 2) with smooth boundary and q > 0 be a constant. For any φ ∈ C ∞ (Ω), there exists a unique λ ∈ R and a function u ∈ C ∞ (Ω) solving where ν is an inward unit normal vector to ∂Ω. Moreover, the solution u is unique up to a constant.

Remark 3.
Integrating two sides in Ω in the formula (13), we obtain
Remark 4. Consider the constant mean curvature equation with the prescribed contact angle boundary value condition where ν is an inward unit normal vector to ∂Ω. Even if the domain Ω is a strictly convex bounded domain in R n with smooth boundary, θ 0 is a constant and it satisfies the compatibility condition λ|Ω| = − cos θ o |∂Ω|, the existence of solution for the equation (14) is a very delicate problem. For n = 2, Giusti [10] got an existence theorem when the boundary curvature of the domain Ω denoted by k, satisfying k < |∂Ω| |Ω| at each point. For more information, one can see the related papers [6], [10] and the book [8].
To help the readers know the method for Theorem 1.2 and Theorem 1.5, we introduce the strategy. Firstly, we give the uniform C 1 estimate (independent of ε) for the solution to mean curvature type equations in Ω, Then by the maximum principle, we can give the C 0 uniform estimate for εu. By the Schauder theory, we get uniform high order estimates. Letting ε → 0, Theorem 1.2 and Theorem 1.5 have been proved.
In this paper, in order to simplify the proof of the theorems, we write O(z) as an expression that there exists a constant C > 0 such that |O(z)| ≤ Cz and C is not related to k, where k is in (11).
When Ω is a strictly convex smooth domain, there exists a defining function h for Ω such that {h ij } n×n ≥ k 0 {δ ij } n×n in Ω for k 0 > 0, h ν = −1 and |Dh| = 1 on ∂Ω. Actually, as h is convex, we know sup Ω |Dh| 2 ≤ 1. In the remaining paragraphs, we will always denote h as the defining function for the strictly convex domain.
In this paper, we proceed as below. In Section 2, in order to prove Theorem 1.1 for mean curvature flow, we shall first give a priori C 0 and C 1 estimates by choosing a new auxiliary function. In Section 3, similarly, we shall show the C 0 estimate and uniform C 1 estimate of mean curvature equation with capillary-type problem so that we can prove Theorem 1.4. In Section 4, adopting the classical method, we will prove the existence of solution of the additive eigenvalue problem, and then will prove Theorem 1.2 and Theorem 1.5. In Section 5, making use of the additive eigenvalue problem, we will describe the long time asymptotic behavior of the corresponding mean curvature flow and finish the proof of Theorem 1.3. Finally, we give some remarks and the comparison with the results in [32] in section 6.
2. Mean curvature flow with general Capillary-type boundary value problem. In this section, we will consider the parabolic equation where Ω is a strictly convex bounded domain in R n with C 3 boundary, ν is an inward unit normal vector to ∂Ω, f satisfy (4) (5) and a ij has been denoted as (10).
We first give C 0 and C 1 estimate for this problem (16) and then get the existence of the solution to the mean curvature flow by the standard theory.
where C 1 is a constant independent of T .
Proof. Following the method in [1,Lemma2.2], it suffices to prove the following: for any fixed T > 0, if u t admits a positive local maximum at some point ( We argue by contradiction. Thus we assume t 0 > 0. It is easy to calculate that u t satisfies the equation Then from f u ≥ 0 and the parabolic maximum principle to obtain that u t attains its maximum on ∂Ω × [0, T ] or on Ω × {t = 0}. Hence x 0 ∈ ∂Ω.
On the other hand, differentiating the boundary condition where C ij u ti is the tangential part vector of Du t on the boundary. The right side of equation (19) is zero at (x 0 , t 0 ). It's a contradiction to the Hopf's lemma. Hence it follows that max where C 1 is a constant which is independent of T .
For interior gradient estimate for mean curvature equation with zero gravity, the auxiliary function with main term like log log |Du| 2 works in Wang's paper [31]. For global gradient estimate for oblique boundary problem, |Du| 2 is not suitable to deal with the boundary. In [20], Lieberman used the auxiliary function with the main term like |D u| 2 to deal with the oblique boundary for q = 0 and q > 1. After that, in [26] , Ma and Xu used the auxiliary function with main term like |D(u + φd)| 2 , where d is the distance function to the bounary and solved the equation for q = 1. In this paper, we are inspired by the auxiliary function |v + φu ν | in Gerhardt's paper [9], where Gerhardt used the integral methods to solve the prescribed contact boundary problem. To solve the boundary problem for 0 < q < 1 , we will introduce the auxiliary function with the main term as v q+1 −(q +1) n l=1 φ(x)u l h l . Actually, this main term also works for q ≥ 1 even for a non-strict domain. As we want to study the asymptotic behavior of the mean curvature flow, it's necessary to get the uniform gradient estimate with respect to time. Next we will apply the maximum principle to prove the gradient estimates for Theorem 1.1 by constructing a new auxiliary function.
Theorem 2.2. Assume that Ω ⊂ R n is a strictly convex bounded C 3 domain, n ≥ 2, and u is the solution to (16). Let ν be an inward unit normal vector to ∂Ω. Suppose f (x, z) ∈ C 1 (Ω × R), and φ(x) ∈ C 2 (Ω) are given functions respectively which satisfy the conditions (4)- (6). Then there exists a positive constant M 1 such that for any fixed T > 0, sup Here M 1 is independent of T and |u| C 0 .
and β is to be determined later.
Suppose that Φ(x, t) attains its maximum at the point (x 0 , t 0 ) ∈ Ω × [0, T ]. Now we divide two cases to prove Theorem 2.2. In the following proof, all computations will be done at (x 0 , t 0 ). If t 0 = 0, then we get the Theorem.
Case1: x 0 ∈ ∂Ω. If Φ attains its maximum at x 0 on ∂Ω, then we have Choose the coordinates in R n such that the positive x n -axis is the interior normal direction to ∂Ω at x 0 . More specifically, u n denote the unit inner normal derivative and u i , 1 ≤ i ≤ n − 1 denote the n − 1 tangential derivatives of u on the boundary. D denotes the derivative in R n and ∇ denotes the tangential derivative on the boundary. We also denote ∇ i (u n ) := u ni for 1 ≤ i ≤ n − 1. So |D u| 2 = n−1 k=1 u 2 k . By the Gauss-Weingarten formula, we get D kn u = u nk +κ kj u j , whereκ kj is the curvature matrix of the boundary.

As
By (21) and (22), we obtain where the second equality holds because we cancelled the term u nn due to h n = −1 and h i = 0 for i = 1, · · · , n − 1 on ∂Ω.
If u n (x 0 , t 0 ) is not zero, then we argue as below. As u n = −φv 1−q on ∂Ω, it implies that |u n | 2 ≤ C(1 + |D u| 2 + u 2 n ) 1−q . By q > 0, For 0 < q < 1, we consider that if |u n | > (100C) 1 2q , then it holds that |D u| 2 u 2 n > 100 1 1−q − 2 > 1 and |D u| 2 ≥ u 2 n . If |u n | ≤ (100C) 1 2q , as it is assumed that v is very large, then one gets |D u| 2 ≥ u 2 n . For q ≥ 1, we have |u n | ≤ C, as we assume v is very large, it holds that |D u| 2 ≥ u 2 n . Above all, |D u| 2 ≥ u 2 n at (x 0 , t 0 ). As we always assume v is very large, it holds that on ∂Ω Therefore, by (27) and (28), it yields that where κ 1 is the smallest principle curvature of the boundary. So, taking 0 < β < 1 4 κ 1 , we get |Du| 2 (x 0 , t 0 ) ≤ C. Case2: x 0 ∈ Ω. It is easy to see that the first and the second derivatives of Φ are as follows and It holds that By rotating the axes, we assume that and the matrix {u ij (x 0 , t 0 )} 2≤i,j≤n is diagonal. Therefore a 11 = 1 v 2 , a ij = 0 for i = j, and a ii = 1 for i ≥ 2.
For i = 1, · · · , n, it holds that By Φ i (x 0 , t 0 ) = 0 and (35), we have for i ≥ 1, For i ≥ 2, it follows that We write O(z) as an expression that there exists a constant C > 0 such that |O(z)| ≤ Cz and C is not related to α, where α is determined later. And we point out that α will be taken as a small constant. We emphasis this notation again and use it to make the expression clearly and simply.
For i = 1, by (36) and (37), it follows that Since we have denoted a ij = δ ij − u i u j 1 + |Du| 2 , the first equation in (16) can be rewritten Differentiating (39) to x l for l ≥ 1, we have n i,j=1 and

JUN WANG, WEI WEI AND JINJU XU
Therefore for l ≥ 1, (40) becomes n i,j=1 To deal withĪ 1 , we denoteĪ 1 = 1 w I 1 and handle I 1 . By the coordinate, in fact, By direct computations, it holds that Based on (42) and (45), we rewrite I 1 in (43) as where and Firstly, from (37) and (38), it is not difficult to obtain C|u ii |.
Next we deal with I 11 . By f u ≥ 0 and |f x | ≤ L 1 , it yields that For J 1 , we get that For J 2 , by Cauchy inequality, it holds that For J 3 , we have By (52)-(54) and q > 0, we obtain Putting I 11 in (55), I 12 +I 13 in (50) into (46), and combiningĪ 1 = 1 w I 1 , it follows thatĪ since it holds that w > 1 C|u ii | ≥ −Cv 1−q .
With the gradient estimates of the solution to the parabolic equation in (1), it becomes uniformly parabolic equation for any fixed T > 0. By the standard theory, we obtain the long time existence of solutions. Theorem 2.3 (Theorem 1.1). Assume that Ω ⊂ R n is a strictly convex bounded C 3 domain for n ≥ 2. Let ν be the inward unit normal vector to ∂Ω. Under the conditions (4)-(6), there exists a unique solution u ∈ C 2,σ (Ω×[0, ∞)) to the problem (1) for some σ ∈ (0, 1).
Using the same auxiliary function without t and almost the same computation, we can also obtain Theorem 2.4. Let Ω be a strictly convex bounded domain in R n with smooth boundary and suppose q > 0 is a constant and k ≥ 0. Let φ ∈ C 2 (Ω), ϕ ∈ C 1 (Ω) and u ∈ C 3 (Ω) satisfy in Ω , where ν is an inward unit normal vector to ∂Ω. Then we have where M 1 is independent of k and depends on Ω, |φ| C 2 (Ω) , |ϕ| C 1 (Ω) .
3. Uniformly global C 0 and C 1 estimate for the proof of Theorem 1.4. In this section, we give C 0 estimate for the solution to the classical mean curvature equation (60) and then we will use the same auxiliary function in Theorem 2.2 to give the C 1 estimate for the solution to equation (60) with more delicate computations.
Proof. We denote the solution to equation (2) by u k . We give the C 0 estimate for the solution. Let g be a smooth function on Ω satisfying ( 1 + |Dg| 2 ) q−1 D ν g < − sup Ω |φ(x)| on ∂Ω for q > 0. Let ζ be a point where g − u k achieves its minimum. If ζ ∈ ∂Ω, then D T g(ζ) = D T u k (ζ) and D ν g(ζ) ≥ D ν u k (ζ), where T denotes the tangent vector to ∂Ω.
For 0 < q ≤ 1, we know that p ( 1 + a 2 + p 2 ) 1−q is monotonously increasing with respect to p for fixed a.
It's a contradiction to the definition of g.
With similar proof to Lemma 3.1 , we know Corollary 1.
Next we will prove the uniform gradient estimate for equation (2).
Let Ω be a strictly convex bounded domain in R n with C 3 boundary and suppose q > 0 is a constant, k ≥ 0 and α ∈ R.. Suppose φ(x) ∈ C 2 (Ω) and Then there exists a positive constant C > 0 such that where C is independent of k and depends on Ω, |φ| C 2 (Ω) .
Proof. We choose the auxiliary functioñ which is similar to the proof in Theorem 2.2. Here we omit the detailed proof.
4. Proof of Theorem 1.4 and Theorem 1.5. In this section, we will give the proof for the existence of solution to equation (60) and the corresponding additive eigenvalue problem, which follows the argument in [22,25] and includes Theorem 1.4 and Theorem 1.5.
Let Ω be a strictly convex bounded domain in R n (n ≥ 2) with smooth boundary, q > 0 be a constant and α ∈ R. For any φ ∈ C ∞ (Ω) and k > 0, there exists a unique function u ∈ C ∞ (Ω) solving (64).
For q > 0, it holds that n l=1 G p l ν l > 0.
According to the paper by Lieberman and Trudinger [21], this capillary-type boundary value problem belongs to oblique boundary value problems. In order to make use of the continuity method to prove the existence for such oblique boundary value problems, we consider a family of capillary-type boundary value problems div( Du where τ ∈ [0, 1]. For τ = 0, u = 0 is the unique solution. And we need to find the solution for τ = 1. By the standard elliptic regularity theory or as in the reference [21], if we can get the a priori estimates for the C 2 (Ω) solution of the equations (65) and (66) sup Ω |u| ≤K 1 , sup Ω |Du| ≤K 2 , whereK 1 ,K 2 are independent of τ . Then we can get the existence. As we have obtained the global C 0 and C 1 a priori estimates for τ = 1, by the continuity method, the existence of such problem has been proved. Next we prove the uniqueness. Assume there exist two different solutions u 1 and u 2 for (11). Letw = u 1 − u 2 andw satisfies Because n l=1 G p l ν l > 0 and k > 0, we knoww = 0 and therefore u 1 = u 2 by the maximum principle and Hopf's lemma. This completes the proof.
By Theorem 4.1, we get Theorem 1.4 with α = 0. Let Ω be a strictly convex bounded domain in R n (n ≥ 2) with smooth boundary, q > 0 be a constant and α ∈ R. For any φ ∈ C ∞ (Ω), there exists a unique λ ∈ R and a function u ∈ C ∞ (Ω) solving where ν is an inward unit normal vector to ∂Ω. Moreover, the solution u is unique up to a constant. Here λ is called the additive eigenvalue.
Proof. We denote by u ε the solution to equation (11) with k = ε. We know that for each ε, the solution to equation (11) exists, which is achieved in Section 3.
Let α = 0, we get Theorem 1.5 by Theorem 4.2. Actually, with the same proof, we can get Theorem 1.2 and we skip the proof. 5. Asymptotic behavior of the mean curvature flow with oblique boundary. In this section, by making use of the additive eigenvalue problem of (8) and following the argument in [25,27], we will study the asymptotic behavior of the mean curvature flow with capillary-type boundary in Theorem 1.3.
By the strong maximum principle and the Hopf's lemma, we know u * is a constant. This makes a contradiction to osc(u * ) = δ.