A New Proof of Gradient Estimates for Mean Curvature Equations with Oblique Boundary Conditions

In this paper, we will use the maximum principle to give a new proof of the gradient estimates for mean curvature equations with some oblique derivative problems. Specially, we shall give a new proof for the capillary problem with zero gravity in any dimension $n\geq 2$ and Neumann problem in $n=2, 3$ dimensions.

In (2), for q = 0, it is corresponding to capillary boundary condition and for q = 1, it is corresponding to Neumann boundary value condition.
The interior gradient estimates and the Dirichlet problem for the prescribed mean curvature equation have been extensively studied. We refer the reader to see the book [4] written by Gilbarg and Trudinger. More other oblique boundary value problems for the second order elliptic equations can be seen in the literature [10] written by Lieberman and the references therein.
For the mean curvature equation with capillary problem, Ural'tseva [15] first got the boundary gradient estimates and the corresponding existence theorem on the positive gravity case (f u ≥C 0 > 0,C 0 is a constant). In the same year, Simon and Spruck [12] and Gerhardt [2] obtained existence theorem on the positive gravity case respectively. For more general quasilinear divergence structure equation with conormal derivative boundary value problem, Lieberman [6] gave the gradient estimate. They all obtained these estimates via test function technique.
Spruck [13] firstly used the maximum principle to obtain boundary gradient estimate in two dimension for positive gravity case. Later, Korevaar [5] generalized his normal variation technique and got the boundary gradient estimates in the positive gravity case in high dimensional cases. In [7,8], Lieberman developed his maximum principle approach on the boundary gradient estimates to the quasilinear elliptic equations with oblique derivative boundary value problem and in [9] he got the maximum principle proof for the gradient estimates on the general quasilinear elliptic equations with capillary boundary value problems in zero gravity case (f u ≥ 0).
Lieberman ([10], in page 360) proved the gradient estimates of a more general class of quasilinear elliptic equations with the boundary condition (2) for q = 0 or q > 1. Recently, for the specific problem (1)-(2), Ma and Xu [11] got the boundary gradient estimates of mean curvature equations with Neumann problem for q = 1 via the maximum principle and obtained an existence result in positive gravity case.
In this paper, we find a new auxiliary function and use the maximum principle to give a unified proof of the gradient estimates for the problem (1)-(2) with q > 1 or q = 0 or q = 1 (n=2, 3) cases. In order to avoid many repeated calculations, here we only give the new proof for q > 1 or q = 0.
The rest of the paper is organized as follows. In section 2, we first give some notations and prove Theorem 1.1 under the help of Lemma 3.5. This lemma will be proved in section 3. Finally we shall show the proof of Theorem 1.2 in section 4.
In this paper, we follow the summation convention: All repeated indices from 1 to n denote summation.
Then it is well known that there exists a positive constant µ 1 > 0 such that d(x) ∈ C 3 (Ω µ1 ). As in [12] or [10] in page 331, we can take γ = Dd in Ω µ1 and note that γ is a C 2 (Ω µ1 ) vector field. As mentioned in [9] and in [10], we also have the following formulas |Dγ| + |D 2 γ| ≤C(n, Ω) in Ω µ1 , As in [10], we define and for a vector ζ ∈ R n , we write ζ for the vector with i-th component c ij ζ j . So Let Then the equations (1)-(2) are equivalent to the followings Now we begin to prove Theorem 1.1. Using the techniques developed by Spruck [13], Lieberman [9] and Wang [16], we shall choose an auxiliary function which contains |D u| 2 and other lower order terms. Then we use the maximum principle for this auxiliary function in Ω µ0 , 0 < µ 0 ≤ µ 1 , µ 0 is to be determined later. At last, we get our estimates.
Proof of Theorem 1.1. Let where α 0 = 2C 0 L 2 + 2C 0 + 2 is a constant, and C 0 = 10n + 10n 2 max x∈∂Ω |Ã|, |Ã| is the modulus of the second fundamental form of ∂Ω. Setting where in q > 1 boundary value case, we choose We now choose µ 0 = min{ 1 2 , µ1 2 }. Assume that ϕ(x) attains its maximum at In the following, we divide three cases to complete the proof of Theorem 1.1. Case I. If ϕ(x) attains its maximum at x 0 ∈ ∂Ω, then we shall get the bound of |Du|(x 0 ).
Case II. If ϕ(x) attains its maximum at x 0 ∈ ∂Ω µ0 Ω, then we shall get the estimates via the standard interior gradient bound [4].
Case III. If ϕ(x) attains its maximum at x 0 ∈ Ω µ0 , then we can use the maximum principle to get the bound of |Du|(x 0 ). Now all computations work at the point x 0 . Case I. Suppose first that x 0 ∈ ∂Ω. We differentiate ϕ along the normal direction.
Applying (9) and (11), it follows that Differentiating (14) with respect to tangential direction, we have It follows that For ψ = ψ(x, u), denote by Since differentiating (21) with respect to x k , we obtain Inserting (22) into (16), we have Since c kl ϕ k = 0, and c kl γ k = 0, we obtain Inserting (24) into (23), we have Putting (25) into (17), combining (16), we have In the following, for the the proof of our theorems in this paper, we only consider special case ψ = ψ(x).
At first for q > 1, since at x 0 , then Thus we complete this proof. So in the following we assume then from |Du| 2 = |D u| 2 + u 2 γ , we have Now assume at x 0 , Inserting (31) into (26), and by the choice of h(u), g(d) in (15), we obtain On the other hand, we have ∂ϕ ∂γ it is a contradiction to (32). So we have This is due to interior gradient estimates. From Remark 1, we have whereM 1 is a positive constant depending only on n, M 0 , µ 0 , L 1 .
In this case, x 0 is a critical point of ϕ. Now we choose the normal coordinate at x 0 by rotating the coordinate system suitably. We may assume that u i (x 0 ) = 0 for 2 ≤ i ≤ n and u 1 (x 0 ) = |Du|(x 0 ) > 0. And we can further assume that the matrix From Gilbarg and Trudinger [4] [page 368, formula (15.38)], we have where C 1 is a positive constant depending on n, L 1 , M 0 . From Case I, suppose that the formula (29) holds, otherwise we finish the proof of Theorem 1.1. Hence, So we have where Ω µ0 (M ) = Ω µ0 {|D u| ≥ M }, M > 10 is a constant; C 3 is a positive constant depending only on n, Since P (x 0 ) ≥ P (x 1 ), then we have It follows that where C 4 is a positive constant depending on n, µ 0 , L 1 , L 2 , M 0 . However Assume |D u|(x 0 ) ≥ M , otherwise we get the estimate. Hence at x 0 , Then we have at x 0 , From the above choices, we shall prove Theorem 1.1 with three steps, as we mentioned before, all the calculations will be done at the fixed point x 0 .
Taking the first derivatives of ϕ, Taking the derivatives again for ϕ i , we have Using the formula (44), it follows that Then we get 0 ≥ a ij ϕ ij =: where and From the choice of the coordinate, we have and Now we first treat the term I 2 .
From the equation (11), taking the first derivatives of |D u| 2 , we have Taking the derivatives of |D u| 2 once more, we have By the equations (48) and (55), we can rewrite I 1 as where In the following, we shall deal with I 11 , I 12 , I 13 and I 14 respectively. For the terms I 11 and I 12 : from (50), we have For the term I 13 : by the equation (13), we have and Differentiating the equation (13), we have From the equation (12), we have By the definition of v, we have Hence, from the equationss (60)-(63), we have By the equation (64), we get For the term I 14 : Combining the equations (57), (58), (65) and (66), it follows that Inserting the equations (52) and (67) into (48), we can obtain the following formula 0 ≥ a ij ϕ ij =: where Q 1 contains all the quadratic terms of u ij ; Q 2 is the term which contains all linear terms of u ij ; and the remaining terms are denoted by Q 3 . Then we have The linear terms of u ij are and the remaining terms are From the estimate on I 2 in (53), we have in the computation of Q 3 , we use the relation D k f = f u u k + f x k and f u ≥ 0, where C 6 is a positive constant which depends only on n, Ω, M 0 , µ 0 , L 1 , L 2 .
Step 2: In this step we shall treat the terms Q 1 , Q 2 using the first order derivative condition ϕ i (x 0 ) = 0, and let By the equations (44) and (54), we have Using the equation (74, we get and Through the equation (76) and the choice of the coordinate at x 0 , we have Using the equations (75) and (77), it follows that where we have let By the equations (59) and (78), we have Now we use the formulas (75)-(78) to treat each term in Q 1 , Q 2 . At first, we treat the first five terms of Q 1 in (69), and get (80)-(84). By the equations (75) and (78), we have From the equations (76) and (77), we get From the equation (77), we have By the equations (77) and (78), it follows that Again by the equations (77) and (10), we get 2 2≤i,j≤n Now we treat the first four terms of Q 2 in (70), and get (85) By the equation (78), we obtain From the equation (77), we have and We treat the term Q 1 using the relations (80)-(84), and use the formulas (85)-(88) to treat the term Q 2 . By the formula on Q 3 in (71), we can rewrite the formula (68) as the following. 0 ≥ a ij ϕ ij =: where J 1 only contains the terms with u ii , the other terms belong to J 2 . Denote here J 11 contains the quadratic terms of u ii (i ≥ 2), and J 12 is the term including linear terms of u ii (i ≥ 2). It follows that where d i =(c 11 ) 2 v 2 + (c 1i ) 2 = (c 11 ) 2 u 2 1 + (c 11 ) 2 + (c 1i ) 2 , i = 2, 3, . . . , n, In addition, We write other terms as J 2 , then Using the formula on Q 3 in (72) and I 2 in (53), we get the following estimate on J 2 , So if we use h(u), g(d) in (15), then we have where C 7 , C 8 and the following C 9 , . . . , C 15 are positive constants which only depend on n, Ω, µ 0 , M 0 , L 1 , L 2 .
Step 3: In this step, we concentrate on J 1 . We first treat the terms J 11 and J 12 and obtain the formula (104), then we complete the proof of Theorem 1.1 through Lemma 3.5.
By the equation (79), we have where we have let We first treat the term J 11 : using the equation (99) to simplify (91), we get and A 1i =(c 11 ) 2 (c 1i ) 2 (e 2 + e i ) + c 11 (c 1i ) 2 + (c 12 ) 2 , Now we simplify the terms in J 12 : by the equation (98), we can rewrite (94) as Using the equations (100) and (103) to treat (90), we have and we also have let For K i and R, using the formulas on D in (99); the formula of A in (73); e i ; d i in (92)-(93), and h(u), g(d) in (15), we have the following estimates Now we use Lemma 3.5, if there is a sufficiently large positive constant C 11 such that then we have where we use the formulas (γ 1 ) 2 = 1 − c 11 , d 2 in (92) and A in (73). Using the estimates on J 1 in (110) and J 2 in (96), from (89) we obtain By the choice of h(u), g(d) in (15), it follows that By (42), (109) and (112), there exists a positive constant C 15 such that So from Case I, Case II, and (113), we have Since ϕ(x) ≤ ϕ(x 0 ), for ∀x ∈ Ω µ0 , there exists M 2 such that where M 2 depends only on n, Ω, µ 0 , M 0 , L 1 , L 2 . So at last we get the following estimate 3. Some lemmas. In this section, we prove the main Lemma 3.5 which has been used to get the important estimate (110). Firstly, we give the definition and some properties of elementary symmetric functions. More details can be seen in Caffarelli-Nirenberg-Spruck [1] or Guan-Ma [3].
Proof of Lemma 3.4. We only need to prove that the following determinate of E is positive.
Case 2: if n = even Since σ 1 (a) = 2≤i≤n a i = c 11 > 0, it follows that then the matrix E is positive definite.
Now we prove the main lemma.
Lemma 3.5. We define (b ij ) as in (101), where d i , e i are defined as in (92)-(93) and A 1i , A 2i , G ij ,Ĝ ij are defined as in (102). And we define b i as in (105), v 2 = 1 + u 2 1 and c 11 ≥ 1 C4 . We study the following quadratic form where K i defined in (106) and we have the estimate (107) for K i . Then there exists a sufficiently large positive constant C 16 which depends only on n, Ω, µ 0 , M 0 , L 1 , L 2 such that if where positive constant C 17 also depends only on n, Ω, µ 0 , M 0 , L 1 , L 2 .