The Neumann problem of Hessian quotient equations

In this paper, we obtain some important inequalities of Hessian quotient operators, and global $C^2$ estimates of the Neumann problem of Hessian quotient equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann problem of Hessian quotient equations.

If l = 0, (1.1) is known as the k-Hessian equation. In particular, (1.1) is the Laplace equation if k = 1, l = 0, and the Monge-Ampère equation if k = n, l = 0. Hessian quotient equations are a more general form of k-Hessian equations, which appear naturally in classical geometry, conformal geometry and Kähler geometry, etc.
For the Dirichlet problem of elliptic equaions in R n , many results are known. For example, the Dirichlet problem of Laplace equation was studied in [5], Caffarelli-Nirenberg-Spruck [1] and Ivochkina [8] solved the Dirichlet problem of the Monge-Ampère equation, and Caffarelli-Nirenberg-Spruck [2] solved the Dirichlet problem of the k-Hessian equation. For the general Hessian quotient equation, the Dirichlet problem was solved by Trudinger in [24]. Also, the Neumann or oblique derivative problem of partial differential equations was widely studied. For a priori estimates and the existence theorem of Laplace equation with Neumann boundary condition, we refer to the book [5]. Also, we can see the recent book written by Lieberman [13] for the Neumann or oblique derivative problem of linear and quasilinear elliptic equations. In 1986, Lions-Trudinger-Urbas solved the Neumann problem of Monge-Ampère equation in the celebrated paper [16]. For related results on the Neumann or oblique derivative problem for some class fully nonlinear elliptic equations can be found in Urbas [25] and [26]. For the the Neumann problem of k-Hessian equations, Trudinger [23] established the existence theorem when the domain is a ball, and he conjectured (in [23], page 305) that one could solve the problem in sufficiently smooth strictly convex domains. Recently, Ma-Qiu [17] gave a positive answer to this problem and solved the the Neumann problem of k-Hessian equations in strictly convex domains.
Following the idea in [20], we can obtain the existence theorem of the classical Neumann problem of Hessian quotient equations Theorem 1.3. Suppose that Ω ⊂ R n is a C 4 strictly convex domain, 0 ≤ l < k ≤ n, ν is the outer unit normal vector of ∂Ω, f ∈ C 2 (Ω) is a positive function and ϕ ∈ C 3 (∂Ω). Then there exists a unique constant c, such that the Neumann problem of the Hessian quotient equation has k-admissible solutions u ∈ C 3,α (Ω), which are unique up to a constant. Remark 1.4. For the classical Neumann problem of Hessian quotient equations (1.3), it is easy to know that a solution plus any constant is still a solution. So we cannot obtain a uniform bound for the solutions of (1.3), and cannot use the method of continuity directly to get the existence. As in Lions-Trudinger-Urbas [16] and Qiu-Xia [20], we consider the k-admissible solution u ε of the equation for any small ε > 0. We need to establish a priori estimates of u ε independent of ε, and the strict convexity of Ω plays an important role. By letting ε → 0 and a perturbation argument, we can obtain a solution of (1.3). The uniqueness holds from the maximum principle and Hopf Lemma.
Remark 1.5. In the recent papers [9,10], Jiang and Trudinger studied the general oblique boundary value problems for augmented Hessian equations with some regular conditions and some concavity conditions. But here, the problems (1.2) and (1.3) do not satisfy the strictly regular condition and the uniform concavity condition. Remark 1.6. As we all know, the Dirichlet problems of Hessian and Hessian quotient equations are solved in strictly (k − 1)-convex domains. For the Neumann problems, we also want to know the existence results in strictly (k − 1)-convex but not convex domains. A special case, that is is solvable in strictly (k − 1)-convex domains, and see [19] for the proof. For general cases, the problem is open.
The rest of the paper is organized as follows. In Section 2, we collect some properties of the lementary symmetric function σ k , and establish some key inequalities of Hessian quotient operators. Following the idea of Lions-Trudinger-Urbas [16] and Ma-Qiu [17], we establish the C 0 , C 1 and C 2 estimates for the Neumann problem of Hessian quotient equations in Section 3, Section 4, Section 5, respectively. At last, we prove Theorem 1.1 and Theorem 1.3 in Section 6.

preliminary
In this section, we give some basic properties of elementary symmetric functions, which could be found in [12], and establish some key inequalities of Hessian quotient operators.
2.1. Basic properties of elementary symmetric functions. First, we denote by σ k (λ |i) the symmetric function with λ i = 0 and σ k (λ |ij ) the symmetric function with λ i = λ j = 0. Proposition 2.1. Let λ = (λ 1 , . . . , λ n ) ∈ R n and k = 1, · · · , n, then We also denote by σ k (W |i) the symmetric function with W deleting the i-row and i-column and σ k (W |ij ) the symmetric function with W deleting the i, j-rows and i, j-columns. Then we have the following identities.
The generalized Newton-MacLaurin inequality is as follows, which will be used all the time.

Key Lemmas.
In the establishment of the a priori estimates, the following inequalities of Hessian quotient operators play an important role.
Directly calculations yield and λ(A) ∈ Γ k with k ≥ 1. Then we have Proof. See Lemma 3.9 in [3] for the proof of (2.11).
We divide into two cases to prove (2.14).
From (2.9) and the generalized Newton-MacLaurin inequality, we can get Remark 2.8. These lemmas play an important role in the establishment of a priori estimates. Precisely, Lemma 2.6 is the key of the gradient estimates in Section 4, including the interior gradient estimate and the near boundary gradient estimate. Lemmas 2.5 and Lemma 2.7 are the keys of the lower and upper estimates of double normal second order derivatives on the boundary in Section 5, respectively.

C 0 estimate
The C 0 estimate is easy. For completeness, we produce a proof here following the idea of Lions-Trudinger-Urbas [16] and Ma-Qiu [17].
Proof. Firstly, since u is subharmonic, the maximum of u is attained at some boundary point x 0 ∈ ∂Ω. Then we can get By the comparison principle, we know u − A|x − x 1 | 2 attains its minimum at some boundary point Here, following the proof of Theorem 3.1, we can easily obtain where M 0 depends on n, k, l, diam(Ω), max ∂Ω |ϕ| and sup Ω f .

Global gradient estimate
In this section, we prove the global gradient estimate, involving the interior gradient estimate and the near boundary gradient estimate. To state our theorems, we denote d(x) = dist(x, ∂Ω), and Ω µ = {x ∈ Ω|d(x) < µ} where µ is a small positive universal constant depending only on Ω. In Subsection 4.1, we give the interior gradient estimate in Ω \ Ω µ , and in Subsection 4.2 we establish the near boundary gradient estimate in Ω µ , following the idea of Ma-Qiu-Xu [18] and Ma-Qiu [17].

4.1.
Interior gradient estimate. The interior gradient estimate is established in [3] as follows where C is a positive constant depending only on n, k, l and |D Hence we can get the interior gradient estimate in Ω \ Ω µ directly.
where M 1 depends on n, k, l, µ, |u| C 0 and |D x f | C 0 .
Since Ω is a C 3 domain, it is well known that there exists a small positive universal constant 0 < µ < 1 10 such that d(x) ∈ C 3 (Ω µ ). As in Simon-Spruck [21] or Lieberman [13] (in page 331), we can extend ν by ν = −Dd in Ω µ and thus ν is a C 2 (Ω µ ) vector field. As mentioned in the book [13], we also have the following formulas where C 0 depends only on n and Ω. As in [13], we define and for a vector ζ ∈ R n , we write ζ ′ for the vector with i-th component n j=1 c ij ζ j . Then we have We consider the auxiliary function It is easy to know G(x) is well-defined in Ω µ . Then we assume that G(x) attains its maximum at a point Hence, we can assume |Du|(x 0 ) > 10n[|ϕ| C 1 (Ω) + sup Ω |u|] in the following. Then we have Then x 0 ∈ Ω \ Ω µ , and we can use the interior gradient estimate, that is from Theorem 4.2, then we can prove (4.4) by a calculation similar with (4.10).
CASE II: We know from (4.8) , and by the Neumann boundary condition, we can get Also by the Neumann boundary condition, we can get (4.14) and (4.15), we We choose So we can prove (4.4) by a calculation similar with (4.10), or x 0 cannot be at the boundary ∂Ω by a contradiction discussion.
CASE III: x 0 ∈ Ω µ . At x 0 , we have 0 < d < µ, and by rotating the coordinate e 1 , · · · , e n , we can assume In the following, we denote λ = ( λ 2 , · · · , λ n ) = (u 22 (x 0 ), · · · , u nn (x 0 )), and all the calculations are at x 0 . So from the definition of w, we know Also we have at x 0 , From the definition of w, we know So we have ( otherwise there is nothing to prove). Moreover, for i = 1, · · · , n, we can get Then we have It is easy to know From the definition of w, we know From Lemma 2.6, we know Then we can get from (4.33), (4.34) and (4.35) So we can prove (4.4) by a calculation similar with (4.10).
As discussed in Remark 1.4, we need to consider the equation (1.4) to prove Theorem 1.3. It is crucial to establish a global gradient estimate of u ε independent of ε, and we need the strict convexity of Ω. Following the idea of [20], we can easily obtain Theorem 4.4. Suppose Ω ⊂ R n is a C 3 strictly convex domain, f ∈ C 1 (Ω) is a positive function, ϕ ∈ C 3 (∂Ω) and u ε ∈ C 3 (Ω) ∩ C 2 (Ω) is the k-admissible solution of Hessian quotient equation (1.4) with ε > 0 sufficiently small, then we have where M 1 depends on n, k, l, Ω, |f | C 1 and |ϕ| C 3 .

Global second derivatives estimate
We now come to the a priori estimates of global second derivatives, and we obtain the following theorem Theorem 5.1. Suppose that Ω ⊂ R n is a C 4 convex and strictly (k − 1)-convex domain, f ∈ C 2 (Ω) is a positive function, ϕ ∈ C 3 (∂Ω) and u ∈ C 4 (Ω) ∩ C 3 (Ω) is the k-admissible solution of Hessian quotient equation (1.2), then we have where M 2 depends on n, k, l, Ω, |u| C 1 , inf f , |f | C 2 and |ϕ| C 3 .
Following the idea of Lions-Trudinger-Urbas [16] and Ma-Qiu [17], we divide the proof of Theorem 5.1 into three steps. In step one, we reduce global second derivatives to double normal second derivatives on boundary, then we prove the lower estimate of double normal second derivatives on the boundary in step two, and at last we prove the upper estimate of double normal second derivatives on the boundary. where C 9 depends on n, k, l, Ω, |u| C 1 , inf f , |f | C 2 and |ϕ| C 3 .

Proof.
Since Ω is a C 4 domain, it is well known that there exists a small positive universal constant 0 < µ < 1 10 such that d(x) ∈ C 4 (Ω µ ) and ν = −Dd on ∂Ω. We define d ∈ C 4 (Ω) such that d = d in Ω µ and denote In fact, ν is a C 3 (Ω) extension of the outer unit normal vector field on ∂Ω.
To prove Lemma 5.3 and Lemma 5.5, we need the following lemma.
Proof. Firstly, we assume min In the following, we assume − min Motivated by Ma-Qiu [17], we consider the test function It is easy to know that P ≤ 0 on ∂Ω µ . Precisely, on ∂Ω, we have d = h = 0, and −Dd = ν, so we can get P (x) = 0, on ∂Ω. On ∂Ω µ \ ∂Ω, we have d = µ, and In the following, we want to prove P attains its maximum only on ∂Ω. Then we can get To prove P attains its maximum only on ∂Ω, we assume P attains its maximum at some point x 0 ∈ Ω µ by contradiction. Rotating the coordinates, we can assume In the following, all the calculations are at x 0 .
Firstly, we have where C 16 is a positive constant under control as follows It holds i∈B d 2 i < 1 = |Dd| 2 , and G is not empty. Hence for any i ∈ G, it holds and from (5.36), we have Also there is an i 0 ∈ G such that where κ max is defined as in (5.25). Direct calculations yield For u i 0 i 0 < 0, we know from Lemma 2.5, . This is a contradiction. So P attains its maximum only on ∂Ω. The proof of Lemma 5.3 is complete.

5.3.
Upper estimate of double normal second derivatives on boundary.
Proof. Firstly, we assume max Motivated by Ma-Qiu [17], we consider the test function It is easy to know P ≥ 0 on ∂Ω µ . Precisely, on ∂Ω, we have d = h = 0, and −Dd = ν, so we can get On ∂Ω µ \ ∂Ω, we have d = µ, and In the following, we want to prove P attains its minimum only on ∂Ω. Then we can get To prove P attains its minimum only on ∂Ω, we assume P attains its minimum at some point x 0 ∈ Ω µ by contradiction. Rotating the coordinates, we can assume In the following, all the calculations are at x 0 .
Firstly, we have where C 19 is a positive constant under control as follows It holds i∈B d 2 i < 1 = |Dd| 2 , and G is not empty. Hence for any i ∈ G, it holds and from (5.58), we have then we can get Also there is an i 0 ∈ G such that where κ max is defined as in (5.25). Direct calculations yield We divide into three cases to prove the result. Without generality, we assume that i 0 = 1 ∈ G, and u 22 ≥ · · · ≥ u nn .
CASE I: u nn ≥ 0. In this case, we have Hence from (5.71) and (5.72) This is a contradiction. CASE II: u nn < 0 and −u nn < c 4 10(4κmax+ 2 n ) u 11 . In this case, we have F ii <0, (5.78) since β ≥ 5n 2 (2κ max + 1 n ) C 9 c 1 . This is a contradiction. So P attains its maximum only on ∂Ω. The proof of Lemma 5.5 is complete.
Following above proofs, we can also obtain the estimates of second order derivatives of u ε in (1.4), and the strict convexity of Ω is important in reducing global second derivatives to double normal second derivatives on boundary. So we have Theorem 5.6. Suppose Ω ⊂ R n is a C 4 strictly convex domain, f ∈ C 2 (Ω) is a positive function, ϕ ∈ C 3 (∂Ω) and u ε ∈ C 4 (Ω) ∩ C 3 (Ω) is the k-admissible solution of Hessian quotient equation (1.4) with ε > 0 sufficiently small,, then we have where M 2 depends on n, k, l, Ω, M 1 , inf f , |f | C 2 and |ϕ| C 3 .

Existence of the boundary problems
In this section we complete the proof of the Theorem 1.1 and Theorem 1.3.
6.1. Proof of Theorem 1.1. For the Neumann problem of Hessian quotient equations (1.2), we have established the C 0 , C 1 and C 2 estimates in Section 3, Section 4, Section 5, respectively. By the global C 2 a priori estimates, Hessian quotient equations (1.2) is uniformly elliptic in Ω. Due to the concavity of Hessian quotient operator [ σ k (λ) σ l (λ) ] 1 k−l in Γ k , we can get the global Hölder estimates of second derivative following the discussions in [14], that is, we can get |u| C 2,α (Ω) ≤ C, (6.1) where C and α depend on n, k, l, Ω, inf f , |f | C 2 and |ϕ| C 3 . From (6.1) one also obtains C 3,α (Ω) estimates by differentiating the equation (1.2) and apply the Schauder theory for linear, uniformly elliptic equations.
Applying the method of continuity (see [5], Theorem 17.28), the existence of the classical solution holds. By the standard regularity theory of uniformly elliptic partial differential equations, we can obtain the higher regularity.
6.2. Proof of Theorem 1.3. The proof is following the idea of Qiu-Xia [20].
By a similar proof of Theorem 1.1, we know there exists a unique k-admissible solution u ε ∈ C 3,α (Ω) to (1.4) for any small ε > 0. Let v ε = u ε − 1 |Ω| Ω u ε , and it is easy to know v ε satisfies |Ω| Ω εu ε + ϕ(x), on ∂Ω. (6.2) By the global gradient estimate (4.36), it is easy to know ε sup |Du ε | → 0. Hence there is a constant c and a function v ∈ C 2 (Ω), such that −εu ε → c, −εv ε → 0, 1 |Ω| Ω εu ε → c and v ε → v uniformly in C 2 (Ω) as ε → 0. It is easy to verify that v is a solution of If there is another function v 1 ∈ C 2 (Ω) and another constant c 1 such that on ∂Ω. (6.4) Applying the maximum principle and Hopf Lemma, we can know c = c 1 and v −v 1 is a constant. By the standard regularity theory of uniformly elliptic partial differential equations, we can obtain the higher regularity.