Oblique boundary value problems for augmented Hessian equations I

In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge-Amp`ere type operators in optimal transportation and geometric optics, the general theory here embraces prescribed mean curvature problems in conformal geometry as well as oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations.


Introduction
In this paper we develop the essentials of a general theory of classical solutions of oblique boundary value problems for certain types of fully nonlinear elliptic partial differential equations, which we describe as augmented Hessian equations. Such problems arise in various applications, notably to optimal transportation, geometric optics and conformal geometry and our critical domain and augmenting matrix convexity notions are adapted from those introduced in [27,36,40] for regularity in optimal transportation. Our main concern here will be with semilinear boundary conditions but we will also cover the nonlinear case for appropriate subclasses of our general operators. The classical solvability of the Neumann problem for the Monge-Ampère equation was proved by Lions, Trudinger and Urbas in [23]. Not only was the approach in [23] special for the Neumann problem, but it follows from the fundamental example of Pogorelov [28] that the result cannot be extended to general linear oblique boundary value problems, [47,42]. On the other hand, the classical Dirichlet problem for basic Hessian equations has been well studied in the wake of fundamental papers by Caffarelli, Nirenberg and Spruck [1] and Ivochkina [8], with further key developments by several authors, including Krylov in [17] and related papers and Trudinger in [34]; (see also [7] for a recent account of the resultant theory under fairly general conditions).
Our main concerns in this paper are second derivative estimates under natural "strict regularity" conditions on the augmenting matrices, together with accompanying gradient and Hölder estimates, which then lead to classical existence theorems. Our theory embraces a wide class of examples which we also present as well as a key application to semilinear Neumann problems arising in conformal geometry. In ensuing papers we consider extensions to weaker matrix convexity conditions as well as the regularity of weak solutions and the sharpness of our domain convexity conditions. Extensions to the Dirichlet problem for our general class of equations are treated in [11]. Overall this paper provides a comprehensive framework for studying oblique boundary value problems for a large class of fully nonlinear equations, which embraces the Monge-Ampère type case in Section 4 in [13] as a special example.
Specifically we study augmented Hessian partial differential equations of the form, where Ω is a bounded domain in n dimensional Euclidean space R n with smooth boundary, Du and D 2 u denote the gradient vector and the Hessian matrix of the solution u ∈ C 2 (Ω), A is a n × n symmetric matrix function defined on Ω × R × R n , B is a scalar valued function on Ω × R × R n and G is a scalar valued function defined on ∂Ω × R × R n . We use x, z, p, r to denote the points in Ω, R, R n and R n×n respectively. The boundary condition (1.2) is said to be oblique, with respect to u ∈ C 1 (Ω), if where ν is the unit inner normal vector field on ∂Ω and β 0 is a positive constant. If G p · ν > 0 on all of ∂Ω × R × R n , we will simply refer to G (or G) as oblique. In the context, we shall use either F or F to denote the general operator in (1.1), and either G or G to denote the boundary operator in (1.2). Letting S n denote the linear space of n × n symmetric matrices, the function F in (1.1) is defined on an open, convex cone Γ in S n , with vertex at 0, containing the positive cone K + . In order to consider F in a very general setting, we assume that F ∈ C 2 (Γ) satisfies a subset of the following properties. F4: F (tr) → ∞ as t → ∞, for all r ∈ Γ.
We say an operator F satisfies the above properties, if the corresponding function F satisfies them. Note that we can take the constant a 0 in F2 and F5 to be 0 or −∞. When F is given as a symmetric function f of the eigenvalues λ 1 , · · · , λ n of the matrix r, with Γ closed under orthogonal transformations, we will refer to F as orthogonally invariant. In this case the above conditions are modelled on the conditions introduced for the study of the Dirichlet problem for the basic Hessian equations, with A = 0, by Caffarelli, Nirenberg and Spruck in [1] and Trudinger in [34]. The standard operators satisfying the above properties are the k-Hessian operators, F k = (S k ) 1 k , k = 2, · · · , n, which satisfy F1-F4, F5 + on Γ k with a 0 = 0, and their quotients F k,l = ( S k S l ) 1 k−l , n ≥ k > l ≥ 1, which satisfy F1-F5 on Γ k with a 0 = 0, where S k denotes the k-th order elementary symmetric function defined by (1.7) S k [r] := S k (λ(r)) = 1≤i 1 <···<i k ≤n λ i 1 · · · λ i k , k = 1, · · · , n, and Γ k denotes the cone defined by (1.8) Γ k = {r ∈ S n | S j [r] > 0, ∀j = 1, · · · , k}.
As usual we set F 0 = 1, so that we can also write the standard k-Hessian F k as the quotient F k,0 . It turns out that the proofs of our results and their underlying ideas are essentially just as complicated for these special cases as for the general situation so that a reader will not miss the main features of our techniques by restricting attention to them. More generally when a 0 = 0 and F is positive homogeneous of degree one, then properties F1, F2, F3 imply F4 and F5. Clearly F4 is obvious and to show F5 we have by the concavity F2, for a positive constant µ and r ∈ Γ, where the equality follows from the homogeneity, which implies r · F r = F (r). (Clearly it is enough to take µ = 1 here but it is convenient to use a general µ for later use).
We also note in general that F2 and F3 imply, for r ∈ Γ and finite a 0 , By the concavity F2, we have, for t > 0, from which (1.10) follows by taking t sufficiently large for the first inequality and sufficiently small for the second. If a 0 ≥ −∞, then (1.10) clearly holds if also F4 is satisfied, (or more generally lim inf F (tr) > −∞ as t → ∞). From (1.9) and (1.10), we then obtain for F (r) ≤ b, for some constant δ 0 , depending on F and b, by taking µ sufficiently large, so that condition F5 is itself a consequence of F2 and F4. As for (1.10) and (1.11), the condition F4 is typically more than we need in general and can be dispensed with in most of our estimates. When considering the equation (1.1), it will be enough to assume instead F (tr) > B(·, u, Du) for sufficiently large t.
In our scenario, we call M [u] := D 2 u − A(·, u, Du) the augmented Hessian matrix. Usually, we denote the elements of M [u] and the matrix F r in F1 by w ij = D ij u − A ij and F ij respectively. A function u is called admissible in Ω (Ω) if (1.12) M [u] ∈ Γ, in Ω, (Ω), so that the operator F satisfying F1 is elliptic with respect to u in Ω (Ω) when (1.12) holds. It is also clear that if M [u] ∈Γ with B ∈ F (Γ) in Ω (Ω) then (1.1) is elliptic with respect to u in Ω (Ω), namely we require B > a 0 in Ω (Ω) for F satisfying F3. An important ingredient for regularity of solutions to equations involving the augmented matrix M [u] is the co-dimension one convexity (strict convexity) condition on the matrix A with respect to p, that is (1.13) A kl ij (x, z, p)ξ i ξ j η k η l ≥ 0, (> 0), for all (x, z, p) ∈ Ω × R × R n , ξ, η ∈ R n , ξ ⊥ η, where A kl ij = D 2 p k p l A ij . As in [36,14,15], we also call the matrix A regular (strictly regular) if A satisfies (1.13). These conditions were originally formulated for optimal transportation problems in the Monge-Ampère case, k = n, in [27] and [40]. The strictly regular condition may also be viewed as a supplementary ellipticity. 3 We now start to formulate the main theorems in this paper. First we state a local/global second derivative estimate which extends the Monge-Ampère case in [27] and whose global version is needed for our treatment of the boundary condition (1.2). Theorem 1.1. Let u ∈ C 4 (Ω) be an admissible solution of equation (1.1) in Ω. Assume one of the following conditions: (i): F1, F2, F3 and F5 + hold; (ii): F1, F2, F3, F5 hold, and B is convex with respect to p.
The estimate (1.14) in Theorem 1.1 in the case Ω ′ = Ω provides us a global estimate which reduces the bound for second derivatives to the boundary. When Ω 0 = Ω we get the usual form of the interior estimate, which is already formulated for case (i) in [36]. A more precise version involving cut-off functions will be presented in Section 2. For the boundary estimates we need to assume appropriate geometric assumptions on the domain Ω. We consider the operator F in (1.1) and domains Ω ⊂ R n with ∂Ω ∈ C 2 and ν denoting the unit inner normal to ∂Ω, δ = D − (ν · D)ν the tangential gradient in ∂Ω, and P = I − ν ⊗ ν the projection matrix onto the tangent space on ∂Ω, where I is the n × n identity matrix. Then we introduce the A-curvature matrix on ∂Ω, We call Ω uniformly (Γ, A, G)-convex with respect to an interval valued function I(x) on ∂Ω, if for all x ∈ ∂Ω, z ∈ I(x), G(x, z, p) ≥ 0 and some µ = µ(x, z, p) > 0. For a given function u 0 , if we take I = {u 0 } in the above definition, then Ω is called uniformly (Γ, A, G)-convex with respect to u 0 . For the cases Γ = Γ k , corresponding to the k-Hessians and their quotients, (1.16) is equivalent to K A [∂Ω](x, z, p) ∈ Γ k−1 . Moreover in the Monge-Ampère case, k = n, we recover our definitions of uniform (A, G)-convexity in [13], which extend the notion of uniform c-convexity with respect to a target domain Ω * in the optimal transportation case as introduced in [40]. When the interval I = R, we will simply call Ω uniformly (Γ, A, G)-convex. This includes the case when A and G are independent of z as then the interval I becomes irrelevant. As in [13], we will assume that the function This includes the quasilinear case, when G pp = 0, where β = G p and ϕ are defined on ∂Ω × R. If G pp (·, u, Du) ≤ 0 on ∂Ω for u ∈ C 1 (Ω) then we say that G is concave in p, with respect to u. Note that we define the obliqueness in (1.3) with respect to the unit inner normal ν, so that our function G keeps the same sign with those in [13] and is the negative of that in [43,40,37]. When G is nonlinear in p we will assume a further structural condition on F . F6: We remark that the Hessian operators F k (k = 1, · · · , n) and the Hessian quotients F n,k (1 ≤ l ≤ n − 1) satisfy F6 in the positive cone K + [45]. Further examples are given in Section 4.2. To complete our hypotheses, we will also assume for the second derivative bounds in this paper that the cone Γ lies strictly in a half space in the sense that r ≤ trace(r)I for all r ∈ Γ, (unless F6 is satisfied). This property is satisfied by the cones Γ k for k ≥ 2, (but excludes the already well known quasilinear case when k = 1). We now state the global second derivative bound which can be viewed as the main result of this paper.
be an admissible solution of the boundary value problem (1.1)- is oblique and concave in p with respect to u satisfying (1.3), and either F5 + holds or B is independent of p. Assume further that either G is quasilinear or F also satisfies F6. Then we have the estimate where C is a constant depending on F, A, B, G, Ω, β 0 and |u| 1;Ω .
Remark 1.1. A stronger condition than regularity of the matrix function A is necessary in the above hypotheses as it is known from the Monge-Ampère case that one cannot expect second derivative estimates for general oblique boundary value problems for A ≡ 0, which is a special case of regular A but not strictly regular, see [42], [47]. We also remark that the alternative condition that B is independent of p may be replaced by D p B sufficiently small, as well as B convex with respect to p, and we will see from our treatment in Section 2 that such a condition is reasonable. Analogously, we may also replace the condition that G is quasilinear by D 2 p G sufficiently small.
Remark 1.2. It should be noted that the feasibility of our uniform convexity condition depends on an effective relationship between the boundary operator G and the curvature matrix K A [∂Ω] to ensure at least that the matrix P (D p A(x, z, p) · ν(x))P is uniformly bounded from below, for all x ∈ ∂Ω, z ∈ I(x), G(x, z, p) ≥ 0. More generally we need to impose a condition on the gradient Du, namely that there exists a sufficiently small boundary In order to apply Theorem 1.2, to the existence of smooth solutions to (1.1)-(1.2), we need gradient and solution estimates. Our conditions for gradient estimates are motivated by the case when F is linear and the corresponding conditions for gradient estimates for uniformly elliptic quasilinear equations, as originally introduced by Ladyzhenskaya and Ural'tseva [6,18]. First we need additional conditions on either A or F which facilitate an analogue of uniform ellipticity.
F7: For a given constant a > a 0 , there exists constants δ 0 , δ 1 > 0 such that F r ij ξ i ξ j ≥ δ 0 + δ 1 T , if a ≤ F (r) and ξ is a unit eigenvector of r corresponding to a negative eigenvalue.
We remark that F7 implies F5, with b = ∞, and moreover the Hessian quotients F k,l , for 0 ≤ l < k ≤ n satisfy F7 in the cone Γ k with constants δ 0 , δ 1 > 0, depending only on k, l and n, [2,33]. We formulate (almost) quadratic growth conditions on A and B as follows.
as |p| → ∞, uniformly for x ∈ Ω, |z| ≤ M for any M > 0. Note that in the analogous natural growth conditions in the uniform elliptic theory, "o" can be weakened to "O" as a continuity estimate is available [6,21]. Also the growth conditions on D z A and D z B in (1.22) are automatically satisfied under standard uniqueness conditions, namely when A and B are non-decreasing in z, that is D z A ≥ 0 and D z B ≥ 0. We now state a gradient estimate for oblique semilinear boundary conditions, that is when β in (1.17) is independent of z so that (1.2) may be written in the form Some variants and extensions, including weaker versions of conditions (1.22), local gradient estimates and extensions to nonlinear G will also be considered in conjunction with our treatment in Section 3.
As we will show in Section 3 the concavity condition F2 in Theorem 1.3 may be removed when F is positive homogeneous of degree one and more generally. Note that the condition on B in case (i) is automatically satisfied when B is convex in p. Also when B is bounded, we do not need to take b = ∞ in F5, while if the constants δ 0 and δ 1 in conditions F5 and F7 are independent of a, the constant C in the estimate (1.24) does not depend on b 0 in cases (i)(b) and (ii). Analogously to the situation with uniformly elliptic equations, we obtain gradient estimates in terms of moduli of continuity when "o" is weakened to "O" in the hypotheses, (1.22) and case (ii), of Theorem 1.3. In particular we will also prove a Hölder estimate for admissible functions in the cones Γ k for k > n/2, when A ≥ O(|p| 2 )I, which extends our gradient estimate in the case k = n in [13], Lemma 4.1. Taking account of this, as well as Theorems 1.2 and 1.3, we have, as an example of our consequent existent results, the following existence theorem for the augmented k-Hessian and Hessian quotient equations. In its formulation we will assume the existence of subsolutions and supersolutions to provide the necessary solution estimates and an appropriate interval I in our boundary convexity conditions. For this purpose we will say that functions u andū, in C 2 (Ω) ∩ C 1 (Ω), are respectively subsolution and supersolution of the boundary value problem (1. Theorem 1.4. Let F = F k,l for some 0 ≤ l < k ≤ n, A ∈ C 2 (Ω × R × R n ) strictly regular inΩ, , G is semilinear and oblique with G ∈ C 2,1 (∂Ω × R × R n ). Assume that u andū, ∈ C 2 (Ω) ∩ C 1 (Ω) are respectively an admissible subsolution and a supersolution of (1.1)-(1.2) with Ω uniformly (Γ k , A, G)-convex with respect to the interval I = [u,ū]. Assume also that A, B and ϕ are non-decreasing in z, with at least one of them strictly increasing, and that A and B satisfy This paper is organized as follows. In Section 2, we first prove the local/global second derivative estimate, Theorem 1.1, as well as an extension to non-constant vector fields, Lemma 2.1. Then in Section 2.2, by delicate analysis of the second derivatives on the boundary, we complete the proof of Theorem 1.2 through Lemmas 2.2 and 2.3 which treat respectively the estimation of non-tangential and tangential second derivatives. In the proof of Lemma 2.3 the strict regularity condition is crucial. In Section 3, we first prove the global gradient estimate, Theorem 1.3, under various more general structural assumptions on F , A and B. Following this, in Section 3.2, we prove the analogous local gradient estimates in Theorem 3.1. In Section 3.3 we derive a Hölder estimate for admissible functions in the cones Γ k for k > n/2, from which we can infer gradient estimates under natural quadratic growth conditions. In Section 4, we prove existence theorems, Theorems 4.1 and 4.2 for semilinear and nonlinear oblique boundary value problems based on the a priori derivative estimates, which include Theorem 1.4 as a special case. We then present in Section 4.2 various examples of operators F, matrices A, and boundary operators G along with the application to conformal geometry, where we relax the umbilic boundary restriction for second derivative estimates in Yamabe problems with boundary as studied in [3,16]. Furthermore we show in Section 4.3 that our theory can be applied to degenerate elliptic equations, where F is only assumed non-decreasing in F1; see Corollary 4.1, and provide a particular example in Corollaries 4.2 and 4.3. In Section 4.4, we conclude this paper with some final remarks which also foreshadow further results.

Second derivative estimates
We introduce some notation and proceed to the second order derivative estimates for admissible solutions u of (1.1)-(1.2). We denote the augmented Hessian M [u] by w = {w ij }, that is As usual we denote the first and second partial derivatives of F at M [u] by F ij and F ij,kl , namely Then for an admissible u, we know from F1 that the matrix {F ij } is positive definite and from F2 that for all {η ij } ∈ S n . Let us also denote T = traceF r = F ii so that by positivity |F r | ≤ T . It will also be convenient here to use (1.20) to express the strict regularity of A, with respect to u, in the form for arbitrary vectors ξ, η ∈ R n , where c 0 and c 1 are positive constants depending on A and sup(|u| + |Du|). Then for any positive symmetric matrix {F ij } with eigenvalues λ 1 , · · · , λ n > 0 and corresponding eigenvectors, φ 1 , · · · , φ n , we can write Local/global second derivative estimates. In this subsection, we derive the local and global second derivative estimates for admissible solutions of equation (1.1), and give the proof of Theorem 1.1. We will need to differentiate the equation (1.1), with respect to vector fields τ = (τ 1 , · · · , τ n ) with τ i ∈ C 2 (Ω), i = 1, · · · , n. We introduce the linearized operators of the operator F and equation (1.1), For convenience below we shall as usual denote partial derivatives of functions on Ω by subscripts, that is Differentiating once we now obtain, Differentiating twice, we then obtain To derive the local and global estimates in Theorem 1.1, we only need τ to be a constant unit vector, in which case the last two terms in (2.6) and (2.7) are not present. Setting we then have from the concavity F2 and strict regularity (2.3), where C is a constant depending on n, |A| C 2 , |B| C 2 and |u| 1;Ω and λ B is the minimum eigenvalue of the matrix D 2 p B. Invoking conditions F5 + , or F5 and convexity of B in p, we then have from the classical maximum principle, under the hypotheses of Theorem 1.1, which implies a global upper bound for D 2 u, since τ can be any unit vector. To get the corresponding local estimate, we fix a function ζ ∈ C 2 (R n ), satisfying 0 ≤ ζ ≤ 1 and define, From the inequality (2.9) we obtain the corresponding inequality for v at a maximum point, namely where C depends additionally on |ζ| 2;Ω , so that extending (2.10), we have The lower bounds follow from the concavity F2 since for a fixed matrix r 0 ∈ Γ, for example r 0 = I, and positive matrix a ij 0 = F r ij (r 0 ), we have (1.1). Taking τ to be an eigenvector of {a ij 0 }, we infer the full bound from the upper bound (2.13). Hence we conclude the estimate (2.15) sup Theorem 1.1 now follows by taking ζ ∈ C 2 0 (Ω 0 ) and ζ = 1 on Ω ′ .

9
The one-sided estimate (2.13) can be extended to non-constant vector fields τ when F is orthogonally invariant. Moreover the relevant calculations will be critical for us in the proof of tangential boundary estimates when G is nonlinear in p.
Lemma 2.1. Assume in addition to the hypotheses of Theorem 1.1 that the operator F is orthogonally invariant. Then the estimate (2.13) holds for any vector field τ with skew symmetric Jacobian, with the constant C depending additionally on |τ | 1;Ω .
Proof. First we note that the Jacobian Dτ = {τ i j } will in fact be a constant skew symmetric matrix so that τ itself is an affine mapping. Consequently the second derivatives of τ in (2.6) and (2.7) will vanish. Our main concern now is to control the third derivatives of u in the last line of (2.7) and for this we adapt the key identities in the proof of Lemma 2.1 in [9], which follow by differentiating F (P α rP t α ), for r ∈ Γ, with respect to α and setting α = 0, where P α = exp(αDτ ) is orthogonal by virtue of the skew symmetry of Dτ . Thus taking r ij = w ij , we have Differentiating (2.16) with respect to τ , we have From (2.17) and (2.18), we then obtain i τ l k w jl by the concavity F2. Substituting into (2.7), using the definition w ij = u ij − A ij and following our previous argument for constant τ , we would obtain the upper bound (2.13), with constant C replaced In order to get the full strength of Lemma 2.1, we need to control the last term in (2.19). Note that this term is nonnegative if Γ = Γ n or in the special case when |Dτ i | = τ 0 , i = 1, · · · , n, for a constant . In general we proceed by calculating Taking account of (2.6), (2.7), (2.19) and (2.20), we then obtain the differential inequality (2.12) with the function u τ τ in (2.11) replaced by the function from which Lemma 2.1 follows.
We remark that for the Monge-Ampère operator, in the form F (r) = log(det r), we can take τ to be any C 2 vector field in (2.13) with the constant C now depending additionally on |τ | 2;Ω . This follows from the identity F ik r kj = δ ij .

2.2.
Boundary second derivative estimates. To prove Theorem 1.2, we have to establish estimates for second derivatives on the boundary ∂Ω under the boundary condition (1.2). First we will consider the non-tangential estimates and as in [13], the geometric convexity hypotheses on the domain Ω in Theorem 1.2 are crucial for this stage. We assume that the functions G(·, z, p) and ν have been extended toΩ, to be constant along normals to ∂Ω in some neighbourhood N of ∂Ω. Differentiating the boundary condition (1.2) with respect to a tangential vector field τ we have and hence we have an estimate for any unit tangential vector field τ , where β = D p G(·, u, Du) and the constant C depends on G, Ω and |u| 1;Ω . The estimation of the pure second order oblique derivatives u ββ is much more complicated.
In general we can only obtain an estimate from above in terms of the tangential derivatives on the boundary. Setting we formulate this as follows.
3), either F5 + holds or B is independent of p and either G is quasilinear or F6 holds. Then for any ǫ > 0, where β = D p G(·, u, Du), and C ǫ is a constant depending on ǫ, F, A, B, G, Ω, β 0 and |u| 1;Ω .
For any function of the form g ∈ C 2 (Ω × R × R n ), by calculations we have , we obtain the differential inequality, where the constant C depends on n, Ω, |g| C 2 , |A| C 1 , |B| C 1 and |u| 1;Ω . Note that when a 0 is finite, (2.25) can be estimated directly from (1.10).
Proof of Lemma 2.2. For any fixed boundary point x 0 ∈ ∂Ω, we consider the function where G is the boundary function in (1.2), and a ≤ 1 is a positive constant. We consider the quasilinear case of G, (1.17), namely G pp = 0. We also consider the case when F5 + holds. By (2.27), Cauchy's inequality and (2.26), we have for any ǫ 1 > 0, where g is relaxed by G inβ ik , F5 + is used in the last inequality, the constants C depend on n, Ω, |G| C 2 , |A| C 1 , |B| C 1 and |u| 1;Ω , and the constant C ǫ 1 depends on ǫ 1 , F and B.
We shall construct a suitable upper barrier forv at the point x 0 . We employ a function of the form  (2.32). By the uniform (Γ, A, G)-convexity of Ω, since |u| and |Du| are bounded in Ω, there exists a small positive constant σ such that for all x ∈ ∂Ω satisfying G(x, u(x), Du(x)) ≥ 0. Reversing the projection onto the tangent space of ∂Ω, we then have for all x ∈ ∂Ω, for a larger constantμ 0 , which implies (h ij − σδ ij ) ∈ Γ for x ∈ ∂Ω provided t ≥μ 0 .
By choosing ρ sufficiently small and then t sufficiently large, we have ( The constants ρ and t are now fixed. Then, by the concavity F2, we have from (2.32), where (1.1) and (1.10) are used in the last inequality. By using F4, we have from (2.35) Thus, we obtain, from (2.29), where the mixed derivative estimate (2.22) and the strict obliqueness (1.3) are used in the second inequality, so the constant C depends also on β 0 . For x ∈ Ω ρ and x ′ the closest point on ∂Ω, we then obtain, We can fix the constant c so that where C now depends on F, A, B, G, Ω, β 0 and |u| 1;Ω , and C ǫ 1 depends additionally on ǫ 1 . For any ǫ > 0, taking a = 1 1+ǫ 1 M 2 and ǫ 1 = ǫ β 0 C for a further constant C in (2.45), we then get from (2.44) and (2.45). Since x 0 is any boundary point, we can take the supremum of (2.46) over ∂Ω to arrive at the desired estimate (2.23). Therefore, we have proved Lemma 2.2 in the case when G is quasilinear and F5 + holds. While in the case G is quasilinear and only F5 holds with B independent in p, the last term in the second line of (2.29) does not appear. So we still arrive at the same estimate (2.23) and Lemma 2.2 is thus proved in the quasilinear case.
Next, we turn to the case that F satisfies F6. Here we may simply take a = 0 in (2.28) and b = 0 in (2.30) so that from (2.27), F6 and Cauchy's inequality in Ω, for any ǫ 1 > 0, where ǫ 1 now comes from the use of both F5 + and F6, the constant C depends on n, Ω, |G| C 2 , |A| C 1 , |B| C 1 and |u| 1;Ω , and the constant C ǫ 1 depends also on ǫ 1 and F . We can then derive the desired estimate (2.23), without the dependence on M ′ 2 , for both F5 + and B independent of p.
It now remains to estimate the pure tangential derivatives on the boundary. In this part, the strictly regular condition on the matrix A is crucial. Actually, we can formulate the pure tangential derivative estimates as follows.
be an admissible solution of the boundary value problem (1.1)-(1.2) in a C 2,1 domain Ω ⊂ R n . Assume that F, A and B satisfy the hypothesis of Theorem 1.1 and G ∈ C 2 (∂Ω × R × R n ) is oblique and concave in p with respect to u satisfying (1.3), and either G is quasilinear or F is orthogonally invariant or F also satisfies condition F6. Then for any tangential vector field τ , |τ | ≤ 1 and constant ǫ > 0, we have the estimate where M + 2 (τ ) = sup ∂Ω u τ τ , and C ǫ is a constant depending on ǫ, F, A, B, G, Ω, β 0 and |u| 1;Ω .
Proof. As usual we extend ν and G smoothly to all ofΩ so that ν and G(·, z, p) are constant along normals to ∂Ω in some neighbourhood of ∂Ω. Suppose that the function takes a maximum over ∂Ω and tangential vectors τ , such that |τ | ≤ 1, at a point x 0 ∈ ∂Ω and vector τ = τ 0 , where c 1 is the constant in the strict regularity condition (2.3). Without loss of generality, we may assume x 0 = 0 and τ 0 = e 1 . Setting we then have, at any point in ∂Ω, where C is a constant depending on G, Ω, β 0 and |u| 1;Ω . Accordingly, there exists a further constant C 1 depending on the same quantities, such that the function, where f is any non-negative function in C 2 (Ω) satisfying f ≥ |τ | 2 on ∂Ω, f (0) = 1. In the case when G is quasilinear, that is β = β(·, u), we may simply estimate 2) twice in a tangential direction τ , with τ (0) = e 1 , we obtain using the concavity of G and (2.22), Consequently for a sufficiently large constant K depending on the same quantities as C 1 , the function must take an interior maximum in Ω, where φ ∈ C 2 (Ω) is a negative defining function for Ω satisfying This effectively reduces our argument to the proof of Theorem 1.1 and we obtain at a maximum point x 0 , using (2.3), where C depends additionally on A and K|φ| 2;Ω . The estimate (2.48) then follows by fixing φ and the constant C 1 in (2.54) so that φ ≥ − ǫ 4K and f ≥ 1 2 in Ω. Instead of adjusting φ we can alternatively maximize v in a sufficiently small strip Ω δ 0 around ∂Ω and apply the interior estimate in Theorem 1.1.
When G is nonlinear in p, the coefficient β 1 in the expansion (2.54) of |τ | 2 depends on Du and cannot be controlled by the argument above. For orthogonally invariant F, this is overcome by using a first order approximation to the tangent vector e 1 at 0. Fixing the x n coordinate in the direction of ν at 0, we then replace e 1 by the vector field Then in place of (2.50), we have so that, both b(0) = 0, δb(0) = 0. Accordingly we then have, in place of (2.52) and (2.53), where C 1 is again a constant depending on G, Ω, β 0 and |u| 1;Ω . Since the vector field ξ has skew symmetric Jacobian Dξ, we can then reduce to the argument of the proof of for any ǫ > 0 and provided |D 2 u| ≥ C ǫ . Here the constant C depends on Ω, |g| C 2 , |A| C 1 , |B| C 1 and |u| 1;Ω while C ǫ depends on ǫ, F and |u| 1;Ω . Now taking f = f (·, u, Du) satisfying f ≥ |τ | 2 on ∂Ω, we then obtain at the maximum point x 0 , corresponding to (2.9), for t ≤ β 0 /4 and C 1 is a large enough constant, depending on G, Ω, β 0 and |u| 1;Ω , to ensure f ≥ 1 2 . Also using F6 in conjunction with (2.3), we only need take c 1 = 0 in (2.49 where C is a constant depending on F, A, B, G, Ω, β 0 and |u| 1;Ω .

Gradient estimates
In this section, we prove various gradient estimates for admissible solutions u of the oblique problem (1.1)-(1.2). We mainly consider the case when the oblique boundary operator G is semilinear and in particular give the proof of Theorem 1.3. We also derive corresponding local gradient estimates as well as an estimate for nonlinear G. As mentioned in the introduction, our conditions on either the matrix A or the operator F enable an analogue of uniform ellipticity. Accordingly we will employ improvements of the methods for uniformly elliptic equations in [21] with a critical adjustment used to supplement the tangential gradient terms in [21], which is similar to that used for gradient estimates in the conformal geometry case in [16]. We also prove a Hölder estimate for Γ = Γ k for k > n/2, from which we infer gradient estimates under natural quadratic growth conditions. 3.1. Global gradient estimates. We consider the case of oblique semilinear G in (1.23) and normalise G by dividing by β · ν ≥ β 0 , so that we can write For convenience, we still use ϕ to denote its normalised form here. The boundary condition (1.2) can thus be written in the form We begin with a preliminary calculation to estimate Lg from below, for g given by is an admissible solution of (1.1). For this estimation we may assume more general growth conditions than (1.22), namely as |p| → ∞, uniformly for x ∈ Ω, |z| ≤ M for any M > 0. Now we calculate, using (2.24), and ω is a positive decreasing function on [0, ∞) tending to 0 at infinity, depending on A, B and M 0 = sup Ω |u|, and C is a constant depending on a kl , b k , c, Ω, A, B and M 0 . For our approach here we will assume {a kl } is positive definite so that {a kl } ≥ a 1 I for some positive constant a 1 . By Cauchy's inequality and the positivity of {a kl }, we then obtain In our proof of Theorem 1.3, we will specifically choose (3.10) where ν and G given by (3.1) are appropriately extended to all ofΩ. For our purposes a convenient extension will be as usual to take ν constant along normals to ∂Ω in a sufficiently small strip {d < ρ} where the distance function d is C 2 smooth and to extend β ′ ∈ C 2 (Ω) and ϕ ∈ C 2 (Ω × R) to vanish for d ≥ ρ so that g = |Du| 2 for d ≥ ρ. Using Cauchy's inequality, we can estimate here (3.9) holds with a 1 = (1 + β ′ 0 ) −2 and C depending on β ′ , ϕ, Ω, A, B and M 0 , with Ω ∈ C 3 . By further use of Cauchy's inequality, we also obtain so that the estimation of Du is equivalent to that of the function g.
With these preparations, we give the proofs of the global gradient estimates.
Proof of Theorem 1.3. We employ auxiliary functions of the form Then we have on ∂Ω, using (3.14) and the boundary condition (3.2). Next by tangential differentiation of (3.2), as in (2.21), we have Plugging (3.16) into (3.15), by Cauchy's inequality and the fact that |δu| ≤ |Du|, we then have at x 0 , We next consider the case that the maximum of v occurs at a point x 0 ∈ Ω. We now take the constant α sufficiently small and fix the defining function φ such that Taking (3.12) into account, these restrictions in (3.19) will enable us to proceed from estimating Du(x 0 ) to an estimate for M 1 . Note that these conditions ultimately depend on the independent choice of the constant K. Since Dv(x 0 ) = 0 and D 2 v(x 0 ) ≤ 0, we have where L is the linearized operator defined in (2.5). Our estimations then reduce to getting an appropriate lower bound for Lη and for this we separate cases (i) and (ii) in Theorem 1.3.

Case (i):
A uniformly regular.
Here we take the "+" sign in η, that is η = e K(u 0 −u) , and for convenience set u 0 = M + 0 = sup Ω u so that η ≥ 1. Our estimation of Lη is motivated by the barrier constructions in [10,12,14,15] for regular A, where the constant u 0 is replaced by an admissible functionū. In particular the reader is referred to Section 2 in [12] for the extension to general operators. First by Taylor's formula, we have wherep = θDu with θ ∈ (0, 1). Since A is uniformly regular, we can estimate by choosing K ≥λ 0 , where µ is a constant depending on A. At this point we introduce a more general condition than the concavity F2 which also includes the homogeneous case. Namely we assume for any constant a > a 0 , there exist non-negative constants µ 0 and µ 1 such that From (1.9), we see that F2 implies (3.24), with µ 0 = max{0, −F (µI)} and µ 1 = µ for any µ > 0. Note that when a 0 > −∞, then (3.24), with µ 0 = max{0, −a 0 } and µ 1 = 0, is immediate from (1.10). Using (3.24) in (3.23) we thus obtain where now µ depends on A, µ 0 and µ 1 .
Assuming now F5, with b = ∞, so that T ≥ δ 0 for B ≥ b 0 > a 0 and supplementing the growth conditions (3.4) by we then have from (3.25), with ω sufficiently small, provided |Du| ≥ C 1 for some sufficiently large constant C 1 , depending on F , A, B and M 0 . Combining On the other hand if F5 + is satisfied, with b = ∞, and "o" is weakened to "O" in (3.26), that is then we have from (3.25), and we arrive again at the inequality (3.28), with large enough K, by using the positivity of F r to estimate F ij u i u j ≥ δ 1 |Du| 2 for some positive constant δ 1 when |r| ≤ C and F (r) ≥ b 0 . Now since v is maximised at x 0 and ω can be made arbitrarily small for large enough |Du|, we must have |Du(x 0 )| ≤ C, noting that α and κ are now fixed by our choice of K above. From (3.12), (3.13) and our restrictions on α and φ, we obtain Case (ii): F7 holds, β = ν.
We take the "−" sign in η so that η = e K(u−u 0 ) and set u 0 = M − 0 = inf Ω u. Assuming growth conditions, as |p| → ∞, uniformly for x ∈ Ω, |z| ≤ M for any M > 0, we then have from (3.6), in place of (3.23), where µ is a constant depending on M 0 . At this stage, anticipating our use of F7, we can fix the constant K so that K ≥ 2n(1 + 2µ)/ min{δ 0 , δ 1 }. In order to get a lower bound for Lη, similar to (3.27), at the point x 0 ∈ Ω where v, given by (3.13), is maximised, we also need to impose our key restrictions in the hypotheses of case (ii) of Theorem 1.3, namely F is orthogonally invariant, β = ν, that is β ′ = 0, and

21
From now on, all the calculations will be made at the maximum point x 0 . Since Dv(x 0 ) = 0, we have On the other hand, we have for i = 1, · · · , n, whereD x = D x + DuD z and in accordance with our extension of G to Ω, ϕ = 0 for d ≥ ρ. Combining (3.36), (3.37) and (3.34), we have for i = 1, · · · , n. Without loss of generality, we can further choose our coordinates so that at x 0 . From our choice of K we then obtain, from (3.32) and (3.41), at x 0 . Note that if F ij w ij ≥ 0, as in (1.10), then we can absorb the term F ij u ij in the last term in (3.32) so that it is not present in (3.42). Furthermore if F is positive homogeneous of degree one, we can replace p · B p in (3.31) by p · B p − B.
Assuming also the growth conditions (3.4), we now combine (3.42) with (3.9), with a 1 = 1, and in general, (when F ij w ij may be unbounded from below), use Cauchy's inequality to control the term F ij u ij . Accordingly we obtain, at the maximum point x 0 , provided, taking account of (3.12) and (3.13), we further restrict α so that αKη ≤ 1 16 . As in case (i), since ω can be made arbitrarily small for large enough |Du| and, α and κ are fixed by our choice of K above, we obtain an estimate |Du(x 0 )| ≤ C and hence as |p| → ∞, uniformly for x ∈ Ω, |z| ≤ M for any M > 0, while if also ϕ = 0 so that g = |Du| 2 , we need only assume, in place of (3.4), Using our barrier constructions in Section 2 of [12] in the proof of case (i') also enables some alternative conditions to uniform regularity which would include the case when A is independent of p. for nonlinear G, when G is uniformly concave in p, that is for all x ∈ ∂Ω, |z| ≤ M , p ∈ R n , any unit vector ξ, and some positive constant σ 0 , depending on the constant M . By virtue of the global bound (3.46), we only need to estimate Du on ∂Ω. Using Taylor's expansion, with θ ∈ (0, 1), we have on ∂Ω, which leads to can assume more generally Replacing ν by β in (3.37), we still obtain w 11 (x 0 ) < 0, if |Du(x 0 )| ≥ C 1 , under condition (3.51), where now C 1 depends also on sup ∂Ω |β − ν|.

Local gradient estimates.
In this subsection we prove local and interior versions of Theorem 1.3 and unlike the global gradient estimates in the previous section we will need the full strength of the growth conditions in (1.22) with respect to the p variables. The local estimates will also provide us with estimates in terms of moduli of continuity of solutions under weaker growth conditions analogous to the uniformly elliptic case in [21]. For the latter estimates we also need to assume in case (ii) a complementary condition to (3.24), namely that there exist non-negative constants µ 0 , µ 1 and µ 2 such that for any r ∈ Γ, Clearly (3.52) is satisfied trivially for positive homogeneous F or if F2 and either a 0 > −∞ or F4 are satisfied by (1.10).
We summarise the results in the following theorem, where for convenience we use balls rather than the domains Ω 0 and Ω ′ in Theorem 1.1. Then for any y ∈Ω, 0 < R < 1 and ball B R = B R (y), we have the estimate Cauchy's inequality, we now obtain in place of (3.9), (3.55) in Ω ∩ S, where ω = ω(|Du|) is a positive decreasing function on [0, ∞) tending to 0 at infinity. With g defined by (3.10), we consider now in place of (3.13), auxiliary functions of the form for some constant c, and at the maximum point x 0 of v in Ω ∩ S, provided ζ(x 0 )|Du(x 0 )| > 1/R and |Du(x 0 )| > 1. For the estimate (3.54), we need to take R sufficiently small so that there exists a cut-off function ζ ∈ C 0 (B R ) ∩ C 2 (S ζ ), 0 ≤ ζ ≤ 1 satisfying (3.58) together with the boundary condition We show how to construct such a function ζ from the function (3.57) at the end of the proof.
From the property (3.60), we now obtain in place of (3.15), ).
Note that in case (ii), we cannot satisfy the further restriction αKη ≤ 1 16 , for large α, so here we use condition (3.52) to control the term F ij u ij in (3.42). We remark that when A is regular such a control can be alternatively achieved through a barrier [12].
To end the proof of Theorem 3.1, we give the key construction of the cut-off function at boundary.
Construction of cut-off function at boundary. We fix a point y ∈ ∂Ω, which we may take to be the origin, and a coordinate system so that ν(0) = e n . Suppose that in some ball ρ} and x ′ = (x 1 , · · · , x n−1 ). By taking ρ sufficiently small, we can assume, for any fixed δ > 0, since β n (0) = 1, Dh(0) = 0. Now we consider a coordinate transformation Again with ρ sufficiently small, we have ψ(0) = 0, ψ(∂Ω ∩ B ρ ) = {x n = 0}, Dψ(0) = I and det Dψ > 0, we obtain by calculation With the help of (3.66), a boundary cut-off function ζ ∈ C 0 (B R ) ∩ C 2 (S ζ ), 0 ≤ ζ ≤ 1 satisfying (3.58) and (3.60) can be constructed. For a fixed point y ∈ ∂Ω, which we may take to be the origin, we make the coordinate transformation x →x = ψ(x) as in (3.65). In thex-coordinate system, we can choose the function then the function ζ =ζ • ψ is the desired cut-off function satisfying the above properties (3.58) and (3.60) as we expected.
Remark 3.4. Note that when β = ν, (3.10) is similar to the corresponding function used for the gradient estimate of Neumann problems in [16], (and more recently for the k-Hessian equations in [25]). In our proof, we use the auxiliary functions (3.13) and (3.56), which are modifications of the auxiliary functions used in Section 3 of [21] for uniformly elliptic equations and for interior gradient bounds for k-Hessian equations in [35]. We remark that we can use alternative functions; in particular functions of the form v = g exp (αη − κφ) and v = ζ 2 g exp (αη − κφ), with appropriately chosen positive constants α and κ, in place of (3.13) and (3.56) respectively. be equivalently formulated as follows, namely A(x, z, p) ≥ −µ 0 (1 + |p| 2 )I, for all x ∈ Ω, |z| ≤ M , p ∈ R n , and some positive constant µ 0 depending on the constant M . The onesided quadratic condition (3.72) has already been used for the gradient estimate in the Monge-Ampère case, for the Dirichlet problem in [14] and the Neumann and oblique problems in [13].
Proof. First we consider the interior case (i). For any ball B R = B R (y) ⊂ Ω, we need to compare the following two functions in B R , where n/2 < k ≤ n, K, a and c are positive constants to be determined. By direct calculation, we first observe that DenotingΦ = Φ + ǫ 2 |x − y| 2 for some positive constant ǫ < 1, then the perturbation functionΦ of Φ satisfies D 2Φ ∈ Γ k in B R . Fixing a constant ρ < min{R, 1}, it is readily checked that F k (Φ) is a strictly decreasing function with respect to |x − y| for x ∈ B R \B ρ , and hence where B ρ := B ρ (y), C(n, k, c) is a positive constant depending on n, k and c. By introducing We also denote For our desired comparison of the functions v in (3.74) and Φ in (3.75) on B R , we shall first compare them on B R \B ρ for a fixed ρ, and then send ρ to 0 in the end. For convenience of later discussion, we now introduce some notation and fix some constants in advance. We denote and (3.81) δ(n, k) = 1/(n − k + 1), n/2 < k < n, We can fix the constant K large such that K > µ 0 /δ(n, k), and fix the constant a such that a > , we have v − Φ ≤ 0 on ∂B R . Now the constant c has been fixed as well. For fixed ρ and c, by choosing ǫ sufficiently small such that the quantity on the right hand side of (3.77) is sufficiently small, we can have 29 whereK m is the constant defined in (3.80). If v − Φ attains its maximum over B R \B ρ at a point . Without loss of generality, by rotation of the coordinates, we can assume which has a negative eigenvalue Φ 11 (x 0 ) when n/2 < k < n and has a null eigenvalue Φ 11 (x 0 ) when k = n. Correspondingly, for small ǫ, the perturbed Hessian D 2Φ (x 0 ) = D 2 Φ(x 0 ) + ǫI is diagonal, and has a negative eigenvalueΦ 11 (x 0 ) when n/2 < k < n and has a least positive eigenvalueΦ 11 (x 0 ) when k = n. Note also that the matrices {F ij k } and {S ij k } are diagonal at x 0 . From the properties of the k-Hessian operator and the Monge-Ampère operator, we have where δ(n, k) is defined in (3.81).
The contradiction from (3.84) and (3.87) shows that v − Φ must take its maximum over B R \B ρ at the boundary ∂B R or ∂B ρ . Therefore, we have We observe that we can choose ǫ as small as we want in (3.87). Letting ǫ → 0 in (3.87), the inequality (3.88) can hold for ρ as small as we want. Sending ρ → 0 and using the forms of v in (3.74) and Φ in (3.75), the right hand side of (3.88) tends to 0. Correspondingly, we have from (3.88), and hence assertion (i) follows from the estimate, c ≤ R −αK M osc B R u.
Remark 3.8. In the above argument, we use the perturbationΦ = c|x − y| α + ǫ 2 |x − y| 2 of the function Φ = c|x − y| α . We remark that there are alternative perturbations that can be used here. For instance, we can choose a perturbation in the form,Φ = c(|x − y| 2 + ǫ 2 ) α 2 for small ǫ.
by using D νũ > 0 on ∂Ω, and choosing the constant c large such that c ≥ aR n k /α. Then (3.91) leads to a contradiction and excludes the case that the maximum of v − Φ occurs at (B R \B ρ ) ∩ ∂Ω. By fixing the defining function φ such that φ > −1/κ, we have κ(φ(x) − φ(y)) > −1 for x ∈ ∂B R ∩ Ω and y ∈ ∂Ω. With this property of the defining function, now by choosing c larger again such that Similarly to (3.88) of the interior case, with µ 0 and K appropriately adjusted, we now have Therefore, by successively passing ǫ and ρ to 0, the same inequality (3.90) holds onΩ R and hence assertion (ii) is proved.
Finally for Dirichlet boundary values, as in case (iii), we suppose again that B R intersects ∂Ω and for all x, y ∈ B R ∩ ∂Ω for some non-negative constant κ = [u] α;B R ∩∂Ω . Assume first that the centre y ∈ ∂Ω. Then proceeding as in the previous case we need to compare v and Φ on (B R \B ρ ) ∩ ∂Ω.
Accordingly we now obtain by taking c larger such that c ≥ κKe With K and c also chosen as in case (i), with B R replaced by Ω R , we arrive again at (3.90) onΩ R .
The general case, y ∈ Ω, in case (iii), now follows by combining the case, y ∈ ∂Ω with the interior estimate, (3.73) in case (i), as in Theorem 8.29 in [6].
For convex domains, Lemma 3.1 extends the gradient bound, Lemma 3.2 in [13], for the case k = n.
More generally it provides a modulus of continuity estimate for solutions of (1.1) that are admissible in Γ k for k > n/2. Combining with the local gradient estimate in Theorem 3.1, the estimate (3.53) can hold by extending "o" to "O" in (1.22) and (3.33). For a convex domain Ω, the estimate (3.54) can still hold for the semilinear Neumann problems in case (ii) of Theorem 3.1 by extending "o" to "O" in (1.22) and (3.33).

Existence and applications
In this section, we present some existence results for classical solutions based on our first and second derivative a priori estimates for admissible solutions for the oblique boundary value problem (1.1)-(1.2). We also give various examples of equations and boundary conditions satisfying our conditions and also show that our theory can be extended to embrace C 1,1 solutions of degenerate equations. 4.1. Existence theorems. We assume that u andū in C 2 (Ω) ∩ C 1 (Ω) are respectively an admissible subsolution and supersolution of the boundary value problem (1.1)-(1.2), satisfying the inequalities (1.25) and (1.26), with F satisfying F1 and G oblique. Under the assumptions A, B and G are nondecreasing in z, with at least one of them strictly increasing, if u ∈ C 2 (Ω) ∩ C 1 (Ω) is an admissible solution of the problem (1.1)-(1.2), by the comparison principle, we have For our purposes here we note that the comparison principle, as formulated in Lemma 3.1 in [13], extends automatically to operators F satisfying F1. Then (4.1) provides the solution bound and the interval I = (u,ū) for the convexity definition (1.16).
With the a priori estimates up to second order, we can formulate existence results for the classical admissible solutions of the oblique boundary value problems (1.1)-(1.2). We consider first the case when the matrix A is strictly regular and the boundary operator G is semilinear. Chapter 9]. The uniqueness readily follows from the comparison principle.
With the above existence result for general operators, the existence for semilinear oblique problem (1.1)-(1.2) of the k-Hessian and Hessian quotient equations, F = F k,l for 0 ≤ l < k ≤ n, Theorem 1.4, is just a special case. The conditions in cases (i), (ii) in Theorem 1.4 agree with those in (i), (ii) in Theorem 4.1, respectively. For case (iii) of Theorem 1.4, the gradient estimate follows from Lemma 3.2 in [13], while second derivative estimate is from Theorem 1.2.
Using Lemma 4.1 in [13] and the nonlinear case in Theorem 1.2, we can extend Theorem 4.1 to cover nonlinear boundary operators in the case where Γ is the positive cone Γ n . For this we also need to assume that G is uniformly oblique in the sense that for all x ∈ Ω, |z| ≤ M , p ∈ R n and positive constants β 0 and σ 0 , depending on the constant M . The following existence result, which is proved similarly to Theorem 4.

33
G ∈ C 2,1 (∂Ω×R×R n ) is concave with respect to p and uniformly oblique in the sense of (4.3), u andū,  (as in [1], [42] or [19]), instead of the method of continuity, provided we have appropriate a priori bounds for solutions. Note that the monotonicity conditions themselves may be relaxed somewhat to get the inequality (4.1). In particular we can strengthen the sub and super solution properties of u andū so that D 2 u ≥ A(x, z, Du(x)) and in Ω, whenever z < u(x), with one of the inequalities in (4.4) strict, and in Ω, G(x, z, Dū(x)) ≤ 0, on ∂Ω, whenever z >ū(x), D 2ū ≥ A(x, z, Dū(x)), with one of (4.5) strict. As a special case, if u = −K, u = K for some constant K, we get |u| ≤ K; (see also [13], Section 3). Under this more general hypothesis we can infer the existence of admissible solutions in Theorems 4.1 and 4.2 provided either A or B or ϕ is strictly increasing in z. The latter condition is needed to avoid compatibility issues, such as mass balance in optimal transportation. Note that the same remarks are also pertinent for the existence theorem, Theorem 1.4.

4.2.
Examples. In this subsection, we present various examples of operators F, matrix functions A and associated oblique boundary operators G which satisfy our hypotheses.
Examples for F. As already indicated in Section 1, our main examples are the k -Hessian operators and their quotients F k,l (0 ≤ l < k ≤ n), as considered in Theorem 1.4. For 0 ≤ l < k ≤ n, F k,l satisfy F1-F5, F7 in Γ k with a 0 = 0. We remark that b in F5 can be a positive constant or +∞. For l = 0, the corresponding k-Hessian operators F k , 2 ≤ k ≤ n satisfy F1-F4, F5 + and F7 in Γ k with a 0 = 0. This time, the operators F k only satisfy F5 + for finite b but not infinite b, while they still satisfy F5 for both finite b and infinite b. Note that the Monge-Ampère operator in the form (det) 1 n is also covered by F k when k = n. Another well known concave form of the Monge-Ampère operator is log(det), which satisfies F1-F4, F5 + in K + with a 0 = −∞. As stated in the introduction, the k-Hessian operators F k (k = 1, · · · , n) and the Hessian quotients F n,l (1 ≤ l ≤ n − 1) satisfy F6 in the positive cone K + . If F is an operator satisfying F1, F2, F3, F5 + , with finite a 0 , then F6 holds in the positive cone K + , since (4.6) where the property of K + is used in the first inequality, (1.10) is used in the second inequality, equation (1.1), finite a 0 and F5 + are used in the last two lines. Such a property was observed by Urbas [42,45] for orthogonally invariant F. Note that the property (4.6) here holds for non-orthogonally invariant F as well.
Instead of the elementary symmetric functions S k , we may also consider functions P k , which are products of k sums of eigenvalues, namely (4.7) P k [r] := P k (λ(r)) = λ is (r), k = 1, · · · , n, defined in the cones where i 1 , · · · , i k ⊂ {1, · · · , n}, λ(r) = (λ 1 (r), · · · , λ n (r)) denote the eigenvalues of the matrix r ∈ P k . In differential geometry, there is a large amount of literature dealing with k-convex hypersurfaces, where the notion k-convexity of a hypersurface, originating from [29,30], is that the sum of any k-principal curvatures at each point is positive. Clearly the associated operators in (4.7) interpolate between the Laplacian, k = n, and the Monge-Ampère operator k = 1. We then obtain another group of examples satisfying the hypotheses of Theorem 1.2, namely (4.9) log(P k [r]) = log[ For 1 ≤ k ≤ n, the functions log(P k ) satisfy F1-F5 in P k with a 0 = −∞. For 1 ≤ k ≤ n − 1, the functions log(P k ) also satisfy F5 + in P k and F6 in the positive cone K + . Moreover, the normalised functionsF k = (P k ) 1 C k n , 1 ≤ k ≤ n, are homogeneous of degree one and satisfy F1, F3-F5 in P k with a 0 = 0, where C k n = n! k!(n−k)! for 1 ≤ k ≤ n. Note that the associated operators also interpolate between the Laplacian F 1 and the normalised Monge-Ampère operator F n . Furthermore from Theorems 1.2 and 1.3, it follows that we can substituteF k for F k and P k for Γ k in case (i) of Theorem 1.4. In the next subsection we will also introduce degenerate versions of these operators.
We also have further examples originating from geometric applications, given by functions, also defined in the cone P k for k = 1, · · · , n. When α = k = 1, F k,−α coincides with the Hessian quotient F n,n−1 and if κ = (κ 1 , · · · , κ n ) denotes the principal curvatures of a hypersurface in R n+1 , then F 1,−1 [κ] is its harmonic curvature while F 1,−2 [κ] is the inverse of the length of the second fundamental form; see [5]. The associated operators are homogeneous and satisfy F1-F5 in P k with a 0 = 0 and either finite or infinite b in F5. where Q i , i = 1, · · · , m are nonsingular matrices and m > 1 is a finite integer. We can define an operator of the form (4.11) F k,V [r] = min Q∈V F k (QrQ −1 ), for k = 1, · · · , n, in the cone Γ k,V = {r ∈ S n | F j (QrQ −1 ) > 0, ∀Q ∈ V, j = 1, · · · , k}. Then the operator in (4.11) provides an example, which is non-orthogonally invariant, but still satisfies our assumptions F1-F5 + and F7. Note that since F k,V is a concave function in r, it has first and second order derivatives almost everywhere in Γ k,V so that the differential inequalities in (1.4), (1.5), as well as condition F5 + , hold in this sense. We can also consider the case of infinite V and replace F k by other functions. The resulting Bellman type augmented Hessian operators can then treated by smooth approximation as in the k = 1 case, (see for example [6]); and we would obtain the existence of C 2,α (Ω) solutions, for some α > 0 in Theorems 1.4 and 4.1. More generally if we drop the smoothness condition F ∈ C 2 (Γ), then we still obtain existence of C 2,α (Ω) solutions, for some α > 0, in Theorems 1.4, 4.1 and Remark 4.1. Here we need the more general C 2,α (Ω) estimate for concave fully nonlinear uniformly elliptic equations from [31].
Examples for A. Examples of strictly regular matrix functions arising in optimal transportation and geometric optics can be found for example in [27,40,38,10,24]. Typically there is not a natural association with oblique boundary operators, except for those coming from the second boundary value problem to prescribe the images of the associated mappings, so that second derivative estimates may depend on gradient restrictions in accordance with Remark 1.2. Moreover the relevant equations typically involve constraints so that we are also in the situation of Remark 1.3. Both these situations will be further examined in ensuing work. However we will give some examples satisfying our hypotheses, where oblique boundary operators arise naturally through our domain convexity conditions.
Conformal geometry. The application to conformal geometry concerns the special case a ij = δ ij , a 0 = 1 in (4.12), that is with the associated semilinear Neumann condition, and is related to the fully nonlinear Yamabe problem with boundary, where Ag = e 2u M [u] is the Schouten tensor of the conformal deformationg = e −2u g 0 and g 0 denotes the standard metric on R n . If F is positive homogenous of degree one, satisfying F3 with a 0 = 0,φ is a positive function on Ω and h a function on ∂Ω, then the problem of finding a conformal metricg on Ω such that F (Ag) =φ, with mean curvatureh on ∂Ω, is equivalent to solving the semilinear Neumann problem, where h 0 denotes the mean curvature of ∂Ω with respect to g 0 , [16]. With Ω,φ andh sufficiently smooth, (4.18) satisfies the hypotheses of the second derivative estimate, Theorem 1.2, if F also satisfies F1 and F2 and Ω satisfies (4.15) with ϕ =he −u − h 0 . Note that our restriction r ≤ trace(r)I on Γ implies thath > 0. However (4.15) does provide some relaxation of the umbilic condition in [16] and related papers, possibly depending on solution upper bounds, and can be extended to more general Riemannian manifolds with boundary. In particular for the cones, Γ 2 and P n−1 , the convexity condition (4.15) is equivalent toh > 0, since δ · ν = (1 − n)h 0 so no geometric conditions are needed. Note that for the local gradient bound, Theorem 3.1, we only need F to satisfy F1 to fulfil the hypotheses of case(i) (and no geometric restrictions on Ω).
Optimal transportation and geometric optics. In optimal transportation problems, the matrix A is generated by a cost function c ∈ C 2 (D), where D is a domain in R n × R n , through the relation (4.19) A(x, p) = c xx (x, Y (x, p)), where the mapping Y ∈ C 1 (U ), for some domain U ∈ R n × R n , is given as the unique solution of (4.20) Here we assume conditions A1, A2 as in [27,40] to guarantee the unique solvability of Y from (4.20).
The strict regularity was introduced as condition A3 in [27]; (see also [36]). More generally the matrices A arise from prescribed Jacobian equations [37] where now the mapping Y ∈ C 1 (U ) is given for a domain U ∈ R n × R × R n satisfying detY p = 0 in U and the matrix A is given by . Mappings Y in geometric optics can also be unified through a notion of generating function [38], which extends that of a cost function to permit the z dependence in Y and provides symmetric matrices in (4.21). For further information and particular examples of strictly regular matrices A the reader is referred to [27,40,38,10,24] and the references therein. As mentioned above in most of these examples there are not natural relationships with semilinear oblique boundary operators so that the situation in Remarks 1.2 and 1.3 is applicable. The natural boundary condition is the prescription of the image Ω * of the mapping T := Y (·, u, Du) on Ω, which implies a boundary condition which is oblique with respect to admissible functions, [40,37]. Once the obliqueness is estimated we are in the situation of Theorem 1.2 and moreover our domain convexity conditions there originate from those used in the optimal transportation and more generally; (see [40,37,24]).
Accordingly we just mention here some examples which fit simply with our hypotheses. First the logarithm cost function, given by c(x, y) = 1 2 log |x − y| for x = y, also generates our example (4.16), [40]. From geometric optics we have the example coming from the reflection of a parallel beam to a flat target, [38,24], (4.22) A(x, z, p) = 1 2z (|p| 2 − 1)I for z > 0. Here there is a constraint, namely u > 0, which is readily handled by taking a logarithm or assuming the subsolution u > 0 inΩ. Then for a semilinear Neumann boundary condition of the form we obtain again that Ω is uniformly (Γ, A, G)-convex with respect to u if and only if (4.15) holds.
Admissible functions. Quadratic functions of the form u 0 = c 0 + 1 2 ǫ|x − x 0 | 2 , will be admissible for the matrices (4.12) and for arbitrary constants c 0 , points x 0 ∈ Ω and sufficiently small ǫ. In general for matrices A arising in optimal transportation and geometric optics the existence of admissible functions is proved in [10].
Nonlinear boundary operators. The capillarity type operators, given by would satisfy our hypotheses for 0 < θ < 1 on ∂Ω. Furthermore for A in the form (4.12) with {a ij } = I, condition (4.15) would at least imply that that Ω is uniformly (Γ, A, G)-convex with respect to u. Note that here and quite generally we cannot have ϕ(·, u) ≥ 0 everywhere on ∂Ω for an admissible function so the basic capillarity condition is ruled out by our concavity condition which requires θ > 0.

Degenerate equations.
In this subsection, we consider the extension of our results to degenerate elliptic equations and in particular apply the classical existence results, Theorem 4.1 and 4.2, to yield the existence of C 1,1 admissible solutions for the oblique boundary value problems. We shall use the following assumption, in place of F1, to describe the degenerate ellipticity: Then using an elliptic regularisation as in [33], we define for a constant ǫ ≥ 0, F 1 (r) = trace(r), approximating operators and cones, (4.26) F ǫ (r) = F (r + ǫF 1 (r)I), Γ ǫ = {r ∈ S n | r + ǫF 1 (r)I ∈ Γ}.
Clearly F ǫ satisfies the ellipticity condition F1 in the cone Γ ǫ , for ǫ > 0 and is also uniformly elliptic there with (4.27) ǫT (r)I ≤ F ǫ r ≤ (1 + ǫ)T (r)I. Moreover if F also satisfies any of conditions F2 to F7, then F ǫ satisfies the same condition in Γ ǫ with relevant constants independent of ǫ, as ǫ tends to 0. Consequently we may replace F by F ǫ and the operator F by F ǫ , for sufficiently small ǫ ≥ 0 in our Hessian and gradient estimates in Sections 2 and 3. To get the lower second derivative bounds in Theorem 1.1 we can simply use T ǫ (r) := trace(F ǫ r ) > 0 in Γ ǫ , while for the lower tangential bounds in Theorem 1.  Proof. We claim thatū and u are respectively supersolution and admissible subsolution of the boundary value problem (1.1)-(1.2) for F replaced by F ǫ for sufficiently small ǫ, depending onū and δ. To prove this we first define the sets where a is constant satisfying a 0 < a < B(·,ū, Dū) in Ω. Then K ǫ is a decreasing family of compact subsets ofΩ approaching K 0 as ǫ approaches zero. Consequently K ǫ ⊂ Ω ′ 0 for sufficiently small ǫ. By the concavity F2, we then have, in K ǫ , for sufficiently small ǫ depending onū and δ. Clearly F ǫ [ū] ≤ B(·,ū, Dū) in Ω ′ ǫ − K ǫ so thatū is a supersolution of the equation, F ǫ = B, for sufficiently small ǫ. Next it follows immediately from the degenerate ellipticity F1 − , that u is an admissible subsolution, for any ǫ ≥ 0 so that our claim is proved.
From the stability property of the theory of viscosity solutions [4], it is readily seen that u ∈ C 1,1 (Ω) is an admissible solution of the problem (1.1)-(1.2).
To illustrate the application of Corollary 4.1, we consider the degenerate elliptic operators m k , given by functions (4.31) m k (r) = min{ k s=1 λ is (r)}, for k = 1, · · · , n, i 1 , · · · , i k ⊂ {1, · · · , n}, in the cones P k introduced in (4.8). As for the examples (4.11), the functions m k for k < n are not C 2 but will still satisfy conditions F1 − , F2, F3, F4, F5 and F7, with a 0 = 0, in P k almost everywhere. As well m k is positive homogeneous of degree one. The operators m k are also related to our examples (4.10) since m k = F k,∞ = lim α→∞ F k,−α . More explicitly the functions F k,−α are monotone decreasing in α and satisfy the inequalities m k < F k,−α < (C k n ) − 1 α m k in P k . We also note that when k = n, m n is the Poisson operator F 1 .
Assume also that A, B and ϕ are non-decreasing in z, with at least one of them strictly increasing, A satisfies the quadratic growth conditions (1.22) and B is independent of p. Then if one of the following further conditions is satisfied: Furthermore the operator m 1 satisfies condition F6 in S n since, from its orthogonal invariance, we have  , Ω a C 3,1 bounded domain in R n , A ∈ C 2 (Ω × R × R n ) strictly regular inΩ, B > 0, ∈ C 2 (Ω × R × R n ), G ∈ C 2,1 (∂Ω × R × R n ) is concave with respect to p and uniformly oblique in the sense of (4.3). Assume that u andū, ∈ C 1,1 (Ω) ∩ C 1 (Ω) are respectively an admissible subsolution of (1.1)-(1.2) and supersolution of (1. We remark that we may also prove Corollaries 4.2 and 4.3 directly from Theorem 4.1 by approximating m k by F k,−α for large α. Also the solutions in Corollaries 4.1, 4.2 and 4.3 will be unique if either A or B are strictly increasing and more generally under appropriate barrier conditions, as considered in Section 2 of [12]. 4.4. Final remarks. The oblique boundary value problem (1.1)-(1.2) for augmented Hessian equations is natural in the classical theory of fully nonlinear elliptic equations. In this paper and its sequel [12], we have treated this problem in a very general setting. Through a priori estimates, we have established the classical existence theorems under appropriate domain convexity hypotheses for both (i) strictly regular A and semilinear or concave G, and (ii) regular A and uniformly concave G. Our emphasis of this paper is the case (i), since the case (ii) is already known in the context of the second boundary value problems of Monge-Ampère equations [44,45] and optimal transportation equations [40,46]. In case (i), the boundary conditions can be any oblique conditions, including the special case of the Neumann problem, while the operators embrace a large class including the Monge-Ampère operator, k-Hessian operators and their quotients, as well as degenerate and non-orthogonally invariant operators.
In part II [12] we treat the case of regular matrices A which includes the basic Hessian equation case, where A = 0 or more generally where A is independent of the gradient variables. A fundamental tool here is the extension of our barrier constructions for Monge-Ampère operators in [14,10] to general operators; (see Remarks in Section 2 of [13]). In general as indicated by the Pogorelov example, [47,42], we cannot expect second derivative estimates for arbitrary linear oblique boundary conditions and moreover the strict regularity of A is critical for our second derivative estimates in Section 2. We remark though that our methods in this paper also show that the strict regularity can be replaced by not so natural, strong monotonicity conditions with respect to the solution variable on either the matrix A or the boundary function ϕ, that is either A z or ϕ z is sufficiently large, and the latter would include the case when A = 0, in agreement with the Monge-Ampère case in [41,47,42]. For Monge-Ampère type operators, we are able to derive the second derivative bound for semilinear Neumann boundary value problem when A is just regular, under additional assumption of the existence for a supersolution u satisfying det(M [ū]) ≤ B(·,ū, Dū) in Ω and D νū = ϕ(·,ū) on ∂Ω, see Jiang et al. [13]. This is an extension of the fundamental result in [23] for the standard Monge-Ampère operator. For the semilinear oblique problem for standard k-Hessian equations, the known results due to Trudinger [32] and Urbas [42], where the second derivative estimates for Neumann problem in balls, and for oblique problem in general domains in dimension two respectively were studied. Recently the Neumann problem for the standard k-Hessian equation has been studied in uniformly convex domains in [26]. However, it is not clear whether there are the corresponding second derivative estimates for admissible solutions of the Neumann problem of the k-Hessian equations in uniformly (k − 1)-convex domains. Also, the second derivative estimates for admissible solutions of the Neumann problem of the augmented k-Hessian equations with regular A in uniformly (Γ k , A, G)-convex domains is still an open problem.
In Section 3, we have established the gradient estimate for augmented Hessian equations in the cones Γ k when k > n/2 under structure conditions for A and B corresponding to the natural conditions of Ladyzhenskaya and Ural'tseva for quasilinear elliptic equations [18,6]. The gradient estimate under natural conditions is also known for k = 1, in [6]. Therefore, it is reasonable to expect gradient estimates, (interior and global), for both oblique and Dirichlet boundary value problems under natural conditions for the remaining cases for operators in the cones Γ k when 2 ≤ k ≤ n/2. An interesting question is whether it can be proved for the basic Hessian operators F k when 2 ≤ k ≤ n/2, which also enjoy L p gradient estimates for p < nk/(n − k), [39]. In [11], we apply our gradient estimates here and general barrier constructions in Section 2 of [12] to study the classical Dirichlet problem for general augmented Hessian equations with only regular matrix functions A. Here as well as our conditions on F in case (ii) of Theorem 1.1, for global second derivative estimates we also need to assume orthogonal invariance and the existence of an appropriate subsolution, as in our previous papers [14,15]. Our barrier constructions in [12] also permit some relaxation of the conditions on F in the regular case, as already indicated in Remark 3.2.
As pointed out in Remark 1.2, our domain convexity conditions require some relationship between the matrix A and the boundary operator G. If we drop these from our hypotheses, we can still infer the existence of classical solutions of the equation (1.1) which are globally Lipschitz continuous and satisfy the boundary condition (1.2) in a weak viscosity sense [4] so that our domain convexity conditions become conditions for boundary regularity. This situation will be further amplified in a future paper, along with examination of the sharpness of our convexity conditions. A preliminary result here for the conformal geometry application is given in [20].