Oblique boundary value problems for augmented Hessian equations I
Abstract
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–Ampère type operators in optimal transportation and geometric optics, the general theory here embraces Neumann problems arising from prescribed mean curvature problems in conformal geometry as well as general oblique boundary value problems for augmented kHessian, Hessian quotient equations and certain degenerate equations.
Keywords
Oblique boundary value problems Augmented Hessian equations Second derivative estimates Gradient estimatesMathematics Subject Classification
35J60 35J251 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 [31, 40, 45] 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 et al. [27]. Not only was the approach in [27] special for the Neumann problem, but it follows from the fundamental example of Pogorelov [32] that the result cannot be extended to general linear oblique boundary value problems [47, 52]. On the other hand, the classical Dirichlet problem for basic Hessian equations has been well studied in the wake of fundamental papers by Caffarelli et al. [1] and Ivochkina [9], with further key developments by several authors, including Krylov [20] and related papers and Trudinger [38]; (see also [8] 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, where remarkably our adaptation of optimal transportation domain convexity from [40, 45] enables us to replace the rather strong umbilic boundary condition for second derivative bounds, assumed in previous work [3, 19], by more general natural convexity conditions, (in the prescribed positive mean curvature case). 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 [14]. 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 [16] as a special example.
 F1
 F is strictly increasing in \(\Gamma \), namely$$\begin{aligned} F_r := F_{r_{ij}} = \left\{ \frac{\partial F}{\partial r_{ij}} \right\} >0, \ \mathrm{in} \ \Gamma . \end{aligned}$$(1.4)
 F2
 F is concave in \(\Gamma \), namelyfor all symmetric matrices \(\{\eta _{ij}\}\in {\mathbb {S}}^n\).$$\begin{aligned} \frac{\partial ^2 F}{\partial r_{ij}\partial r_{kl}} \eta _{ij} \eta _{kl} \le 0, \ \mathrm{in} \ \Gamma , \end{aligned}$$(1.5)
 F3
 \(F(\Gamma )=(a_0, \infty )\) for a constant \(a_0\ge \infty \) with$$\begin{aligned} \sup _{r_0\in \partial \Gamma }\limsup _{r\rightarrow r_0} F(r) \le a_0. \end{aligned}$$(1.6)
 F4

\(F(tr)\rightarrow \infty \) as \(t\rightarrow \infty \), for all \(r\in \Gamma \).
 F5

For given constants a, b satisfying \(a_0<a<b\), there exists a constant \(\delta _0>0\) such that \({\mathscr {T}}(r): =\mathrm{{trace}}(F_r)\ge \delta _0\) if \(a< F(r)< b\).
 \(\mathbf{F5}^{+}\)

\({\mathscr {T}}(r) \rightarrow \infty \) uniformly for \(a \le F(r) \le b\) as \(r\rightarrow \infty \).
As for (1.10) and (1.11), the condition F4, (and the endpoint \(\infty \) in F3), is typically more than we need in general and can be dispensed with in most of our estimates. When considering the Eq. (1.1), it will be enough to assume instead \(F(tr) > B(\cdot ,u,Du)\) for \(r\in \Gamma \) and sufficiently large t, (depending on r), in accordance with [10].
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 [31] and whose global version is needed for our treatment of the boundary condition (1.2).
Theorem 1.1
 (i)
F1, F2, F3 and \(\hbox {F5}^{+}\) hold;
 (ii)
F1, F2, F3, F5 hold, and B is convex with respect to p.
 F6

\({\mathcal {E}}_2: = F_{r_{ij}} r_{ik}r_{jk} \le o( r){\mathscr {T}}\), uniformly for \(a_0 <a \le F(r) \le b\), as \(r\rightarrow \infty .\)
Theorem 1.2
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 \(A\equiv 0\), which is a special case of regular A but not strictly regular, see [47, 52]. We also remark that the alternative condition that B is independent of p may be replaced by \(D_pB\) sufficiently small, as well as B convex with respect to p, and we will see from our treatment in Sect. 2 that such a condition is reasonable. Analogously, we may also replace the condition that G is quasilinear by \(D^2_pG\) sufficiently small.
Remark 1.2
Remark 1.3
We may assume more generally that the matrix function A and scalar function B are only defined and \(C^2\) smooth on some open set \({\mathcal {U}} \subset {\mathbb {R}}^n \times {\mathbb {R}} \times {\mathbb {R}}^n\), with A strictly regular and \(B > a_0\) in \({\mathcal {U}}\). Then Theorems 1.1 and 1.2 will continue to hold provided the one jet \(J_1 =J_1[u](\Omega ) = (\cdot , u, Du)(\Omega )\) is strictly contained in \({\mathcal {U}}\), with the constants C in the estimates (1.14) and (1.18) depending additionally on dist\((J_1, \partial {\mathcal {U}})\). This remark is particularly pertinent to examples arising from optimal transportation or geometric optics where often the resultant Monge–Ampère type equations are subject to constraints on \(J_1[u]\) and moreover such constraints may determine an appropriate constant \(\mu _0\) in (1.19).
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 quadratic structure conditions for gradient estimates for uniformly elliptic quasilinear equations, as originally introduced by Ladyzhenskaya and Ural’tseva [7, 21]. 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 \(\delta _0, \delta _1>0\) such that \(F_{r_{ij}}\xi _i\xi _j \ge \delta _0 + \delta _1 {\mathscr {T}}\), if \(a\le F(r)\) and \(\xi \) is a unit eigenvector of r corresponding to a negative eigenvalue.
Theorem 1.3
Theorem 1.4
This paper is organised as follows. In Sect. 2, we first prove the local/global second derivative estimate, Theorem 1.1, as well as an extension to nonconstant vector fields, Lemma 2.1. Then in Sect. 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 nontangential and tangential second derivatives. In the proof of Lemma 2.3 the strict regularity condition is crucial. In Sect. 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 Sect. 3.2, we prove the analogous local gradient estimates in Theorem 3.1. In Sect. 3.3 we derive a Hölder estimate for admissible functions in the cones \(\Gamma _k\) for \(k>n/2\), from which we can infer gradient estimates under natural quadratic growth conditions. In Sect. 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 Sect. 4.2 various examples of operators \({\mathcal {F}}\), matrices A, and boundary operators \({\mathcal {G}}\) 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, 19]. Furthermore we show in Sect. 4.3 that our theory can be applied to degenerate elliptic equations, where F is only assumed nondecreasing in F1, with resultant solutions \(u \in C^{1,1}({\bar{\Omega }})\); see Corollary 4.1, and provide a particular example in Corollaries 4.2 and 4.3. As indicated at the end of Section 4.3, these results also embrace degeneracies when \(b=a_0\). In Sect. 4.4, we conclude this paper with some final remarks which also foreshadow further results.
We also point out here that in our formulations and proofs we have generally assumed for simplicity that the functions A, B, G and domains \(\Omega \) are \(C^k\) smooth for appropriate k, although typically this can be replaced by \(C^{k1,1}\) smooth, as indicated in Remark 4.2. Since our main concern here is a priori estimates for classical solutions the reader may simply assume that all functions and domains are \(C^\infty \) smooth.
2 Second derivative estimates
2.1 Local/global second derivative estimates
The onesided estimate (2.13) can be extended to nonconstant vector fields \(\tau \) when \({\mathcal {F}}\) 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 \({\mathcal {F}}\) is orthogonally invariant. Then the estimate (2.13) holds for any vector field \(\tau \) with skew symmetric Jacobian, with the constant C depending additionally on \(\tau _{1;\Omega }\).
Proof
We remark that for the Monge–Ampère operator, in the form \(F(r) =\log (\det r)\), we can take \(\tau \) to be any \(C^2\) vector field in (2.13) with the constant C now depending additionally on \(\tau _{2;\Omega }\). This follows from the identity \(F^{ik}r_{kj} = \delta _{ij}\).
2.2 Boundary second derivative estimates
Lemma 2.2
Proof of Lemma 2.2
Remark 2.1
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. We can formulate the pure tangential derivative estimates as follows.
Lemma 2.3
Proof
With the local/global second derivative estimate in Theorem 1.1, and the boundary estimates in (2.22), Lemmas 2.2 and 2.3, we are now ready to prove the global second derivative estimate (1.18) in Theorem 1.2.
Proof of Theorem 1.2
Remark 2.2
3 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 \({\mathcal {G}}\) 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 \({\mathcal {G}}\). 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 [25] with a critical adjustment used to supplement the tangential gradient terms in [25], which is similar to that used for gradient estimates in the conformal geometry case in [19]. We also prove a Hölder estimate for \(\Gamma = \Gamma _k\) for \(k >n/2\), from which we infer gradient estimates under natural quadratic growth conditions.
3.1 Global gradient estimates
With these preparations, we give the proofs of the global gradient estimates.
Proof of Theorem 1.3
Case (i): A uniformly regular.
Case (ii): F7 holds, \(\beta =\nu \).
Remark 3.1
Remark 3.2
Remark 3.3
3.2 Local gradient estimates
We summarise the results in the following theorem, where for convenience we use balls rather than the domains \(\Omega _0\) and \(\Omega ^\prime \) in Theorem 1.1.
Theorem 3.1
Proof
To end the proof of Theorem 3.1, we give the key construction of the cutoff function at boundary.
Remark 3.4
Note that when \(\beta =\nu \), (3.10) is similar to the corresponding function used for the gradient estimate of Neumann problems in [19], (and more recently for the kHessian equations in [30]). In our proof, we use the auxiliary functions (3.13) and (3.58), which are modifications of the auxiliary functions used in Section 3 of [25] for uniformly elliptic equations and for interior gradient bounds for kHessian equations in [39]. We remark that we can use alternative functions; in particular functions of the form \(v = g \exp {(\alpha \eta  \kappa \phi )}\) and \(v=\zeta ^2 g \exp {(\alpha \eta  \kappa \phi )}\), with appropriately chosen positive constants \(\alpha \) and \(\kappa \), in place of (3.13) and (3.58) respectively.
Remark 3.5
Remark 3.6
Remark 3.7
3.3 Hölder estimates
Lemma 3.1
 (i)
\(B_R\subset \Omega \), \(u\in C^0({\bar{\Omega }}_R)\) and the constant C depends on \(n,k,\mu _0\), \(\mathop {\mathrm{osc}}\limits _{\Omega _R}u\) and diam \(\Omega \);
 (ii)
\(\Omega \in C^2\) is convex, \(u\in C^1({\bar{\Omega }}_R)\) and C depends additionally on \(\Omega \) and \(\inf \limits _{B_R\cap \partial \Omega } D_\nu u\);
 (iii)
\(u\in C^0({\bar{\Omega }}_R)\cap C^{0,\alpha }({\bar{B}}_R\cap \partial \Omega )\) and C depends additionally on \([u]_{\alpha ; B_R\cap \partial \Omega }\).
Proof
Remark 3.8
In the above argument, we use the perturbation \({{\tilde{\Phi }}}=cxy^\alpha + \frac{\epsilon }{2}xy^2\) of the function \(\Phi =cxy^\alpha \). We remark that there are alternative perturbations that can be used here. For instance, we can choose a perturbation in the form, \({{\tilde{\Phi }}}= c(xy^2+\epsilon ^2)^{\frac{\alpha }{2}}\) for small \(\epsilon \).
The general case, \(y\in \Omega \), in case (iii), now follows by combining the case, \(y\in \partial \Omega \) with the interior estimate, (3.76) in case (i), as in Theorem 8.29 in [7]. \(\square \)
For convex domains, Lemma 3.1 extends the gradient bound, Lemma 3.2 in [16], for the case \(k = n\). More generally it provides a modulus of continuity estimate for solutions of (1.1) that are admissible in \(\Gamma _k\) for \(k>n/2\). Combining with the local gradient estimate in Theorem 3.1, the estimate (3.55) can hold by extending “o” to “O” in (1.22) and (3.33). For a convex domain \(\Omega \), the estimate (3.56) 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).
4 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
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 \({\mathcal {G}}\) is semilinear.
Theorem 4.1
 (i)
A is uniformly regular, \({\mathcal {F}}\) satisfies F5, with \(b=\infty \) and \(B p\cdot D_pB\le o(p^2)\) in (1.22);
 (ii)
\(\beta =\nu \), \({\mathcal {F}}\) is orthogonally invariant and satisfies F7 and either (a) \(A = o(p^2)\) in (1.22) or (b) \(\Gamma \subset \Gamma _k\) with \(k>n/2\) and \(\Omega \) is convex,
 (iii)
Proof
With the above existence result for general operators, the existence for semilinear oblique problem (1.1)–(1.2) of the kHessian and Hessian quotient equations, \({\mathcal {F}}=F_{k,l}\) for \(0\le l<k\le n, k>1\) in 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 [16], while second derivative estimate is from Theorem 1.2. In the special case when \(k = 1\), equation (1.1) reduces to a quasilinear Poisson equation, as the matrix A can then be absorbed in the scalar B and considerably more general results for arbitrary smooth domains \(\Omega \) follow from the classical Schauder theory [7]. In particular we need only assume the quadratic growth, \(B= O(p^2)\) as \(p \rightarrow \infty \), uniformly for \(x\in \Omega \), \(z\le M\) for any \(M > 0\), and under reduced smoothness hypotheses, \(B\in C^{0,\alpha }({\bar{\Omega }}\times {\mathbb {R}}\times {\mathbb {R}}^n)\), \(\partial \Omega \in C^{1,\alpha }\), \(\beta \in C^{1,\alpha }(\partial \Omega )\), \(\varphi \in C^{1,\alpha }(\partial \Omega \times {\mathbb {R}})\), we infer the existence of a unique classical solution \(u\in C^{2,\alpha }({\bar{\Omega }})\) of the semilinear oblique problem (1.1)–(1.2).
Theorem 4.2
Assume that F satisfies conditions F1–F4 and F6 in the positive cone \(\Gamma _n\), \(\Omega \) is a bounded \(C^{3,1}\) domain in \({\mathbb {R}}^n\), \(A\in C^2({\bar{\Omega }}\times {\mathbb {R}}\times {\mathbb {R}}^n)\) is strictly regular in \({\bar{\Omega }}\), \(B > a_0, \in C^2({\bar{\Omega }}\times {\mathbb {R}}\times {\mathbb {R}}^n)\), \(G\in C^{2,1}(\partial \Omega \times {\mathbb {R}}\times {\mathbb {R}}^n)\) is concave with respect to p and uniformly oblique in the sense of (4.5), \({\underline{u}}\) and \({\bar{u}}\), \(\in C^2(\Omega )\cap C^1({\bar{\Omega }})\) are respectively an admissible subsolution and a supersolution of the oblique boundary value problem (1.1)–(1.2) with \(\Omega \) uniformly (A, G)convex with respect to the interval \({\mathcal {I}} = [{\underline{u}}, {\bar{u}}]\). Assume also that A, B and \(G\) are nondecreasing in z, with at least one of them strictly increasing and A satisfies the quadratic growth condition (3.75). Assume either \(\hbox {F5}^{+}\) holds or B is independent of p. Then there exists a unique admissible solution \(u \in C^{3,\alpha }({\bar{\Omega }})\) of the boundary value problem (1.1)–(1.2), for any \(\alpha <1\).
Remark 4.1
Remark 4.2
By further approximating G and \(\Omega \) in the proof of Theorems 4.1 and 4.2, we need only assume \(G\in C^{2}(\partial \Omega \times {\mathbb {R}}\times {\mathbb {R}}^n)\) and \(\partial \Omega \in C^{3}\), (in accordance with the smoothness hypotheses of Theorems 1.2 and 1.3), to obtain admissible solutions \(u \in C^{3,\alpha }( \Omega )\cap C^{2,\alpha }({\bar{\Omega }})\), for all \(\alpha < 1\).
We can also avoid approximating A and B in the proof of Theorems 4.1 and 4.2 by using the linear \(L^p\) regularity theory, ([7], Chapter 9), which implies, under our smoothness hypotheses on A and B, that a \(C^{2,\alpha }\) solution of (1.1)–(1.2) lies in the Sobolev space \(W^{4,p}_{\mathrm{{loc}}}(\Omega )\), for any \(p <\infty \). Then we may carry out the derivation of the estimate (4.2) by applying the AleksandrovBakel’man maximum principle to our differential inequalities, rather than the classical comparison arguments. Moreover through this approach or by further refinement of our approximation arguments, we need only assume \(A,B \in C^{1,1}({\bar{\Omega }}\times {\mathbb {R}}\times {\mathbb {R}}^n)\), (with our convexity conditions on A holding a.e.), \(G\in C^{1,1}(\partial \Omega \times {\mathbb {R}}\times {\mathbb {R}}^n)\) and \(\partial \Omega \in C^{2,1}\) to obtain the estimate (4.2) and thereby the existence of a unique admissible solution \(u \in C^{3,\alpha }( \Omega )\cap C^{2,\alpha }({\bar{\Omega }})\), for all \(\alpha < 1\).
4.2 Examples
In this subsection, we present various examples of operators \({\mathcal {F}}\), matrix functions A and associated oblique boundary operators \({\mathcal {G}}\) which satisfy our hypotheses.
Examples for A. Examples of strictly regular matrix functions arising in optimal transportation and geometric optics can be found for example in [11, 28, 31, 42, 45]. 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.
Admissible functions. Quadratic functions of the form \(u_0 = c_0 + \frac{1}{2}\epsilon xx_0^2\), will be admissible for the matrices (4.15) and for arbitrary constants \(c_0\), points \(x_0 \in \Omega \) and sufficiently small \(\epsilon \). In general for matrices A arising in optimal transportation and geometric optics the existence of admissible functions is proved in [11].
4.3 Degenerate equations
 \(\mathbf{F1}^{}\)
 F is nondecreasing in \(\Gamma \), namelyand \({\mathscr {T}}(r): =\mathrm{{trace}}(F_r) > 0 \) in \(\Gamma \).$$\begin{aligned} F_r := F_{r_{ij}} = \left\{ \frac{\partial F}{\partial r_{ij}} \right\} \ge 0, \ \mathrm{in} \ \Gamma , \end{aligned}$$(4.28)
Corollary 4.1
In the hypotheses of Theorem 4.1 assume that condition F1 is weakened to condition F1\(^\), \(\partial \Omega \in C^3\), \(\beta \in C^{2}(\partial \Omega )\), \(\varphi \in C^{2}(\partial \Omega \times {\mathbb {R}})\) with the subsolution and supersolution conditions strengthened so that at least one of the inequalities (1.25) holds strictly in \(\Omega \). Then there exists an admissible solution \(u \in C^{1,1}({\bar{\Omega }})\) of the boundary value problem, (1.1)–(1.2).
Proof
Finally in the case that \({\underline{u}}\) is a strict subsolution of (1.1), we replace B by \(B+\delta \) for sufficiently small positive \(\delta \) and let \(\delta \rightarrow 0\). \(\square \)
By suitable approximation of the “minimum” function we then obtain from Corollary 4.1 the following analogue of Theorem 1.4.
Corollary 4.2
Corollary 4.3
Let \({\mathcal {F}} = {\mathfrak {M}}_1\), \(\Omega \) a bounded \(C^{3}\) domain in \({\mathbb {R}}^n\), \(A\in C^2({\bar{\Omega }}\times {\mathbb {R}}\times {\mathbb {R}}^n)\) strictly regular in \({\bar{\Omega }}\), \(B > 0, \in C^2({\bar{\Omega }}\times {\mathbb {R}}\times {\mathbb {R}}^n)\), \(G\in C^{2}(\partial \Omega \times {\mathbb {R}}\times {\mathbb {R}}^n)\) is concave with respect to p and uniformly oblique in the sense of (4.5). Assume that \({\underline{u}}\) and \({\bar{u}}\), \(\in C^{2}(\Omega )\cap C^1({\bar{\Omega }})\) are respectively an admissible subsolution of (1.1)–(1.2) and supersolution of (1.1)–(1.2) with at least one of them strict and \(\Omega \) is uniformly (A, G)convex with respect to the interval \({\mathcal {I}} = [{\underline{u}}, {\bar{u}}]\). Assume also that A, B and \(G\) are nondecreasing in z, with at least one of them strictly increasing, A satisfies the quadratic growth conditions (3.75) and B is independent of p. Then there exists an admissible solution \(u \in C^{1,1}({\bar{\Omega }})\) of the boundary value problem (1.1)–(1.2).
Proof
Note that if we use the functions \(F_{1,\alpha }\), instead of their normalisations, in the proof of Corollary 4.3 we can cover the case that \({\underline{u}}\) is an admissible strict subsolution of (1.1)–(1.2), without having to modify B. Using the approximations \(F_{k,\alpha }\) in the proof of Corollary 4.2, we can similarly adjust the proof of Corollary 4.2.
Also the solutions in Corollaries 4.1, 4.2 and 4.3 will be unique if either A or B are strictly increasing, in which case the strictness of the subsolution or supersolution condition is not explicitly needed in their hypotheses. When only \(\varphi \) or \(G\) is strictly increasing we need to utilize appropriate barrier constructions, as considered in Section 2 of [13], to prove uniqueness. In particular if \({\underline{u}} \in C^2({\bar{\Omega }})\) is a strict admissible subsolution of (1.1), the uniqueness, along with the more general comparison principles, follows from Part (ii) of Lemma 2.1 in [13].
To conclude this section we also point out that our estimates and techniques also embrace degenerate elliptic equations where \(B \ge a_0 > \infty \), with weakly admissible solutions \(u\in C^{1,1}(\Omega )\) now satisfying \(M[u] \in {\bar{\Gamma }}\), a.e. in \(\Omega \), using property F5 with \(a = a_0\), (which as shown in Sect. 4.2 is also a consequence of conditions F1\(^\), F2 and F3). As remarked in Sect. 1, we can also assume \(a_0 = 0\) in this case and, as shown there, concave onehomogeneous functions F will be typical examples. In particular the constants C in the estimates (1.14), (1.18), (1.24), (2.23), (2.49) and (3.55) will not depend on a lower bound for \(B(\cdot ,u,Du)\), (provided in the cases where \(\hbox {F5}^{+}\) is assumed, we also assume \(a=a_0 > \infty \)). Our resultant existence results, Corollaries 4.1, 4.2 and 4.3 now extend similarly, under the strictness of the subsolution \({\underline{u}}\), in their hypotheses and as above, the resultant \(C^{1,1}\) solutions are unique. Note that in the totally degenerate case, \(B = 0\) in \(\Omega \), an admissible function will furnish a suitable strict subsolution, (after subtraction of a sufficiently large constant), when \(G\) is strictly increasing. More details of these arguments, as well as extensions to \({\underline{u}}\) and \({\bar{u}} \in C^{1,1}\), will be provided in conjunction with our treatment of weak solutions in [15].
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 [13], 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 \({\mathcal {G}}\), and (ii) regular A and uniformly concave \({\mathcal {G}}\). Our emphasis in 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 [49, 50] and optimal transportation equations [45, 51]. 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, kHessian operators and their quotients, as well as degenerate and nonorthogonally invariant operators.
In part II [13] 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 [11, 17] to general operators; (see Remarks in Section 2 of [16]). In general as indicated by the Pogorelov example [47, 52], 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 Sect. 2. We remark though that our methods in this paper, as further developed in [13], 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 \(\varphi \), that is either \(A_z\) or \(\varphi _z\) is sufficiently large, and the latter would include the case when \(A = 0\), in agreement with the Monge–Ampère case in [46, 47, 52]. 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 of an admissible supersolution \({\bar{u}}\), satisfying \(\det (M[{\bar{u}}]) \le B(\cdot , {\bar{u}}, D{\bar{u}})\) in \(\Omega \) and \(D_\nu {\bar{u}}= \varphi (\cdot ,{\bar{u}})\) on \(\partial \Omega \); (see Jiang et al. [16]). This is an extension of the fundamental result in [27] for the standard Monge–Ampère operator, (although the supersolution hypothesis is not needed in [27] and more generally when \(D_{px}A = 0\) and \(D_{pz}A=0\) [16]). For the semilinear oblique problem for standard kHessian equations, the known results due to Trudinger [36] and Urbas [47], 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 kHessian equation has been studied in uniformly convex domains in [29]. However, it would be reasonable to expect there are corresponding second derivative estimates for admissible solutions of the Neumann problem for kHessian equations in uniformly \((k1)\)convex domains. More generally, the second derivative estimate for admissible solutions of the Neumann problem of the augmented kHessian equations, with only regular A, in uniformly \((\Gamma _k, A, G)\)convex domains is still an open problem.
In Sect. 3, we have established the gradient estimate for augmented Hessian equations in the cones \(\Gamma _k\) 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 [7, 21]. The gradient estimate under natural conditions is also known for \(k=1\) and is a special case of the uniformly elliptic case [7, 25]. Therefore, it would be interesting to prove 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 \(\Gamma _k\) when \(2\le k\le n/2\), and in particular for the basic Hessian operators \(F_k\) when \(2\le k\le n/2\), which also enjoy \(L^p\) gradient estimates for \(p < nk/(nk)\) [44]. In [14], we apply our gradient estimates here and general barrier constructions in Section 2 of [13] 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 [17, 18]. Our barrier constructions in [13] 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 \({\mathcal {G}}\). If we drop these from our hypotheses, we can still infer the existence of classical solutions of the Eq. (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 is further amplified in [15], with extensions as well to strong solutions in the degenerate cases. Further investigations involve the examination of the sharpness of our boundary convexity conditions for global second derivative bounds. A preliminary result here for the conformal geometry application is already given in [24]. When these degenerate we would still expect to have regularity in the umbilic case, as in [3, 19], which would embrace for example the case \({\tilde{h}} = 0\) in (4.21).
Note As indicated above, this paper is the first in a series of papers on oblique boundary value problems for augmented Hessian equations and its original version was posted in late 2015 [12]. Its immediate sequel [13] and their application to the Dirichlet problem [14] were originally planned in 2015 as part of the present paper but separated in order not to delay for too long the propagation of the main result, Theorem 1.2 and its ramifications.
Notes
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