On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones

In this paper, we study a class of fully nonlinear elliptic equations on annuli of metric cones constructed from closed Sasakian manifolds and derive the a priori estimates assuming the existence of subsolutions. Moreover, such a priori estimates can be applied to certain degenerate equations. A condition for the solvability of Dirichlet problem for non-degenerate fully nonlinear elliptic equations is discovered. Furthermore, we also discuss degenerate equations.


1.
Introduction. Sasakian manifolds, which can be viewed as the odd dimensional counterparts of Kähler manifolds, are odd dimensional Riemannian manifolds whose metric cone (C(M ),ḡ) = (M × R + , r 2 g + dr 2 ) admits a Kähler structure. These manifolds can be used to construct new Einstein manifolds in geometry, and play play an important role in AdS/CFT correspondence in Mathematical physics. We refer the readers to [18,20,42,43] for relevant works on Sasakian geometry from mathematicians and physicists.
Equation (1.1) shall be of the form: There is a smooth and symmetric function f defined on an open symmetric convex cone Γ with Γ n ⊆ Γ ⊂ Γ 1 , such that where λ(g) = (λ 1 , · · · , λ n ) are the eigenvalues of the real (1, 1)-form g with respect toω, Γ k = {λ ∈ R n : S i (λ) > 0 for 1 ≤ i ≤ k} and S k (λ) is the k-th elementary symmetric polynomial of λ ∈ R n .
The study of this type of equations can trace back to Ivochkina [36] where the author treated some special cases, and Caffarelli, Nirenberg and Spruck [5] in which the authors dealt with the Dirichlet problem for Hessian equations on the bounded domains Ω ⊂ R n . As in [5], the function f shall satisfy the following fundamental conditions,
Furthermore, we assume that For any σ < sup Γ f and λ ∈ Γ we have lim t→+∞ f (tλ) > σ. (1.5) Such condition is closely related to (1.12) below and allows one to derive the gradient estimates using the blow-up argument presented in [9,50]. Moreover, Székelyhidi [50] proved that if f satisfies (1.2)-(1.5) then there exists a positive constant depending only on σ and f (may depend on δ ψ,f ) such that In case of equation ( [σ n (λ)] 1/n ) which is related to the representation of Ricci form of the Kähler manifold. A related problem is the well known Calabi conjecture which requires one to solve the smooth non-degenerate complex Monge-Ampère equation, c.f. Calabi [8]. Yau [56] solved it and showed that every closed Kähler manifold of C 1 (M ) ≤ 0 admits a unique Kähler-Einstein metric contained in the given Kähler class. The same result on a Kähler manifold with C 1 (M ) < 0 was obtained by Aubin [1] independently. Another fundamental work about the complex Monge-Ampère equation is due to Caffarelli, Kohn Nirenberg and Spruck [7] who dealt with the Dirichlet problem for the complex Monge-Ampère equation on a strictly pseudoconvex domain in C n . This work was extended to general bounded domains Ω ⊂ C n by Guan [22], in which the author used the admissible subsolution satisfying (1.8) below to relax the restrictions to ∂Ω. This notion of the subsolution plays crucial roles in various works, c.f. [9,23,30,33]. It would be worthwhile to note that the Dirichlet problem is not always solvable without assuming subsolutions.
By contrast with the real setting, it turns out to be a rather challenging task to derive a priori estimates for second derivatives of solutions of fully nonlinear elliptic equations containing gradient terms in complex setting. The underlying reason is the two different types of complex derivatives. For more information, we refer the reader to [17,26,44,51] for related works.
We state our main results as follows.
We now turn to the second order boundary estimates sup M |∇ 2 u|ḡ ≤ C. (1.10) In [5], Caffarelli, Nirenberg and Spruck derived (1.10) for the Dirichlet problem in a bounded domain Ω ⊂ R n f (λ(∇ 2 u)) = ψ in Ω, assuming that ∂Ω satisfies certain curvature condition and hypotheses on f : for every C > 0 and compact subset K ⊂ Γ there is a constant R = R(C, K) such that This result was extended by Li [39] to the general case. Furthermore, condition (1.11) was removed by Trudinger [55]. Since then boundary estimates (1.10) were extended to very general Hessian equations, c.f. [22,23,24,31,37] and reference therein.
Suppose, in addition condition (1.7) holds or ψ is a constant function, that a k (z, ψ) = a k (z) and there is a uniform constant C II depending not on (δ ψ,f ) −1 such that sup Then the constant C in (1.14) does not depend on (δ ψ,f ) −1 .
Combining Theorem 1.1 and Theorem 1.14, the quadratic power of gradient term can dominate the Laplacian (1.16) Therefore, the gradient estimate can be obtained by the blow-up argument presented in [9,50].
The following condition allows one to construct the desired basic admissible subsolutions of Dirichlet problem (1.1).
Given a smooth real (1, 1)-form g, we set where P * D C is the pullback contact bundle D C = D 1,0 ⊕D 0,1 and P : C(M ) → M is the natural projective map. In particular, where α ∈ (0, 1), and λ (U [v] T ) = (λ 1 , · · · , λ n−1 ) are the eigenvalues of U [v] T with respect to r 2 ω T . For Dirichlet problem (1.1), we assume that there exists a function v ∈ C α (χ) with the boundary value condition v| ∂M = ϕ, such that where θ n = dr + √ −1rη andθ n = dr − √ −1rη. Then we can construct a basic admissible subsolution of Dirichlet problem (1.1). A typical example of (1.17) is the function f satisfying (1.11). We shall point out that the assumption (λ (U [v] T ), R) ∈ Γ and Lemma 6.1 below ensure that the subsolution u constructed in (6.1) is admissible. Please refer to Section 6 for more details.
In this paper, we also study the following Dirichlet problem of degenerate fully nonlinear elliptic equations where ϕ a , ϕ b ∈ C 4,γ B (M ) and ψ ∈ C 2,γ B (M ) with δ ψ,f ≥ 0 for some γ ∈ (0, 1). The main theorem for degenerate equations can be stated as follows.
. Clearly, the geodesic equation in H satisfies all the above assumptions. This paper is organized as follows. In Section 2, we briefly outline some knowledge of Sasakian manifolds and some useful lemmas. In Section 3 we establish second order estimates for admissible solutions. In Section 4 we derive a priori boundary estimates precisely if equation (1.1) can be rewritten as the form (1.13). Gradient estimate will be given in Section 5 where we use the blow-up argument. Finally, basic admissible subsolutions are constructed under condition (1.17). Moreover, the existence of the solutions can be proved by the method of continuity.

2.
Preliminaries. In this section, we briefly outline some knowledge of Sasakian manifolds and some useful lemmas.
2.1. Sasakian manifold. We provide some background of Sasakian manifolds here. We also refer the reader to [4,21] for more extensive treatment on the subject.
Kähler. An important fact is that Φ determines a complex structure on the contact sub-bundle D = ker{η}. Furthermore, (D, Φ| D , dη) provides M with a transverse Kähler structure admitting a Kähler form 1 2 dη and a metric g T defined by g T (·, ·) = 1 2 dη(·, Φ·). The complexification D C of the sub-bundle D can be decomposed into its eigenspaces with respect to Φ| D as D C = D 1,0 ⊕ D 0,1 . It is easy to see that the exterior differential preserves basic forms. The transverse complex structure follows the splitting of the complexification of the bundles of the sheaf of germs of basic p-forms where i ξ is the contraction with the Reeb field ξ and L ξ is the Lie derivative with respect to ξ.
The Sasakian structure (ξ, η, Φ, g) of (M, g) determines the following almost complex structure on C(M ): which turns (C(M ),ḡ, J) into a Kähler manifold. From now on, P * η and P * dη will be used to denote pull-backs by η and dη, respectively, where P : C(M ) → M is the projective map.
As in the Kähler setting, the Sasakian metric can be locally generated by a free real function of 2(n − 1) variables. More precisely, for any p ∈ M , there is a local basic function h and a local coordinate chart (z 1 , · · · , z n−1 , x) ∈ C n−1 × R on a small neighborhood U around p such that 2) and D ⊗ C is spanned by Thus Moreover, one can change the local coordinates to normal coordinates such that We refer the reader to Godlinski, Kopczynski and Nurowski [21] for more details.
Then the transverse Kähler form is given by For the normal local coordinate chart (z 1 , · · · , z n−1 , x) on a Sasakian manifold (M, η, ξ, Φ, g), set and where h is the Sasakian potential function (which is basic). Then It is easy to verify that (2.10) Set We know that {θ i ,θ j } is the dual basis of {X i ,X j }.
To overcome the difficulty due to the two different types of complex derivatives, as in Guan and Zhang [33], we establish the following lemma.
. Thus ξu ≡ 0 inM by the maximum principle for linear elliptic equations.

FULLY NONLINEAR ELLIPTIC EQUATIONS ON METRIC CONES 5715
Throughout this paper we shall use the following notations: for a given Hermitian The matrix {F ij } has eigenvalues f 1 , · · · , f n and is positive definite by assumption (1.2), while (1.3) implies that F is a concave function of a ij . Moveover, when A is diagonal, so is {F ij (A)}, see e.g. [5]. Furthermore, In local complex coordinates z = (z 1 , · · · , z n ), we shall use notations such as 3. Second order estimates. In this section we give the proof of Theorem 1.1 which derives second order estimates. Firstly, we give C 0 -estimate and boundary gradient estimate assuming the existence of admissible subsolution u. Let h be a C 2 solution to where ∆ḡ is the standard Laplacian with respect to the Levi-Civita connection. We refer the readers to [52] for the solvability of (3.1). By the maximum principle where the constant C * depends only on u and h. Next we prove Theorem 1.1.
Proof of Theorem 1.1. Firstly, Lemma 2.3 concludes that u is basic. Denote the eigenvalues of the matrix A = {A i j } = {ḡ iq U jq } by (λ 1 , · · · , λ n ), and λ 1 :M → R is the largest eigenvalue and λ 1 ≥ λ 2 · · · ≥ λ n at each point. We want to apply the maximum principle to H, where {ḡ ij } = {ḡ pq } −1 and φ is the test function to be chosen later. Suppose H achieves its maximum at an interior point p 0 = (x 0 , r 0 ), x 0 ∈ M, a < r 0 < b.
We choose a normal and holomorphic local coordinate around p 0 , say (w 1 , · · · , w n ), such that under this coordinate, at p 0 From now on, we denote by In what follows, we use derivatives with respect to the Chern connection ofḡ and the computations will be given at the origin p 0 , and we assume Since the eigenvalues of A do not need to be distinct at the point p 0 , H may only be continuous. To circumvent this difficulty we use the perturbation argument used by Collins, Jacob and Yau [11], which is a modification of the treatment of Székelyhidi [50]. To do this, we can suppose that B is sufficiently small. Moreover, in our fixed local coordinates, B is a constant diagonal matrix B p q with real entries satisfying B 1 1 = B n n = 0 and B n−1 n−1 < · · · < B 2 2 < 0. Then we define the matrix A = A + B with the eigenvaluesλ = (λ 1 , · · · ,λ n ). At the origin p 0 ,λ 1 = λ 1 ,λ i = λ i +B i i if i ≥ 2 and the eigenvalues ofÃ define C 2 -functions near the origin. Noticẽ H =λ 1 e φ also achieves its maximum at the origin p 0 (we may assume λ 1 (p 0 ) =λ 1 (p 0 ) > 1). (3.5) By straightforward calculations, one obtains whereλ pq,rs 1 c.f [11,50]. Evaluating this expression at p 0 , and using that B is constant, B 1 1 = 0, and that we are working in normal coordinates forḡ, we have We need to estimateλ 1 −λ p near the origin p 0 for p > 1. Since Γ n ⊆ Γ ⊂ Γ 1 , we may assume λ i > 0(otherwise we are done). Then |λ i | ≤ (n − 1)λ 1 for all i, and so (λ 1 −λ p ) −1 ≥ (nλ 1 ) −1 . As in [11,50], we know that Then there exists two uniform constant C 0 and C 1 depending only on known data under control such that where we use the inequality such as |h + g| 2 ≥ 1 2 |h| 2 − |g| 2 . We compute (3.12) Differentiating equation (1.1) twice (using the covariant derivative), we have at p 0 In this local holomorphic coordinates (w 1 , · · · , w n ), there is a transform T of vectors (2.3) and (2.8) such that (X 1 , · · · , X n )T = (∂ 1 , · · · , ∂ n ), (3.13) here we use (2.4) and (2.9). Near p 0 , the transform T allows one to present ∂ ∂w i by the linear combinations of X i thanks to (2.3) and (2.8), i.e. ∂ ∂w j = n k=1 T k j X k and we may set X j = n k=1 S k j ∂ k , where T k j S l k = δ jl . Note that u is basic, ∂u ∂r = 2X n u = 2X n u, then by (3.4) one derives where C A is a positive uniform constant depending on supM |T |ḡ and other known data.
We know that [X α , Then by straightforward calculations, one obtains X kXl X n u = X n X kXl u, Note that ). Applying (3.13) and (3.15), one obtains Commuting derivatives and Denote by L the linearized operator of equation (1.1), Then we have by (3.5) and (3.15) By (3.17), (3.4) and (3.14), we obtain where C 1 depends on the constant C A in (3.14) and other known data. We set Ψ : [0, +∞) → R, where A ≥ 1, N ∈ N to be chosen later. Let As in [35], we set Then we know that Differentiating equation (1.1) one obtains Then by Cauchy-Schwarz inequality (3.23) Note that in this computationλ 1 denotes the largest eigenvalue of the perturbed endomorphismÃ = A + B. At the origin p 0 where we compute,λ 1 coincides the largest eigenvalue of A with respect to the metric {ḡ ij }, but at nearby points it is a small perturbation. We would take B → 0, and obtain the above differential inequality (3.23) for the largest eigenvalue of A as well, but this would only hold in a viscosity sense because the largest eigenvalue of A may not be C 2 at the origin p 0 , if some eigenvalues coincide.
Case I. Assume that δλ 1 ≥ −λ n (0 < δ 1 2 ). Set Clearly 1 ∈ J. The identity (3.5) implies that for fixed k one has (3.24) The assumption δλ 1 ≥ −λ n implies that (3.25) We use some of the good term Ψ F kk |(u − u) k | 2 in (3.23) to control the last term of (3.24). So we choose small δ such that For k ∈ J, we know that F kk ≤ δ −1 F 11 . Then

From (3.5) one derives
where we use the elementary inequality for convenience. We briefly outline a useful property of the symmetric function f as follows, c.f. [24].
By the concavity of f , we have Then where ε is the positive constant in Lemma 2.2 determined by f and β. Choosing Hence (3.27) follows that λ 1 ≤ CK.
Note that Combining it with (3.27) and (3.28), we derive that Case II. We assume that δλ 1 < −λ n with the constants A, N and δ fixed as in previous case. In particular, (3.31) Then by (3.23) Note that F nn ≥ 1 n F iī and Thus by (3.28), one obtains λ 1 ≤ CK. 4. Boundary estimates for second derivatives. In this section we study the second order boundary estimates for the admissible solution u via constructing barrier functions. Given a point p ∈ ∂M , let ρ(z) = distM (z, p) be the distance function from z to p and Ω δ = {z ∈ Int(M ) : ρ(z) < δ}, 0 < δ 1.
Denote by σ the distance function to ∂M with respect toḡ.
To construct a barrier function, we will follow Guan [22] and employ a barrier function of the form where u ∈ C 2 B (M ) is the admissible subsolution of Dirichlet problem (1.1). The following lemma was first proved in [22] for domains in C n and holds for general compact Hermitian manifolds with smooth boundaries.
where κ is the constant in (1.6), and ε is the constant in Lemma 2.2.
Note that where ε is the positive constant in Lemma 2.2. If t ≤ ε 2α0 , then Fix t small enough so that t ≤ ε 2α0 . In case II,

Lemma 4.2.
There is a positive constant C 0 depending on |χ| C 0,1 (M ) , |ϕ| C 2,1 (M ) and other known data such that Proof. In Int(M ), one has where we use the fact that u is basic, [X n , D] = 0 and X i is the vectors defined in where C is a positive constant depending on |u| C 2 (M ) , |ϕ| C 2,1 (M ) , |ψ| C 0,1 (M ) and other known data. Suppose in addition that f satisfies (1.7) or ψ is a constant function. Then the constants in (4.6) and (4.5) depend not on κ −1 (κ is the constant in (1.6)).
Proof. The second order boundary estimates for pure tangential derivatives is standard. To be precise, there exists a uniform positive constant C 1 depending on sup ∂M | ∂(u−u) ∂r |ḡ and other known data under control such that sup Combining it with (3.3), the gradient on boundary is uniform. We thus have (4.5).
The tangential-normal case will be proved by constructing barrier functions. This type of construction of barrier functions follows from [22,29,34].
Next we will complete the proof of Theorem 1.2.
Proof of Theorem 1.2. From the proof of Proposition 1, we only need considering pure tangential derivative case. And we also use the notations and computations there. Given p = (q, r p ) ∈ ∂M , r p = a or b.
Therefore, it follows from (4.8), (4.5) and (4.6) that U nn ≤ C(1 + supM |∇u| 2 g ). Hence (1.14) holds. 5. Gradient estimate and blow-up argument. In [50], Székelyhidi proved a Liouville type theorem which extended that of Dinew and Kolodziej [14]. Based on the blow-up argument in [9,50], we obtain the gradient estimate as follows. Moreover, assumption (1.5) is crucial for the blow-up argument in Székelyhidi [50]. Suppose N i := supM |∇u i |ḡ → +∞. Let N i = |∇u i (z i )|ḡ. From the bound of sup ∂M |∇u|ḡ, it follows that there is a constant n 0 , such that for any i ≥ n 0 , z i ∈ Int(M ). By (5.2), we obtain We choose a convergent subsequence of {z i }, also denote by {z i }, such that z i → z. Suppose z ∈ ∂M . Then by the argument used in [9], we have a contradiction. Therefore, z stays in the interior ofM . In this case, we use the argument of Székelyhidi [50] where assumption (1.5) is imporatnt. By combining the blowup argument with a Liouville type theorem in [50], we obtain that sup M |∇u| g ≤ C. Note that in the blow-up argument the only difference from the setup here is the linear term ∂u ∂r in U ij . However, the order of ∂u ∂r is only one which ensures the terms containing ∂u ∂r converge to zero uniformly on compact sets under the rescaling procedure of Székelyhidi [50]. Thus the blow-up argument in [50] still works. Therefore we obtain (5.1). We refer the reader to [9,50] for more details.
6. Existences and basic subsolutions. In this section we solve the equations assuming condition (1.17) holds. We construct basic admissible subsolutions under condition (1.17), and the existences of Dirichlet problems can be derived via the method of continuity and a priori estimates above.
Applying Lemma 6.1, we know that u ∈ C 4,α B (M ) and λ(U [u]) ∈ Γ for A 1 thanks to v ∈ C α (χ). Moreover, u is a basic admissible subsolution of Dirichlet problem (1.1) for large A if in addition condition (1.17) holds.
6.2. Solving equations. In this subsection u is the basic admissible subsolution constructed in (6.1). Based on a priori estimates as previous, we solve the equations by the method of continuity.  Clearly 0 ∈ I by taking u 0 = u. One can verify that (i) u is the basic admissible subsolution along the whole method of continuity.
(ii) The right hand side of (6.2) is basic.
So we can apply the a priori estimates above. Lemma 2.3 implies that u t is basic if it exists. By applying (3.3), Theorems 1.1 and Proposition 2, one derives a uniform bound for complex Hessian of basic admissible solution u t of the Dirichlet problem. Hence, the equations become to be uniform elliptic and concave because of conditions (1.2), (1.3) and (1.4). The argument used by Guan and Li [25] gives us a uniform bound of real Hessian. Thus the theory of Evans-Krylov [15,38] can be applied to derive |u t | C 2,α (M ) ≤ C. The higher regularities can be derived by classical Schauder theory. Therefore our estimates conclude that I is closed. The openness is follows from implicit function theorem and the a priori estimates above, hence then I = [0, 1]. So there is a basic admissible solution u ∈ C 4,α (M ) of Dirichlet problem (1.1). The uniqueness of the solution u follows from maximum principle.