The Neumann Problem of Complex Special Lagrangian Equations with Supercritical Phase

Chuanqiang Chen,Xinan Maand Wei Wei

1 Department of Applied Mathematics,Zhejiang University of Technology,Hangzhou,310023,Zhejiang,China

2 School of Mathematical Sciences, University of Science and Technology of China,Hefei,230026,Anhui,China

3 School of Mathematical Sciences, University of Science and Technology of China,Hefei,230026,Anhui,China

Received 22 August 2017;Accepted(in revised version)8 June 2018

Abstract. Inspired by the Neumann problem of real special Lagrangian equations with supercritical phase, we consider the Neumann problem of complex special Lagrangian equations with supercritical phase in this paper,and establish the global C2 estimates and the existence theorem by the method of continuity.

Key Words: Special Lagrangian equation,Neumann problem,supercritical phase.

1 Introduction

As we all know,the real special Lagrangian equation is

For the special Lagrangian equations with supercritical phase,Yuan obtained the interior C1estimate with Warren in [29] and the interior C2estimate with Wang in [28].Recently Collins-Picard-Wu[8]obtained the existence theorem of the Dirichlet problem.

Moreover, for the Dirichlet problem of elliptic equations in Rn, many results are known. For example, the Dirichlet problem of Laplace equation is studied in [7,11],Caffarelli-Nirenberg-Spruck [2] and Ivochkina [15] solved the Dirichlet problem of Monge-Amp`ere equation,and Caffarelli-Nirenberg-Spruck[4]solved the Dirichlet problem of k-Hessian equation. After the pioneering works of Caffarelli et al., the Dirichlet problem of the general Hessian quotient equation was solved by Trudinger in[27]. For more information about the related subjects,we refer to the citations of[2,4,15].

Also, the Neumann or oblique derivative problem of partial differential equations was widely studied. For a priori estimates and the existence theorem of Laplace equation with Neumann boundary condition,we refer to the book[11].Also,we can see the recent book written by Lieberman[21]for the Neumann and the oblique derivative problems of linear and quasilinear elliptic equations. In 1986,Lions-Trudinger-Urbas solved the Neumann problem of Monge-Amp`ere equation in the celebrated paper [23]. Recently, Ma-Qiu[24]solved the the Neumann problem of k-Hessian equations, and Chen-Zhang[6]generalized the result to the the Neumann problem of Hessian quotient equations. For the Neumann problem of special Lagrangian equations with supercritical phase, Chen-Ma-Wei [5] got the existence theorem. In [16,17], Jiang-Trudinger studied the general oblique boundary value problems for augmented Hessian equations with some regular condition and some concavity condition.

At the same time,the complex equations have attracted a variety of mathematicians and many excellent works have been done. The complex Monge-Amp`ere equations are definitely one of the most important equations in partial differential equation and the geometry. In [1], Bedford and Taylor studied the Dirichlet problem of complex Monge-Amp`ere equations by using the Perron-Bremermann family method, and got the existence and uniqueness of the weak solutions and a global Lipschitz regularity for plurisubharmonic solution when Ω is a bounded strictly pseudoconvex domain in Cn. In[3],Caffarelli et al. studied the classical solution on strongly pseudoconvex domains. Their work was extended to arbitrary bounded domains in Cnby Guan in[12]under some subsolution condition. Recently,Fu-Yau equation on compact K¨ahler manifolds arises much attention, which was introduced in[10]. The Fu-Yau equation was solved in dimension 2 by Fu-Yau and was recently extended to higher dimensions in some cases in [25] by Phong,Picard and Zhang. In high dimensions,the equation is actually a 2-Hessian type equation. Complex Hessian equations have been studied extensively by many authors in recent years. On Cn,the Dirichlet problem of complex Hessian equation was studied by Li [20] under some strict subsolution condition. On compact manifolds there were the existence of admissible solution in [9,14]. For the Neumann problem of complex Monge-Amp`ere,Li proved the existence under some other conditions in[19].

Naturally, we want to know how about the problem of complex special Lagrangian equations of the form

In this paper, we establish global C2estimate of the Neumann problem of special Lagrangian equations with supercritical phase and obtain the existence theorem as follows:

As Li mentioned in his paper[19], because the Neumann boundary condition is not invariant under the holomorphic changes of variables,the strictly pesudoconvex domain may not be enough and we need the extra curvature condition of the domain.

Furthermore,we can obtain the existence theorem of the classical Neumann problem of complex special Lagrangian equation as below

The rest of the paper is organized as follows. In Section 2,we collect some properties of the special Lagrangian equation and establish the C0estimate for the Neumann problem of special Lagrangian equation. The C1and C2estimates are established in Section 3,Section 4,respectively. At last,we prove Theorem 1.1 and Theorem 1.2 in Section 5.

2 Some properties and C0 estimate

In this section, we give some properties of the special Lagrangian equation with supercritical phase and establish the C0estimate.

2.1 Some properties

So λn-1+λn＞0,which implies(2.2a),(2.2b)and(2.2c)hold.

Moreover,

and

so we can get

Thus,we complete the proof.

where

and

From the complex special Lagrangian equation(2.1),we know

and

From the concavity lemma(Lemma 2.2 in[8]),we know

Hence(2.3)holds.

2.2 C0 estimate

The C0estimate is easy. For completeness,we produce a proof here following the idea of Lions-Trudinger-Urbas[23]and Ma-Qiu[24].

where

Proof. From (2.2c), we know u is subharmonic. So the maximum of u is attained at a boundary point z0∈∂Ω. Then we can get

Hence

Without loss of generality,we assume 0∈Ω and denote

Then we have

By the comparison principle,we know u-B|z|2attains its minimum at a boundary point z1∈∂Ω. Then

Hence

Thus,we complete the proof.

3 Global gradient estimate

In this section,we prove the global gradient estimate by following the idea of Li[19].

Proof. In order to prove(3.1),it suffices to prove

Hence we can get from Property 2.1

where

For fixed ξ=ξ0,W(z,ξ0)achieves its maximum at the same point z0∈Ω and we can easily get that at z0,

where C1is a positive constant depending only on n,ε,|u|C0,|φ|C2and|ν|C2.

we can get

This is a contradiction. So z0∈∂Ω. Then we continue our proof in the following three cases.

(a)If ξ0is normal at z0∈∂Ω,then

Then we can easily get(3.2).

(b) If ξ0is non-tangential at z0∈∂Ω, then we can write ξ0=ατ+βν, where τ ∈S2n-1is tangential at z0, that is〈τ,ν〉=0, α=〈ξ0,τ〉＞0, β=〈ξ0,ν〉＜1, and α2+β2=1. Then we have

so

Then we can easily get(3.2).

(c)If ξ0is tangential at z0∈∂Ω,we may assume that the outer normal direction of Ω at z0is(0,···,0,1). By a rotation,we assume that ξ0=(1,···,0)=e1. Then we have

By the boundary condition,we know

Following the argument of[19],we can get

From(1.3),it holds κmin+ε＞0. So

Then we can easily get(3.2).

and following the proof of Theorem 3.1,we can choose K0and K1independent of ε,and obtain the global gradient estimate

4 Global second derivatives estimate

We now come to the a priori estimate of global second derivatives. Firstly,we prove the estimate of double normal second derivatives on boundary, and then we complete the proof of global second derivatives estimate.

4.1 Estimate of double normal second derivatives on boundary

where k0is a positive constant depending only on Ω and Inis the n×n identity matrix.

Denote

where

Following the idea of[19],we choose the auxiliary function

where K1＞0 is sufficiently large so that

and

Let

It is easy to know

where

So we have Dννu(z0)≥-C8.

The same argument for

can give

This completes the estimates of the double normal derivative on the boundary.

4.2 Estimate of global second derivatives

Proof. From (2.2c), we know Δu=4∑ni=1ui¯i=4(λ1+···+λn)＞0. By the argument in Li’s[19],we know that we only need to prove that

As the real case in[23],we use the auxiliary function

For any z ∈Ω, we rotate the coordinates such that{∂i¯ju(z)} is diagonal with λi=ui¯iand λ1≥λ2≥···≥λn. Then we have

Hence we can get from Property 2.1

For any fixed ζ ∈Sn-1,we have

It is easy to know

Choose

then we can get

Then we continue our proof in the following two cases following the idea of[19].(a) If ζ0is non-tangential at z0∈∂Ω, then we can write ζ0=ατ+βν, where τ ∈S2n-1is tangential at z0,that is〈τ,ν〉=0,α=〈ζ0,τ〉,β=〈ζ0,ν〉/=0,and α2+β2=1. Then we have

hence

From the definition of Q(z0,ζ0),we know

and we can prove(4.14).

(b)If ξ0is tangential at z0∈∂Ω,then we have

By the boundary condition,we know

Following the argument of[19],we can get

From(1.3),it holds 2κmin+ε＞0. So

Then we can easily get(4.14).

Remark 4.1. As the discussions in Remark 3.1,to prove Theorem 1.2,we need to consider the complex special Lagrangian equation(1.2)with ε→0. Under the assumptions of Theorem 1.2,and for any sufficiently small ε,we can obtain the following proof of Theorem 4.1 and Theorem 4.2,

5 Existence of the boundary problems

In this section we complete the proofs of the Theorem 1.1 and Theorem 1.2.

5.1 Proof of Theorem 1.1

Applying the method of continuity (see [21]), the existence of the classical solution holds. By the standard regularity theory of uniformly elliptic partial differential equations,we can obtain the higher regularity.

5.2 Proof of Theorem 1.2

Applying the maximum principle and Hopf Lemma, we can know β=β1and v-v1is a constant. By the standard regularity theory of uniformly elliptic partial differential equations,we can obtain the higher regularity.

Acknowledgements

Research of the first author was supported by ZJNSF No. LY17A010022 and NSFC No.11771396. Research of the second author and third author was supported by NSFC No. 11471188 and Wu Wen-Tsun Key Laboratory of Mathematics in USTC. Research of the third author was supported by China Scholarship Council.