The exterior Dirichlet problem for the homogeneous complex k-Hessian equation

Theorem 1.1

Assume 1 ≤ k < n . Let Ω be a smoothly strongly pseudoconvex domain in C n such that 0 ∈ Ω and Ω ¯ is holomorphically convex in ball centered at 0. There exists a unique k-subharmonic solution u ∈ C 1 , 1 ( Ω c ¯ ) of equation (1.3). Moreover, there exists uniform constant C such that, for any z ∈ Ω c ¯ , the following holds

Lemma 2.1

S k 1 k is a concave function in Γ k . In particular, log S k is concave in Γ k .

Proposition 2.2

We have the following two inequalities:

1. If λ ∈ Γ k , then

   S k 2 ( λ ∣ i ) S k − 1 ( λ ∣ i ) ≥ k + 1 k n − k n − k − 1 S k + 1 ( λ ∣ i ) ;

2. If λ ∈ Γ k , then

   S k ( λ ∣ i ) S k − 1 ( λ ∣ i ) ≤ 1 k n − k n − 1 S 1 ( λ ∣ i ) .

Proof

Since λ ∈ Γ k , we have S k − 1 ( λ ∣ i ) > 0 . The first inequality follows from Newton inequality. Now, we prove (b). Since λ ∈ Γ k , we have S h ( λ ∣ i ) > 0 , ∀ h = 0 , 1 , … , k − 1 . If S k ( λ ∣ i ) ≤ 0 , (b) holds naturally. If S k ( λ ∣ i ) > 0 , the second inequality follows from the generalized Newton-MacLaurin inequality.□

Proposition 2.3

Let λ = ( λ 1 , … , λ n ) ∈ Γ ¯ k , and λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n . Then, there exists θ = θ ( n , k ) > 0 such that

from which it follows

Proposition 2.4

Let λ = ( λ 1 , … , λ n ) ∈ Γ ¯ k , and λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n . Then, there exists θ = θ ( n , k ) > 0 such that

Moreover, for any δ ∈ ( 0 , 1 ) , there exists K > 0 such that if

we have

Definition 2.5

Let α be a real ( 1 , 1 ) -form in U , a domain of C n . We say that α is k -positive in U if the following inequalities hold:

Definition 2.6

Let U be a domain in C n .

1. A function u : U → R ∪ { − ∞ } is called k -subharmonic if it is subharmonic, and for all k -positive real ( 1 , 1 ) -form α 1 , … , α k − 1 in U ,

   d d c u ∧ α 1 ∧ ⋯ ∧ α k − 1 ∧ ω n − k ≥ 0 .

   The class of all k -subharmonic functions in U will be denoted by Sℋ k ( U ) .

2. A function u ∈ C 2 ( U ) is called k-subharmonic (strictly k-subharmonic) if λ ( ∂ ∂ ¯ u ) ∈ Γ ¯ k ( λ   ( ∂ ∂ ¯ u ) ∈ Γ k ).

Lemma 2.7

Let U be a bounded domain in C n , u , v ∈ Sℋ k ( U ) ∩ C ( U ¯ ) satisfy

Then, u ≤ v in U .

Definition 2.8

ρ is called a defining function of C 1 domain U , if U = { z : ρ ( z ) < 0 } and ∣ D ρ ∣ ≠ 0 on ∂ U .

Definition 2.9

A C 2 domain U is called pseudoconvex (strictly pseudoconvex) if it is Levi pseudoconvex (strictly Levi pseudoconvex). That is, for a C 2 defining function of U defined in a neighborhood of U , the Levi form at every point z ∈ ∂ U defined by

is nonnegative (positive). T ∂ U , z h ≔ { ξ ∈ C n ∣ ∑ j ∂ ρ ∂ z j ξ j = 0 } is the holomorphic tangent space to ∂ U at z .

Definition 2.10

A C 2 domain U is called k -pseudoconvex (strictly k -pseudoconvex) if for a C 2 defining function of U defined in a neighborhood of U ,

where ( z 1 , … , z n − 1 ) is a holomorphic coordinate system of T ∂ U , z h near z .

Lemma 2.11

Suppose that U is a bounded smooth domain in C n . For h , g ∈ C m ( U ) , m ≥ 2 , for all δ > 0 , there exists H ∈ C m ( U ) such that

1. H ≥ max { h , g } and

   H ( z ) = h ( z ) , if h ( z ) − g ( z ) > δ , g ( z ) , if g ( z ) − h ( z ) > δ ;

2. There exists ∣ t ( z ) ∣ ≤ 1 such that

   { H i j ¯ ( z ) } ≥ 1 + t ( z ) 2 g i j ¯ + 1 − t ( z ) 2 h i j ¯ , for a l l z ∈ { ∣ g − h ∣ < δ } .

Lemma 2.12

Let Ω 0 and Ω 1 be smooth, strongly pseudoconvex domain in R n with Ω 1 ⊂ ⊂ Ω 0 . Assume that Ω 1 is holomorphically convex in Ω 0 . Then, there exists a strictly plurisubharmonic function u ̲ ∈ C ∞ ( Ω ¯ ) with Ω ≔ Ω 0 ⧹ Ω ¯ 1 satisfying

where ρ 0 and ρ 1 are defining functions of Ω 0 and Ω 1 and τ and K are uniform constants.

Theorem 2.13

Let Ω 0 and Ω 1 be smooth, strongly pseudoconvex domains and assume that Ω 1 is holomorphically convex in Ω 0 . There exists a unique solution u ∈ C 1 , 1 ( Ω ¯ ) of equation (2.7).

Lemma 3.3

There exists a uniform constant C such that

Proof

Let h be the unique solution of the problem

By the maximal principle, we have u ̲ ≤ u ≤ h . This gives uniform C 0 estimates.

Let F i j ≔ ∂ ∂ u i j log H k ( u ) = ∂ ∂ u i j S k ( ∂ ∂ ¯ u ) .

Then,

Thus, we have

Since u ̲ ≤ u ε ≤ h in Ω and u ̲ = u ε = h on ∂ Ω , we have

where ν is the unit outer normal to ∂ Ω (unit inner normal to ∂ Ω 1 and unit outer normal to ∂ Ω 0 ). Thus, we have

Lemma 3.4

There exists a uniform constant C such that

Proof

Denote by D ξ u = u ξ . Then,

Hence,

This implies ∀ i , j = 1 , … , n ,

On the other hand, Δ u ( z ) > 0 implies

Then,

Thus, we have

So we need to prove the second-order estimate on the boundary ∂ Ω . Here, we use the method by Guan [13,16,17] and Li [30].

Tangential derivative estimates on ∂ Ω

Consider a point p ∈ ∂ Ω . Without loss of generality, let p be the origin. Choose the coordinate z 1 , … , z n such that the x n axis is the inner normal direction to ∂ Ω at 0. Suppose

Denote by t ′ = ( t 1 , … , t 2 n − 1 ) . Then, around the origin, ∂ Ω can be represented as a graph

Since

we have

It follows by gradient estimate that

Tangential-normal derivative estimates on ∂ Ω

We use Guan’s method [13,14,16]. Our barrier function here is simpler than before since u is constant on the boundary and the right-hand side of the approximating equation is a sufficiently small constant ε .

To estimate u t α t n ( 0 ) for α = 1 , … , 2 n − 1 and u t n t n ( 0 ) , we consider the auxiliary function

on Ω δ = Ω ∩ B δ ( 0 ) with constants N , t , and δ to be determined later. The following lemma proven in [14] is needed.

The following three lemmas were proven by Guan in [16].

At any boundary point p ∈ ∂ Ω , we may choose coordinates z 1 , … , z n with the origin p such that the positive x n axis is the interior normal direction to ∂ Ω at p . Let ϱ be a defining function of Ω , that is, ϱ < 0 in Ω , ϱ = 0 , D ν ϱ = 1 , on ∂ Ω , where ν is a unit outer normal to ∂ Ω . We may assume that, ∂ ϱ ∂ x j ( 0 ) = 0 for 1 ≤ i ≤ n − 1 and ∂ ϱ ∂ y j ( 0 ) = 0 for all 1 ≤ i ≤ n . Moreover, around the origin, we can write

where ∣ Q ( z ) ∣ ≤ C ∣ z ∣ 3 . Let

Let

Then,

So T = ∂ ∂ t α + a α ∂ ∂ x n is a tangential vector to ∂ Ω near the origin. We write

Let

Then,

So

Note that

and

We then have

and

Let

By Lemmas 3.5, 3.6, and 3.8, we see that

and

when A 1 ≫ A 2 ≫ A 3 ≫ 1 . Therefore,

In particular, from (4.30), we know

Double normal derivative estimates on ∂ Ω

For any fixed p ∈ ∂ Ω , we choose the coordinate such that p = 0 , ∂ Ω ⋂ B r ( 0 ) = ( t ′ , φ ( t ′ ) ) , and ∇ φ ( 0 ) = 0 .

Case 1: x 0 ∈ ∂ Ω 0 .

Let ρ 0 be a defining function of Ω 0 , which is strictly plurisubharmonic in a neighborhood of Ω 0 . So

Then, we have

On the other hand, we have

Thus,

and

Since ρ 0 is strictly plurisubharmonic in Ω ¯ 0 , we have

Case 2: x 0 ∈ ∂ Ω 1 .

Note that u ≥ u ̲ near ∂ Ω 1 , u = u ̲ , and 0 < u ̲ ν ≤ u ν on ∂ Ω 1 , ν is the unit outer normal to ∂ Ω 1 , there exists a smooth function g such that u = g u ̲ near ∂ Ω 1 , and g ≥ 1 outside of Ω nearby ∂ Ω 1 . So ∀ 1 ≤ i , j ≤ n − 1 ,

Note that u ̲ = τ ρ 1 near ∂ Ω 1 , where ρ 1 is a given strictly plurisubharmonic function in a neighborhood Ω and τ a constant independent of ε and R as taken in Lemma 2.12. We also have

Let c 0 = min { c 1 , c 2 } (see (3.11) and (3.12)), we have

Then, we obtain

where C is a uniform constant. On the other hand, u n n ¯ ( 0 ) ≥ ∑ i = 1 n − 1 u α α ¯ ( 0 ) ≥ − C . In conclusion, we have ∣ u n n ¯ ( 0 ) ∣ ≤ C .

In conclusion, we obtain the uniform C 2 estimate.□