A C
2,α,β
 estimate for complex Monge–Ampère type equations with conic sigularities

Liding Huang; Gang Tian; Jiaxiang Wang

doi:10.1515/ans-2023-0113

Artikel Open Access

A C ^2,α,β estimate for complex Monge–Ampère type equations with conic sigularities

Liding Huang , Gang Tian und Jiaxiang Wang

Veröffentlicht/Copyright: 13. März 2024

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Abstract

In this paper, we give an alternative approach to the C ^2,α,β estimate for complex Monge-Ampère equations with cone singularities along simple normal crossing divisors.

Keywords: complex; Monge–Ampere equations; estimate; inverse Sobolev inequality

1 Introduction

In [1], the second-named author proves the existence of Kähler–Einstein metrics on Fano manifolds by using a continuity method through deforming conic angle. In one step in the proof, one needs a C ^2,α-type estimate for complex Monge–Ampère equations on complex spaces with conic angles along a smooth divisor. This was achieved in [1] and [2] by adopting the method developed in [3]. In this note, we will extend the arguments in [2] or [1] to complex Monge–Ampere equation with conic singularities in more general cases and obtain corresponding C ^2,α-type estimate.

Let D _i = {z ⁱ = 0} for i = 1, …, l be hyperplanes in C n = C l × C n − l . A standard conic metric with cone angles 2 π β ≔ 2 π β 1 , … , 2 π β l along D ₁, …, D _l is given by

ω β = − 1 ∑ i = 1 l | z i | 2 β i − 2 d z i ∧ d z i ̄ + ∑ j = l + 1 n d z j ∧ d z j ̄ ,

where β _i ∈ (0, 1) for each i. Let B _R(p) be a geodesic R-ball centered at p with respect to ω _β. We consider the equation

(1.1) ( − 1 ∂ ∂ ̄ u ) n = e h ( z ) ω β n ,

where h is a smooth function.

Locally, a Kähler–Einstein metric equation with cone angle 2π β , where β ≔ β 1 , … , β l and each β _i ∈ (0, 1)), is equal to solving (1.1) in a unit ball B ₁(o) for a point o. We call the metric − 1 ∂ ∂ ̄ u equivalent to w _β on B _R(o) if it satisfies

(1.2) K − 1 ω β ≤ − 1 ∂ ∂ ̄ u ≤ K ω β , x ∈ B R ( o ) ,

where K is a uniform constant.

It is important to establish a priori C ^2,α,β (see the definition (2.4)) estimate for solutions of (1.1) when one uses the continuity method and deforms the angles β. For the non-singular case, i.e. l = 0, such an estimate is simply the C ^2,α-estimate which had been obtained by using Calabi’s method (cf. [4]) and the Evans-Krylov method (see [5], [6]), an alternative proof was independently obtained in [3]. When l = 1 and 0 < β ₁≕β < 1, Jeffres-Mazzeo-Rubinstin [2] as well as Tian [7] applied the arguments in [3] to obtain the C ^2,α,β estimate. Tian’s method in [3] is to derive a local L ²-estimate for the third order derivative of solutions u. Then, as a consequence of Morrey’s lemma, they established a C ^2,α,β-estimate for solutions for any α ∈ ( 0,1 ) ∩ ( 0 , 1 β − 1 ) along a divisor. The estimate depends on α , inf Δ β h , ‖ h ‖ C 0,1 , β , K , β , where ‖ ⋅ ‖ C 0,1 , β denotes a Liptschitz norm with respect to ω _β. Huang [8] improved the L ² estimate of the third derivative and the C ^2,α,β estimate in Tian [7] so that it depends only on ‖ h ‖ C 0,1 , β .

There are other arguments for the C ^2,α,β estimate for solutions of (1.1). Brendle [9] proved the C ^2,α,β estimate for β < 1 2 by the estimate of the third order derivative. In [10], Guenancia-P a ̌ un applied the arguments of Evans-Krylov to obtain the C ^2,α,β estimate for some α and their estimate depends on the second order derivative of h. Note also that it is hard to compute α precisely in Evans-Krylov type arguments. Chen-Wang [11], [12] used the blow-up argument to establish the C ^2,α′^,β estimate for α′ < α under the condition u ∈ C ^2,α,β, which depends on ‖ h ‖ C α and J. Chu [13] improved Chen-Wang’s result such that α′ = α. Huang-Zhou [14] proved the properties of Green functions for conic linear elliptic equation and proved the C ^2,α,β estimate by the Green functions.

In this paper, we give an alternative approach by proving the L ² estimate of the third derivatives and the C ^2,α,β estimate along simple normal crossing divisors. More precisely, we use the integral method to prove

Theorem 1.1

Assume that u ∈ C ^1,1(B ₁(o) \ D)⋂PSH(B ₁(o)) is a solution of (1.1) which satisfies (1.2), where D = ∪_i D _i. Suppose that h ∈ C ¹(B ₁(o) \ D) and for some p ₀ > 2n, there exists A > 0,

‖ ∇ h ‖ L p 0 ( B 1 ( o ) , ω β n ) ≤ A , ‖ h ‖ L ∞ ≤ A .

Then, for any α < min 1 − 2 n p 0 , min i β i − 1 − 1 , there exists a r 0 ∈ ( 0 , 1 2 ) and C _α such that, for any x ∈ B 1 2 ( o ) and 0 < r ≤ r ₀, we have

(1.3) ∫ B r ( x ) | ∇ ( − 1 ∂ ∂ ̄ u ) | 2 ω β n ≤ C α r 2 n − 2 + 2 α ,

where C _α depends on n, α, K, β , A and ∇ is the Chern connection with respect to ω _β.

By the above theorem, we immediately obtain the following theorem.

Theorem 1.2

Let u ∈ C ^1,1(B ₁(o) \ D)⋂PSH(B ₁(o)) be a solution of (1.1) which satisfies (1.2). Suppose that h ∈ C ¹(B ₁(o) \ D) and for some p ₀ > 2n

‖ ∇ h ‖ L p 0 ( B 1 ( o ) , ω β n ) ≤ A , ‖ h ‖ L ∞ ≤ A .

Then, for any α ∈ 0 , min 1 − 2 n p 0 , min i β i − 1 − 1 , there exists C > 0 such that

[ − 1 ∂ ∂ ̄ u ] C α , β B 1 2 ( o ) ≤ C ,

where C depends on n, β , K, A, α.

Furthermore, for certain types of general fully nonlinear PDEs (see (4.1) in Section 4), with assumption (1.2), we can also use the same arguments to obtain the W ^3,2-estimate in the form of (1.3), and therefore the C ^2,α,β-estimate in terms of C ^1,1.

The organization of paper is as follows. In Section 2, we state some conventions and notations. In Section 3, we prove L ² estimate of the third derivative. In Section 4, we will briefly describe how to obtain the estimate for more general fully nonlinear PDEs.

2 Preliminary

2.1 Coordinate systems

For convenience, we introduce conic coordinates ( w 1 , … , w n ) ∈ C l × C n − l and denote the conic R-ball B β ( R ) by

B β ( R ) = ( w 1 , … , w n ) ∈ C l × C n − l ∑ j = 1 n | w j | 2 ≤ R 2 , w i = | w i | e − 1 θ i , − β i π ≤ θ i < β i π , 1 ≤ i ≤ l , w j = z j , l + 1 ≤ j ≤ n .

Then there is an isometric branch map ψ defined by

ψ : B R ( o ) → B β ( R ) ( z 1 , … , z n ) ⟼ ( w 1 , … , w n ) = ( z 1 ) β 1 β 1 , … , ( z l ) β l β l , z l + 1 , … , z n .

Then we can write the standard conic metric as

g β = ∑ j = 1 n d w i 2 .

This means

(2.1) B β ( r ) = ( z 1 , … , z l , z ′ ) ∑ i = 1 l | z i | 2 β i β i 2 + | z ′ | 2 ≤ R 2 ,

where z ′ = z l + 1 , … , z n and | z ′ | 2 = ∑ l + 1 n | z j | 2 .

If we use polar coordinates r i , θ i , z ′ , where r i = | z i | β i β i , z i = β i r i 1 β i e − 1 θ i , i = 1, …, l, and each θ _i ∈ [0, 2π], then

(2.2) g β = ∑ i = 1 l d r i 2 + β i 2 r i 2 d θ i 2 + ∑ j = l + 1 n | d z j | 2 .

Notice that in the polar coordinate system, we have

β min 2 g Euc ≤ g β ≤ g Euc ,

where β _min≔ min{β ₁, …, β _l}, and g Euc = ∑ i = 1 l d r i 2 + r i 2 d θ i 2 + ∑ j = l + 1 n | d z j | 2 is Euclidean metric in polar coordinate system.

Now, let us recall some definitions from [15]. For α < min β i − 1 − 1,1

(2.3) C , α , β ( Ω ) ≔ f sup x ≠ y | f ( x ) − f ( y ) | d ω β ( x , y ) α < + ∞ ,

where d ω β ( x , y ) is the distance between x and y with respect to ω _β. Furthermore, we denote

C 1 , α , β ( Ω ) ≔ u ∈ C , α , β ( Ω ) ∂ i β u ∈ C , α , β ( Ω ) ,

C 2 , α , β ( Ω ) ≔ u ∈ C 1 , α , β | ‖ u ‖ C 2 , α , β ( Ω ) < ∞ ,

where

(2.4) ‖ u ‖ C 2 , α , β ≔ ‖ u ‖ C 1 , α , β + ∑ i , j ‖ ∂ i β ∂ j ̄ β u ‖ C , α , β ( Ω ) .

The following lemma is similar to Lemma 2.3 in [7] and can be proved by standard arguments.

Lemma 2.1

There is a constant C _β, which depends on β such that for any smooth function v on B β ( 1 ) with boundary conditions:

(2.5) v w 1 , … , r i e − 1 2 π β i , … , w l , w ′ = e − 1 2 π ( 1 − β i ) v w 1 , … , r i , … , w l , w ′

for each 1 ≤ i ≤ l, we have

(2.6) ∫ B β ( 1 ) | v | 2 d w ∧ d w ̄ ≤ C β ∫ B β ( 1 ) | d v | 2 n n + 1 d w ∧ d w ̄ n + 1 n .

Moreover, if v ∈ L 2 ( B β ( 1 ) ) is a harmonic function on B β ( 1 ) , then for any 0 < r < 1, there is a constant C = C( β , n) such that

(2.7) ‖ d v ‖ L 2 ( B β ( r ) ) 2 ≤ C r 2 n − 4 + 2 β max − 1 ‖ d v ‖ L 2 ( B β ( 1 ) ) 2

where β _max≔ max{β ₁, …, β _l}.

Proof

Let’s see what the boundary condition (2.5) means. Recalling z i = β i w i 1 β i , we have

∂ v ∂ z i … , r i e − 1 2 π β i , … = ∂ v ∂ w i ∂ w i ∂ z i … , r i e − 1 2 π β i , … = e − 1 2 π ( 1 − β i ) ∂ v ∂ w i … , r i , … ⋅ β i w i 1 − β i − 1 = e − 1 2 π ( 1 − β i ) ∂ v ∂ w i ∂ w i ∂ z i … , r i , … ⋅ e − 1 2 π ( β i − 1 ) = ∂ v ∂ w i ∂ w i ∂ z i … , r i , … = ∂ v ∂ z i … , z i , … ,

which allows us to consider v as a function on the ball with standard conic metric. Then by standard Sobolev embedding, we get the first part of the lemma.

Let ζ i ≔ w i 1 − β i β i , which satisfy the boundary condition (2.5), and if we write v as a function of ζ ¹, …, ζ ^l, w ^l+1, …, w ⁿ, we can find that v ( … , ζ i e − 1 2 π ( 1 − β i ) , … ) = e − 1 2 π ( 1 − β i ) v ( … , ζ i , … ) . Hence we can extend v to be a function on a ball which solves the Laplace equation with respect to the Euclidean metric. Note that Δ ∂ v ∂ ζ i 2 ≥ 0 , and

∂ v ∂ ζ i = 1 − β i β i ∂ v ∂ w i ( w i ) 2 − 1 β i .

This implies,

(2.8) sup B β ( r ) ∂ v ∂ w i 2 ≤ C r 2 1 β i − 2 ∫ B β ( 1 ) ∂ v ∂ ζ i 2 d w ∧ d w ̄ ,

and thereby completing the proof.□

Next, we recall two elementary facts, which are shown in [3].

Lemma 2.2

(Divergence free [3], Lemma 2.1). For each j, ∑ i U i i j ̄ ≔ ∑ i ∂ U i j ̄ ∂ z i = 0 , where all derivatives are covariant with respect to ω _β.

Lemma 2.3

([3], Lemma 2.2). For any positive λ ₁, …, λ _n, we have

(2.9) λ λ k u k k ̄ − det u i j ̄ − ( n − 1 ) λ ≤ C ∑ i , j u i j ̄ − λ i δ i j 2 ,

where λ = λ ₁ … λ _n, and C is a constant depending only on λ _i and ‖ u i j ̄ ‖ ∞ .

3 The proof of the Theorem 1.1

In this section, we prove Theorem 1.1 by following the arguments in Section 2 of [7] (see also [3]).

In terms of the coordinate system w ¹, …, w ⁿ, (1.1) becomes

(3.1) det ∂ 2 u ∂ w i ∂ w ̄ j = e h ( z ) .

By standard arguments, we can simply suppose u ∈ C ⁴(B ₁(o) \ D)⋂PSH(B ₁(o)), whereas the constants in our derivation below depend only on ‖ u ‖ C 1,1 in (1.3). By direct computations, we can derive

(3.2) u i j ̄ u i j ̄ p q ̄ = u i l ̄ u k j ̄ u i j ̄ p u k l ̄ q ̄ + h p q ̄ .

Now, we write

∂ k β = z k 1 − β k ∂ ∂ z k , 1 ≤ k ≤ l , ∂ ∂ z k , l + 1 ≤ k ≤ n .

We denote by Δ_u the Laplace operator with respect to − 1 ∂ ∂ ̄ u and ∇^u the gradient operator with respect to − 1 ∂ ∂ ̄ u . Assume γ = (γ ¹ … γ ⁿ) is a unit vector, then we let u γ ≔ ∑ k = 1 k = n γ k ∂ k β u , and U i j ̄ = det ( u i j ̄ ) u i j ̄ , where ( u i j ̄ ) is the inverse matrix of ( u i j ̄ ) .

To establish the C ^2,α,β estimate, we need to use the Harnack inequality for u γ γ ̄ . For this we need some regularity of u γ γ ̄ , especially near the divisors D ₁, …, D _l. The next Lemma can be found in [14].

Lemma 3.1

([14], Lemma 6.1). Under the assumption of Theorem 1.1, u γ γ ̄ ∈ W loc 1,2 , β ( B 1 ( o ) ) .

Lemma 3.2

Under the assumption of Theorem 1.1, u γ γ ̄ is a subsolution of equation

(3.3) U i j ̄ u γ γ ̄ i j ̄ = e h h γ γ ̄ − | h γ | 2 , in B r ( o ) , for any r < 1 .

Proof

Denote w = u γ γ ̄ . By (3.2), we obtain

(3.4) e h u i j ̄ w i j ̄ − e h h γ γ ̄ + | h γ | 2 ≥ 0 .

Thus, by integration by parts, we obtain

(3.5) 0 ≤ ∫ B 1 ( o ) ( e h u i j ̄ w i j ̄ − e h h γ γ ̄ + e h | h γ | 2 ) χ δ η ω β n = ∫ B 1 ( o ) − e h u i j ̄ w i ( η χ δ ) j ̄ + e h h γ ( χ δ η ) γ ̄ + e h | h γ | 2 χ δ η ω β n .

Let δ → 0, the second line of above formula tends to

∫ B 1 ( o ) ( − e h u i j ̄ w i ( η ) j ̄ + e h h γ ( η ) γ ̄ + e h | h γ | 2 η ) ω β n .

By Lemma 3.1 and definition of subsolution, u γ γ ̄ is a subsolution.□

Next, we are going to show the following estimate in our setting.

Lemma 3.3

([7], Lemma 2.4). There is some q > 2, which may depend on β , ‖ u i j ̄ ‖ L ∞ , and ‖ ∇ h ‖ L 2 n , such that for any B _2r(y) ⊂ B ₁(o), we have

(3.6) r − 2 n ∫ B r ( y ) ( 1 + | ∇ 3 u | 2 ) q 2 ω β n 2 q ≤ C r − 2 n ∫ B r ( y ) ( 1 + | ∇ 3 u | 2 ) ω β n ,

where C denotes a uniform constant.

Proof

First we assume y = x. Define λ i j ̄ by

(3.7) λ i j ̄ = r − 2 n ∫ B r ( x ) u i j ̄ ω β n , i , j = 1 , … , n .

By using unitary transformations if necessary, we may assume λ i j ̄ = 0 for any i ≠ j and i, j ≥ l + 1. By Laplacian estimate (1.2), we have

K − 1 δ i j ≤ λ i j ̄ ≤ K δ i j ,

where I is the identity matrix. Choose a cut-off function η : B r ( x ) → R satisfying:

η ( z ) = 1 , ∀ z ∈ B 2 r 3 ( x ) , η ( z ) = 0 , ∀ z ∈ B r ( x ) \ B 5 r 6 ( x ) , | ∇ η | ≤ C r , | ∇ 2 η | ≤ C r 2 .

Using Lemma 2.2 and Lemma 3.2, we can deduce

(3.8) ∫ B r ( x ) U i j ̄ ∑ k = 1 n λ λ k u k k ̄ − det u p q ̄ − ( n − 1 ) λ η i j ̄ ω β n = ∫ B r ( x ) ∑ k = 1 n λ λ k U i j ̄ u k k ̄ i j ̄ η − det u p q ̄ U i j ̄ η i j ̄ ω β n ≥ ∫ B r ( x ) u i q ̄ u p j ̄ u i j ̄ k u p q ̄ k ̄ e h η ω β n + ∫ B r ( x ) e h h k k ̄ η − | h k | 2 ω β n − C r 2 n − 2 ≥ c ∫ B r 2 ( x ) | ∇ 3 u | 2 ω β n − ∫ B r ( x ) e h h k η k ̄ ω β n − C r 2 n − 2 ≥ c ∫ B r 2 ( x ) | ∇ 3 u | 2 ω β n − C r 2 n − 2 ,

where in the last inequality, we used

| ∫ B r ( x ) e h h k η k ̄ ω β n | ≤ C ‖ ∇ η ‖ L 2 n 2 n − 1 ‖ ∇ h ‖ L 2 n ≤ C r 2 n − 2 .

On the other hand, by Lemma 2.3, we can deduce

∫ B r 2 ( x ) | ∇ w | 2 ω β n ≤ C r 2 n − 2 + r − 2 ∫ B r ( x ) ∑ i , j = 1 n | u i j ̄ − λ i δ i j | 2 ω β n .

Using the Sobolev inequality Lemma 2.1, we get

∫ B r 2 ( x ) ( 1 + | ∇ 3 u | 2 ) ω β n ≤ r − 2 ∫ B r ( x ) ( 1 + | ∇ 3 u | 2 ) n n + 1 ω β n n + 1 n .

Then (3.6) follows from Gehring’s inverse Hölder inequality.□

Now, we are in a position to prove the following estimate. Note that the index 2 n 1 − 2 p 0 > 2 n − 2 in the following lemma.

Lemma 3.4

(Tian [7], Lemma 2.5). Under the assumption of Theorem 1.1, for any y ∈ B 1 2 ( o ) , B _4r(y) ⊂ B ₁(o) and σ < r, there is a q > 2 such that

(3.9) ∫ B σ ( y ) | ∇ 3 u | 2 ω β n − C r 2 n 1 − 2 p 0 ≤ C σ r 2 n − 4 + 2 β max − 1 + r ( 2 − 2 n ) ∫ B r ( y ) | ∇ 3 u | 2 ω β n q − 2 q ∫ B r ( y ) | ∇ 3 u | 2 ω β n ,

where q depends on n, K, ‖ h ‖ L ∞ , β and ‖ ∇ h ‖ L 2 n .

Proof

Let v = v i j ̄ i , j = 1 , … , n satisfy

(3.10) ∑ k = 1 n λ λ k v i j ̄ , k k ̄ = 0 , in B r ( x ) , v i j ̄ = u i j ̄ , on ∂ B r ( x ) .

Multiplying (3.10) by w ̂ i j ̄ ≔ u i j ̄ − v i j ̄ , we get

∫ B r ( x ) ∑ k = 1 n λ λ k ∂ v i j ̄ ∂ w k 2 ω β n ≤ ∫ B r ( x ) ∑ k = 1 n λ λ k ∂ u i j ̄ ∂ w k 2 ω β n .

This implies

(3.11) ∫ B r ( x ) | ∇ v i j ̄ | 2 ω β n ≤ C ∫ B r ( x ) | ∇ u i j ̄ | 2 ω β n .

Multiplying (3.2) by w ̂ p q ̄ and by integration by parts, we have

− ∫ B r ( x ) u i j ̄ u i p q ̄ w ̂ p q ̄ j ̄ ω β n ≤ ∫ B r ( x ) u i l ̄ u k j ̄ u i j ̄ p u k l ̄ q ̄ w ̂ p q ̄ − ∫ B r ( x ) w ̂ p q ̄ p h q ̄ .

Note that

∫ B r ( x ) w ̂ p q ̄ p h q ̄ ω β n ≤ 1 2 ∫ B r ( x ) | ∇ w ̂ p q ̄ | ω β n + 1 2 ∫ B r ( x ) | ∇ h | 2 ω β n ≤ 1 2 ∫ B r ( x ) | ∇ w ̂ p q ̄ | ω β n + C r 2 n − 2 .

Hence, we can deduce,

(3.12) ∫ B r ( x ) | ∇ w ̂ p q ̄ | 2 ω β n ≤ C r 2 n 1 − 2 p 0 + ∫ B r ( x ) | w ̂ | + | u i j ̄ − λ i δ i j | 2 | ∇ u i j ̄ | 2 ω β n .

By the Poincaré inequality, we have

(3.13) ∫ B r ( x ) | u i j ̄ − λ i δ i j | 2 | ∇ u i j ̄ | 2 ω β n

Without loss of generality, we may assume that q ≥ 2(q − 2). Since w ̂ i j ̄ vanishes on ∂B _r(x), its L ²-norm is controlled by ‖ ∇ w ̂ i j ̄ ‖ L 2 , and therefore, by ‖ ∇ u i j ̄ ‖ L 2 . Then we have

∫ B r ( x ) | w ̂ i j ̄ | ⋅ | ∇ u i j ̄ | 2 ω β n ≤ r 2 n r − 2 n ∫ B r ( x ) | ∇ u i j ̄ | q ω β n 2 q ⋅ r − 2 n ∫ B r ( x ) | w ̂ i j ̄ | q q − 2 ω β n q − 2 q ≤ C r − 2 n ∫ B r ( x ) | w ̂ i j ̄ | q q − 2 ω β n q − 2 q ∫ B r ( x ) 1 + | ∇ u i j ̄ | 2 ω β n ≤ C r 2 − 2 n ∫ B r ( x ) | ∇ w ̂ i j ̄ | q q − 2 ω β n q − 2 q ∫ B r ( x ) 1 + | ∇ u i j ̄ | 2 ω β n ≤ r 2 − 2 n ∫ B r ( x ) | ∇ u i j ̄ | q q − 2 ω β n q − 2 q ∫ B r ( x ) 1 + | ∇ u i j ̄ | 2 ω β n .

Now we apply Lemma 2.1 to our v i j ̄ above. We can deduce,

(3.15) ∫ B σ ( x ) | ∇ v i j ̄ | 2 ω β n ≤ C σ r 2 n − 4 + 2 β max − 1 ∫ B r ( x ) | ∇ v i j ̄ | 2 ω β n .

We observe,

∫ B σ ( x ) | ∇ u i j ̄ | 2 ω β n ≤ 2 ∫ B r ( x ) | ∇ w ̂ i j ̄ | 2 ω β n + 2 ∫ B σ ( x ) | ∇ v i j ̄ | 2 ω β n .

Hence, by (3.11) and (3.15), we get

(3.16) ∫ B σ ( x ) | ∇ u i j ̄ | 2 ω β n ≤ ∫ B r ( x ) 2 | ∇ w ̂ i j ̄ | 2 + C σ r 2 n − 4 + 2 β max − 1 | ∇ u i j ̄ | 2 ω β n ,

thereby completing the proof.□

Lemma 3.5

For any ɛ ₀ > 0, there is an m ∈ N* which satisfies that for any r ̃ > 0 and B r ̃ (y) ⊂ B ₁(0), there is a r ∈ [2^−m r ̃ , r ̃ ] such that

(3.17) r 2 − 2 n ∫ B r ( y ) | ∇ w | 2 ( − 1 ∂ ∂ ̄ u ) n ≤ ε 0 ,

where m depends on ϵ 0 , K , ‖ h ‖ W 1 , p 0 .

Proof

By directly calculating, we have

(3.18) Δ u △ u = ∑ k u i q ̄ u p j ̄ u i j ̄ k u p q ̄ k ̄ + Δ h ,

where Δ_u is the Laplacian operator with respect to − 1 ∂ ∂ ̄ u , Δ is the Laplacian operator with respect to ω _β. We assume that η ≥ 0 satisfying

η = 1 x ∈ B r 2 ( y ) 0 x ∈ B r ( y ) \ B 3 r 4 ( y ) ,

| ∇ η | ≤ C r , and | Δ u η | ≤ C r 2 .

By (1.2) and (3.18), we have

(3.19) ∫ B r ( y ) η | ∇ w | 2 ω β n ≤ C ∫ B r ( y ) η ∑ k u i q ̄ u p j ̄ u i j ̄ k u p q ̄ k ̄ ( − 1 ∂ ∂ ̄ u ) n = C ∫ B r ( y ) η △ u △ u − η Δ h ( − 1 ∂ ∂ ̄ u ) n = C ∫ B r ( y ) η △ u ( △ u − M r ) − η Δ h ( − 1 ∂ ∂ ̄ u ) n ≤ C ∫ B r ( y ) | △ u η | ( M r − △ u ) + < ∇ η , ∇ h > + η | ∇ h | ω β 2 ( − 1 ∂ ∂ ̄ u ) n ≤ C r 2 n 1 − 2 p 0 + C r − 2 ∫ B r ( y ) ( M r − △ u ) ω β n ,

where ∇_u is the gradient operator with respect to − 1 ∂ ∂ ̄ u , M r = sup B r ( y ) △ u . By the weak Harnack inequality, we obtain

(3.20) r − 2 n ∫ B r ( y ) ( M r − △ u ) ω β n ≤ C inf B r 2 ( y ) ( M r − △ u ) + r 2 − 4 n p 0 .

Therefore,

r 2 − 2 n ∫ B r ( y ) | ∇ ( − 1 ∂ ∂ ̄ u ) | 2 ω β n ≤ C M r − M r 2 + r 2 − 4 n p 0 .

If the formula (3.17) is not true, we choose r = r ̃ , 2⁻¹ r ̃ , …, 2^−k r ̃ , so

k ϵ 0 ≤ C M r ̃ − M 2 − k r ̃ + 2 r ̃ 2 − 4 n p 0 .

When k is sufficient large, we get a contradiction.□

In the following, we give the proof of Theorem 1.1.

The proof of Theorem 1.1. We denote H ( r ) = ∫ B r ( y ) | ∇ w | 2 ω β n , σ = λr where λ < 1. In addition, we choose r satisfying Lemma 3.5. By Lemma 3.4 and 3.5, we have, for any α < α ′ ∈ 0 , min 1 − 2 n p 0 , min i β i − 1 − 1 ,

(3.21) σ 2 − 2 n H ( σ ) − C r 2 − 4 n p 0 λ 2 − 2 n ≤ C σ r 2 n − 4 + 2 β max − 1 σ 2 − 2 n + σ 2 − 2 n r 2 − 2 n H ( r ) q − 2 q H ( r ) = C λ 2 β max − 1 − 2 r 2 − 2 n + λ 2 − 2 n r 2 − 2 n r 2 − 2 n H ( r ) q − 2 q H ( r ) ≤ C λ 2 β max − 1 − 2 + ε 0 λ 2 n − 2 r 2 − 2 n H ( r ) .

Therefore, we conclude,

(3.22) σ 2 − 2 n H ( σ ) + σ 2 α ′ ≤ C λ 2 β max − 1 − 2 + ε 0 λ 2 n − 2 r 2 − 2 n H ( r ) + C r 2 − 4 n p 0 − 2 α ′ λ 2 − 2 n + λ 2 α ′ r 2 α ′ .

We choose λ, ϵ ₀, r ₀ such that

θ ≔ 2 ( 1 + C ) λ 2 α ′ < 1 ,

ϵ 0 ≤ 2 − 1 C − 1 θ λ 2 n − 2 ,

and

C r 0 2 − 4 n p 0 − 2 α ′ λ 2 − 2 n + λ 2 α ′ ≤ θ .

Then for r < r ₀, we have

σ 2 − 2 n H ( σ ) + σ 2 α ′ ≤ θ ( r 2 − 2 n H ( r ) + r 2 α ′ ) .

By the iteration as in [16, Lemma 8.23], we obtain (1.3).

4 General fully nonlinear equations

For more general types of fully nonlinear PDEs, we can also prove Theorem 1.1 under condition (1.2). Denote A [ u ] ≔ ω β − 1 − 1 ∂ ∂ ̄ u , which is positively bounded from below and above under condition (1.2). Consider the equation

(4.1) F ( A [ u ] ) = h ( z , u ) ,

where h(z, t) is a smooth function with respect to the variable z, t. F(A) = f(λ ₁, …, λ _n) is a smooth symmetric function of eigenvalues of A, which is defined on an open symmetric cone Γ ⊂ R n , with vertex at the origin, and containing Γ n ≔ { λ 1 > 0 , … , λ n > 0 } ⊂ R n . We also suppose f satisfy:

∂ f ∂ λ i > 0 for each 1 ≤ i ≤ n, f is concave in Γ, and ∂ 2 f ∂ λ i ∂ λ j < 0 in Γ n ≔ { λ 1 > 0 , … , λ n > 0 } ⊂ R n ;
sup_∂Γ f < inf h;
f is homogeneous of degree 1 and f > 0;
∑ j = 1 n ∂ ∂ z ̄ j ∂ F ( A [ u ] ) ∂ u i j ̄ = 0 .

Then we can change the inequality in Lemma 2.3 to be

(4.2) ∑ i ∂ f ∂ λ i u i i ̄ − f ( λ ) ≤ C 1 + ∑ i , j | u i j ̄ − λ i δ i j | 2 .

Differentiating (4.1) twice we can get,

(4.3) ∂ F ( A [ u ] ) ∂ u i j ̄ u i j ̄ p p ̄ + ∂ 2 F ( A [ u ] ) ∂ u i j ̄ ∂ u k l ̄ u i j ̄ p u k l ̄ p ̄ = h p p ̄ + h t p u p ̄ + h t p ̄ u p + h t t u p u p ̄ + h t u p p ̄ .

The right-hand side is bounded by (1.2). By assumption (1), we have

∂ 2 F ( A [ u ] ) ∂ u i j ̄ ∂ u k l ̄ u i j ̄ p u k l ̄ p ̄ < − c | ∇ 3 u | 2 = − c ( K ) | ∇ 3 u | 2 < 0 ,

and inequality (3.8) is still valid in this setting. Then, we can prove the C ^2,α,β-estimate in terms of K in the same way. The estimate depends on ‖ h t t ‖ L ∞ , ‖ u ‖ L ∞ , ‖ ∇ u ‖ L ∞ . Here h _t(z, t) means the partial derivative with respect to t. As an example, we can choose f ( λ ) = σ k ( λ ) 1 k = ∑ 1 ≤ i 1 < … < i k ≤ n λ i 1 … λ i k 1 k , i.e., the corresponding equation is the complex Hessian equation

(4.4) ( − 1 ∂ ∂ ̄ u ) k ∧ ω β n − k = e h ( z , u ) ω β n − k .

Corresponding author: Gang Tian, BICMR and SMS, Peking University, Beijing 100871, P.R. China, E-mail: gtian@math.pku.edu.cn

Research ethics: Not applicable.
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: The authors states no conflict of interest.
Research funding: None declared.
Data availability: Not applicable.

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Received: 2023-05-19

Accepted: 2023-10-15

Published Online: 2024-03-13

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2023-0113

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complex; Monge–Ampere equations; estimate; inverse Sobolev inequality

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A C 2,α,β estimate for complex Monge–Ampère type equations with conic sigularities

Artikel

Abstract

1 Introduction

Theorem 1.1

Theorem 1.2

2 Preliminary

2.1 Coordinate systems

Lemma 2.1

Proof

Lemma 2.2

Lemma 2.3

3 The proof of the Theorem 1.1

Lemma 3.1

Lemma 3.2

Proof

Lemma 3.3

Proof

Lemma 3.4

Proof

Lemma 3.5

Proof

4 General fully nonlinear equations

References

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Artikel in diesem Heft

Artikel in diesem Heft

A C ^2,α,β estimate for complex Monge–Ampère type equations with conic sigularities