A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure

Zhengmao Chen

doi:10.1515/ans-2022-0068

Article Open Access

A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic L_p Minkowski problem for log-concave measure

Zhengmao Chen

Published/Copyright: June 13, 2023

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From the journal Advanced Nonlinear Studies Volume 23 Issue 1

Abstract

In the present article, we prove the existence and uniqueness of smooth solutions to an anisotropic L p Minkowski problem for the log-concave measure. Our proof of the existence is based on the well-known continuous method whose crucial factor is the a priori bounds of an auxiliary problem. The uniqueness is based on a maximum principle argument. It is worth mentioning that apart from the C 2 bounds of solutions, the C 1 bounds of solutions also need some efforts since the convexity of S cannot be used directly, which is one of great difference between the classical and the anisotropic versions. Moreover, our result can be seen as an attempt to get new results on the geometric analysis of log-concave measure.

Keywords: log-concave measure; anisotropy; L p Minkowski problem; Monge-Ampère equation; the continuous method

MSC 2010: 35J96; 53C42

1 Introduction

The main focus of this article is on the integral geometry of log-concave measure.

We first provide the definition of log-concave measure.

Definition A.1 (Log-concave measure (see [11,37])). A measure μ is called log-concave if its density d μ ( x ) d x is log-concave, i.e., d μ ( x ) d x = e − φ ( x ) for some convex function φ , which means that

(1.1) μ ( E ) = ∫ E e − φ ( x ) d x

for every Borel set E ⊆ R n + 1 and some convex function φ .

Since the constant value function φ ≡ 0 is convex, the standard Lebesgue measure is the most trivial example for log-concave measure. Besides this example, there are many other important examples for the log-concave measure, which are listed as follows.

Examples A.2 (Log-concave measure).

(i) Gauss measure. The ( n + 1 ) -dimensional Gauss measure is defined as follows:

(1.2) d γ n = 1 ( 2 π ) n + 1 2 e − ∣ x ∣ 2 2 d x ,

which characterizes the Gaussian generalized random processes in stochastic analysis (see [17], pp. 246–261).

(ii) Gibbs measure of some nonlinear Schrödinger equation. The Gibbs measure P ( d u ) of some nonlinear Schrödinger equation is defined as follows:

(1.3) P ( d u ) = e − H ( u ) d u ,

where

(1.4) H ( u ) = 1 2 ∫ R n ( ∣ ∇ u ∣ 2 + V ( x ) u 2 ( x ) ) d x .

That is, H ( u ) is the Hamilton functions for the Schrödinger equation with unit mass and positive potential function V ,

(1.5) i ∂ t u = − Δ u + V ( x ) u ,

(see a similar description of [14]).

It may be interesting to mention that some of classical concepts and results in integral geometry have been generalized to the log-concave measure, such as the support function, mean width, and Steiner-type formulas, [3,26,37]. Moreover, the convexity of φ can be used to deduce some interesting geometric inequalities for the measure d μ , such as Brunn-Minkowski inequality, Prékopa-Leindler inequalities, or Blaschke-Santaló inequalities, see [4,6,11,13,16,38]. With the help of these geometric inequalities, it is natural to pose the L p Minkowski problem for log-concave measure, see [11,12,24,25,28,37]. In a united way, the works [11,12,24,28] can be formulated in the following way:

Problem A.3 (Minkowski-type problem). For any fixed n ≥ 1 and p ∈ R , given any Borel measure ϕ ( x ) d x that is supported in N ⊆ R n + 1 , find a convex function h such that

(1.6) ( ∇ h ) ♯ ( ϕ ( x ) d x ) = h 1 − p e − φ ( ∣ y ∣ 2 ) d y .

In particular, if N = S n , φ vanishes, h is the support function of a hypersurface M ⊆ R n + 1 , Problem A.3 is associated with the following classical Minkowski problem that were introduced by Schneider [38] and Lutwak [29] for p = 1 and general p , respectively.

Problem A.4 (Classical L p Minkowski problem). For any fixed n ≥ 1 and p ∈ R , given any Borel measure μ that is supported on the unit sphere S n , under what conditions, there exists a (unique) convex hypersurface M ⊆ R n + 1 such that

(1.7) ν ˜ ♯ ( S ˜ 1 − p d σ ) = d μ ( ξ ) ,

where ν ˜ , S ˜ , and d σ are the standard unit normal mapping, support function, and surface measure of M , respectively.

It is easy to see that in smooth frame, suppose that d μ is absolutely continuous with respect to the spherical Lebesgue measure d ξ and

(1.8) d μ d ξ = ϕ ( ξ ) ,

then the Problem A.4 is equivalent to the following Monge-Ampère equation on S n :

(1.9) S ˜ 1 − p κ ˜ = ϕ ( ξ ) ,

where κ ˜ is the standard Gaussian curvature of M . It follows from (1.9) that Problem A.1 is equivalent to a geometric problem prescribed the reciprocal of Gaussian curvature. In particular, if p = 1 , Problem A.4 was posed and solved by Minkowski [31,32] for the discrete measure or the measure with continuous density. Aleksandrov, Fenchel, and Jensen [38] extended the works of Minkowski [31,32] to the general Borel measure independently by the approximation argument. By the theory of Monge-Ampère equation, Lewy [27], Nirenberg [33], Cheng and Yau [9], Pogorelov [35], and Caffarelli [7,8] resolved the classical problem. For general p , Problem A.4 was posed and solved by Lutwak [29]. For more interesting results on Problem A.4, see. e.g., [5,10,30]. One of the advances to Problem A.4 is to analyze some similar problems when one may replace the surface measure d σ by other geometric measures deduced by Steiner formulas, such as k -surface area measure, k -curvature measure, integral Gaussian curvature, and their L p or dual versions, see [19,21–23,29,34,38] and so on.

If N = R n + 1 , Problem A.3 is the so-called L p Minkowski problem for log-concave measure, see [11,12,25,37]. If e − φ ( ∣ y ∣ 2 ) = 1 ( 2 π ) n + 1 2 e − ∣ y ∣ 2 2 , N = S n , and h is the support function of a convex hypersurface M ⊆ R n + 1 , Problem A.3 is L p Minkowski problem for Gaussian measure, see [24,28].

In the point of view of the development of geometric analysis, the main focus of the this article is to analyze Problem A.3 when the target geometry N enjoys more interesting metric structure.

Among them, there is an interesting metric space, which can be called the anisotropic version of classical metric space. Recently, some interesting geometric and analysis results have been extended to the anisotropic frame, such as the Moser-Trudinger inequalities [40,43], Brunn-Minkowski inequality for Finsler-Laplacian [39], geometric flows [1,15,36,42], and so on.

The main focus of the this article is on Problem A.3 in the frame of anisotropy, which can be stated as follows.

Problem A.5 (Anisotropic L p Minkowski for the log-concave measure). For any fixed n ≥ 1 and p ∈ R , find a strictly convex domain M ⊆ R n + 1 with anisotropic normal mapping ν , such that

(1.10) ν ♯ ( S 1 − p e − φ ( ρ 2 ) d σ ) = ϕ ( ξ ) d ξ , ∀ ξ ∈ W ,

where W ⊆ R n + 1 is a Wulff shape, S and ρ are the anisotropic support function and anisotropic radial function of a strictly convex hypersurface M ⊆ R n + 1 , respectively, and ϕ is a fixed smooth convex function.

In smooth case, equation (1.10) is equivalent to the following prescribed anisotropic Gaussian curvature problem on Wulff shape W ⊆ R n ,

(1.11) S 1 − p e − φ ( ρ 2 ) K aniso = ϕ ( ξ ) ,

where K aniso is the anisotropic Gaussian curvature. Direct calculus show that equation (1.11) is equivalent to the following Monge-Ampère equation in Wulff shape W ⊆ R n + 1 :

(1.12) S 1 − p e − φ ( ρ 2 ) det S i j − 1 2 Q i j k S k + δ i j S = ϕ ( ξ )

for any ξ ∈ W , where S and ρ are the anisotropic support function and anisotropic radial function of a strictly convex hypersurface M ⊆ R n + 1 , respectively, and φ is a smooth convex function (see Lemma 2.5 in Section 2).

The left-hand side of equation (1.12) is called the density of the L p anisotropic surface area measure for log-concave measure e − φ ( ∣ x ∣ 2 ) d x in the present article.

In particular, if p = 1 and the factor e − φ ( ∣ x ∣ 2 ) vanishes, Problem A.5 was first posed and solved by Xia [41], which can be stated as follows:

Theorem A.6[41]. For any fixed n ≥ 1 and 0 < ϕ ∈ C 4 ( W ) , then there exists a (unique) strictly convex function S such that

(1.13) det S i j − 1 2 Q i j k S k + δ i j S = ϕ ( ξ )

and

(1.14) 0 < c − 1 ≤ ‖ S ‖ C 2 , τ ( W ) ≤ c < ∞ .

The main focus of this article is on the a priori bounds, existence, and uniqueness of smooth solutions to Problem A.5 for general p and log-concave measure e − φ ( ∣ x ∣ 2 ) d x , and the main result can be stated as follows.

Theorem 1.1

For any fixed n ≥ 1 and p > n + 1 , there exist positive constants c and τ and a positive solution S ∈ C 2 , τ ( W ) to the equation (1.12) satisfying

(1.15) 0 < c − 1 ≤ ‖ S ‖ C 2 , τ ( W ) ≤ c < ∞ ,

where τ ∈ ( 0 , 1 ) , c is independent of S, φ : ( 0 , ∞ ) ↦ ( 0 , ∞ ) and ϕ : W ↦ ( 0 , ∞ ) , and the following conditions hold:

(A.1.) 0 < ϕ ∈ C 4 ( W ) , φ is a nonnegative, radially symmetric, increasing, smooth, and convex function in W , 0 < φ ∈ C 4 ( R ) , and

(1.16) ‖ ϕ ‖ C 4 ( W ) + ‖ φ ‖ C 4 ( R ) < ∞ .

(A.2.)

(1.17) lim t → ∞ t n + 1 − p e φ ( t 2 ) = 0 , lim t → 0 t n + 1 − p e φ ( t 2 ) = ∞ .

(A.3.) There exists a δ 0 > 0 such that

(1.18) min t > 0 φ ′ ( t ) ≥ δ 0 > 0 .

Our proof of uniqueness part is based on delicate analysis of the linearized problem to problem (1.12) and a Maximum principle. The proof of existence part is based on the powerful continuous method. We let the set of the positive continuous function on W be C + ( W ) and

(1.19) C = S ∈ C 2 , τ ( W ) : S i j − 1 2 Q i j l S l + δ i j S n × n is positive definite .

The main ingredient is the a priori bounds of solutions to the following auxiliary problem for any S ∈ C :

(1.20) S 1 − p e − φ ( ρ 2 ) det S i j ( ξ ) − 1 2 Q i j k S k + δ i j S ( ξ ) = t ϕ ( ξ ) + ( 1 − t ) e − φ ( 1 )

for t ∈ [ 0 , 1 ] , where ρ 2 = ∣ ∇ S ∣ 2 + S 2 .

Remark 1.2

It may be worth mentioning that the convexity of S cannot be used to deduce the C 1 bounds due to the presence of the term 1 2 Q i j k S k , which is a major difference between the classical and anisotropic Minkowski problems. Similar difficulties also arise in the prescribed curvature measure, which is to the following fully nonlinear equation on S n :

(1.21) σ k λ − ρ i j + 2 ρ ρ i ρ j + δ i j ρ = u ( ρ 2 + ∣ ∇ ρ ∣ 2 ) n + 1 2 ϕ ( ξ ) .

However, by some transforms v = 1 ρ , the matrix − ρ i j + 2 ρ ρ i ρ j + δ i j ρ becomes 1 v 2 ( v i j + δ i j v ) . Then, the C 1 bounds of v can be deduced by the convexity of v , see [20]. By v ρ = 1 , we can obtain C 1 bounds of ρ ; similar technique can also be referred to [2]. However, such an idea cannot be used here, and therefore, we need to find a good test function to obtain C 1 bounds.

Remark 1.3

Since there is no Brunn-Minkowski-type inequality for the log-concave measure in the frame of anisotropy, our proof of uniqueness follows from the delicate analysis of the linearized problem to problem (1.12), which is motivated by Huang and Zhao [23]. It may be interesting to prove the Brunn-Minkowski-type inequality for the log-concave measure in the frame of anisotropy and give a direct proof for the uniqueness.

This rest part of the present article is arranged as follows: in Section 2, we recall some knowledge on anisotropic differential and convex geometry; in Section 3, we prove the a priori bounds of S to the problem (1.20); and in Section 4, we prove Theorem 1.1.

2 Anisotropic differential and convex geometry

In Section 2, we list some basic differential geometry and convex geometry, which are needed in the present article and can be referred to [1,15,36,41].

Definition 2.1

[41] A function F : R n + 1 ↦ [ 0 , ∞ ) is called a Minkowski norm if

F is a norm of R n + 1 , i.e., F is convex, 1-homogeneous function satisfying F ( x ) > 0 when x ≠ 0 ;
F ∈ C ∞ ( R n + 1 ⧹ { 0 } ) ;
F satisfies a uniformly elliptic condition in the sense that there exists λ and Λ such that 1 ≤ Λ λ < ∞ ,
(2.1) λ ∣ ζ ∣ 2 ≤ ∂ 2 ∂ x i ∂ x j 1 2 F 2 ( x ) ζ i ζ j ≤ Λ ∣ ζ ∣ 2
for any ζ = ( ζ 1 , ζ 2 , … , ζ n + 1 ) ∈ R n + 1 .

Definition 2.2

[41] The dual norm of F is defined as follows:

(2.2) F 0 ( ξ ) = sup x ≠ 0 x ⋅ ξ F ( x )

for any ξ ∈ R n + 1 .

Lemma 2.3

[41]

For any x , ξ ∈ R n + 1 ,
(2.3) Σ i ∂ F ∂ x i ( x ) x i = F ( x ) , Σ j ∂ F 0 ∂ ξ j ( ξ ) ξ j = F 0 ( ξ ) ,
For any x , ξ ∈ R n + 1 ⧹ { 0 } ,
(2.4) Σ i ∂ 2 F ∂ x i ∂ x j ( x ) x i = 0 = Σ i ∂ 2 F 0 ∂ ξ i ∂ ξ j ( ξ ) ξ i
for any fixed j ∈ { 1 , 2 , … , n + 1 } ,
For any x , ξ ∈ R n + 1 ⧹ { 0 } ,
(2.5) F 0 ( D F ( x ) ) = 1 = F ( D F 0 ( ξ ) ) ,
For any x , ξ ∈ R n + 1 ⧹ { 0 } ,
(2.6) F ( x ) D F 0 ( D F ( x ) ) = x , F 0 ( ξ ) D F ( D F 0 ( ξ ) ) = ξ ,
where
(2.7) D F = ∂ F ∂ x 1 , … , ∂ F ∂ x n + 1 , D F 0 = ∂ F 0 ∂ ξ 1 , … , ∂ F 0 ∂ ξ n + 1 .

Definition 2.4

[41] We let F is a Minkowski norm defined in Definition 2.1. A Wulff shape W ⊆ R n + 1 is a subset of R n + 1 , which is defined as follows:

(2.8) W = { x ∈ R n + 1 : F ( x ) = 1 } .

The anisotropic unit outer normal is defined as follows:

(2.9) ν ≜ ∇ F 0 ( ν ˜ ) ,

where ν ˜ is the standard unit outer normal and F 0 is the so-called dual norm of F .

Lemma 2.5

[41]

The metric G associated with the norm F is defined as follows:
(2.10) G ( x ) ( ξ , η ) ≜ Σ i j G i j ( x ) ξ i η j = Σ i j ∂ 2 ∂ x i ∂ x j 1 2 F 2 ( x ) ξ i η j
for any x ∈ R n + 1 and ξ , η ∈ T x R n + 1 . We let g = G ( ν ) ∣ T x M for any x ∈ M .
The anisotropic support function of a strictly convex hypersurface M is defined as follows:
(2.11) S ( ξ ) = sup y ∈ M G ( ξ ) ( ξ ⋅ y )
for any ξ ∈ R n + 1 .
The anisotropic radial function ρ : W ↦ M of M is defined as follows:
(2.12) ρ ( ξ ) = ∇ ( R n + 1 , G ) S = Σ i n ∇ e i S + S e 0 ,
where { e i } i = 1 n is a local orthonormal frame field with respect to g on W . Furthermore,
(2.13) ρ 2 ( ξ ) = ∣ ∇ S ∣ 2 ( ξ ) + S 2 ( ξ )
for any ξ ∈ W .
The anisotropic Gaussian curvature of M satisfies
(2.14) 1 K aniso = det S i j − 1 2 Q i j l S l + δ i j S ,
where
(2.15) Q i j l ( x ) = ∂ 3 ∂ x i ∂ x j ∂ x l 1 2 F 2 ( x )
for any fixed i , j , l ∈ { 1 , 2 , … , n + 1 } and x ∈ R n + 1 .

Remark 2.6

If F ( x ) = ∣ x ∣ , this is the classical norm in Euclidean space and the Wulff shape W is the sphere S n , G i j = δ i j and S ( x ) = sup y ∈ M x ⋅ y .

More interesting differential geometric theory on the Wulff shape can be referred to [1,15,36,41].

3 A priori bounds of S

In Section 3, we consider the a priori bounds of solutions to the Monge-Ampère equation (1.12) on W .

We let the set of the positive continuous function on W be C + ( W ) and

C = S ∈ C 2 , τ ( W ) : S i j − 1 2 Q i j l S l + δ i j S n × n is positive definite .

This main result of this section can be stated as follows.

Theorem 3.0

For any fixed n ≥ 1 and p > n + 1 , we let S ∈ C ∩ C + ( W ) be a solution to (1.12). Suppose that the conditions (A.1)–(A.3) hold. Then, there exists a positive constant c, independent of S, such that

(3.1) 0 < c − 1 ≤ ‖ S ‖ C 2 , τ ( W ) ≤ c < ∞ ,

where τ ∈ ( 0 , 1 ) .

Now, we divide the proof of Theorem 3.0 into the following lemmas.

Lemma 3.1

For any fixed n ≥ 1 and p > n + 1 , we let S ∈ C ∩ C + ( W ) be a solution to (1.12). Suppose that the conditions (A.1)–(A.3) hold. Then, there exists a positive constant c such that

(3.2) 0 < c − 1 ≤ S ( ξ ) ≤ c < ∞ ∀ ξ ∈ W .

Proof

We consider the following extremal problem:

(3.3) R = max ξ ∈ W S ( ξ ) .

It follows from the compactness of W and the continuity of S that there exists ξ 1 ∈ W such that

(3.4) R = S ( ξ 1 ) .

It follows from the equation (1.12) that at the point ξ = ξ 1 ,

(3.5) R n + 1 − p e φ ( R 2 ) ≥ ϕ ( ξ 1 ) ≥ min ξ ∈ W ϕ ( ξ ) > 0 .

Combining this and condition (A.2) that there exists a positive constant c > 0 such that

(3.6) R ≤ c < ∞ .

We next consider the following extremal problem:

(3.7) r = min ξ ∈ W S ( ξ ) .

Adopting a similar argument, we also see that there exists a positive constant c > 0 such that

(3.8) r ≥ c > 0 .

Equations (3.6) and (3.8) yield the desired conclusion of Lemma 3.1.□

Before getting the estimates of high-order term of S , for any fixed i , j ∈ { 1 , 2 , … , n } , we let

(3.9) u i j = S i j − 1 2 Q i j k S k + δ i j S .

Then, equation (1.12) becomes

(3.10) det ( u i j ) = ϕ ( ξ ) S p − 1 e φ ( ρ 2 ) ≜ B , ∀ ξ ∈ W .

Lemma 3.2

(3.11) 0 ≤ ∣ ∇ S ( ξ ) ∣ ≤ c ∀ ξ ∈ W .

Proof

The proof is based on maximum principle. We let G = e − 2 α S v = e − 2 α S ∣ ∇ S ∣ 2 , where α > 0 to be chosen. Suppose that sup G is achieved at the point ξ = ξ 3 ∈ W . Then, at ξ = ξ 3 ,

(3.12) 0 = G i = 2 e − 2 α S ( Σ l S l S l i − α S i v )

for any fixed i ∈ { 1 , 2 , … , n } and ( G i j ) n × n is nonpositive. Direct calculation deduces that

G i j = 2 e − 2 α S ( Σ l S l j S l i + Σ l S l S l i j − α v S i j − 2 α Σ l S l S l j S i )

at the point ξ = ξ 3 for any fixed i , j ∈ { 1 , 2 , … , n } . For any fixed i , j ∈ { 1 , 2 , … , n } , we let u i j be the function defined in (3.9) and

(3.13) F i j = ∂ ∂ u i j det ( u i j ) .

It is easy to see that ( F i j ) n × n is positive. Therefore, we have

(3.14) 0 ≥ Σ i j F i j ( Σ l S l j S l i + Σ l S l S l i j − α v S i j − 2 α Σ l S l S l j S i ) = Σ i = 1 4 I i

at the point ξ = ξ 3 , where

(3.15) I 1 = Σ i j l F i j S l i S l j , I 2 = Σ i j l F i j S l S l i j ,

and

(3.16) I 3 = − α v Σ i j F i j S i j , I 4 = − 2 α Σ i j l F i j S l S l j S i .

Without loss of generalization, we may assume that v ( ξ 3 ) = ∣ ∇ S ( ξ 3 ) ∣ 2 ≫ 1 . Otherwise, inequality (3.11) is trivial.

By choosing suitable coordinate, we may assume that

(3.17) S i = δ i 1 v

at the point ξ = ξ 3 for any fixed i ∈ { 1 , 2 , … , n } . This means that ( F i j ) n × n is diagonal at the point ξ = ξ 3 . Moreover, it follows from (3.12) and (3.17) that

(3.18) S 1 i = α v δ i 1

for any fixed i ∈ { 1 , 2 , … , n } .

We now obtain the bounds of Σ i = 1 4 I i . At the first step, we first analyze the term I 1 = Σ i j l F i j S l i S l j . It is easy to see that

(3.19) I 1 = Σ i j l F i j S l i S l j = F 11 S 11 2 = α 2 F 11 v 2

at the point ξ = ξ 3 .

We next estimate the term I 2 = Σ i j l F i j S l S l i j . Since det ( u i j ) = B , we have

(3.20) Σ i F i i u i i t = B t

for any fixed t ∈ { 1 , 2 , … , n } . Therefore, multiplying S t on both sides of (3.20) and taking sum for the index t , we obtain

(3.21) Σ i t F i i S t u i i t = Σ t B t S t .

By Ricci identity,

(3.22) Σ t i S t ( S t i i − S i i t ) = Σ i t R i t S i S t = R 11 v ≥ 0

at the point ξ = ξ 3 due to the convexity of W . Combining (3.17), (3.21), and (3.22), we have

(3.23) I 2 = Σ i F i i S 1 S 1 i i = v R 11 Σ l F l l + Σ i F i i S 1 ( S i i ) 1 = v R 11 Σ l F l l + Σ i F i i S 1 u i i + 1 2 Q i i 1 S 1 − S 1 = v R 11 Σ l F l l + Σ i F i i S 1 ( u i i ) 1 + Σ i F i i S 1 1 2 Q i i 1 S 1 − S 1 = v R 11 Σ i F i i + B 1 S 1 + Σ i F i i S 1 1 2 Q i i 1 S 1 − S 1 = v R 11 Σ i F i i + I 21 + I 22 ≥ I 21 + I 22 ,

where

(3.24) I 21 = B 1 S 1 , I 22 = Σ i F i i S 1 1 2 Q i i 1 S 1 − S 1

We first estimate the term I 21 = B 1 S 1 . By the definition of B , we have

(3.25) B 1 S 1 = ( e φ ( ρ 2 ) S p − 2 ( ( p − 1 ) ϕ ( ξ ) S 1 + S ϕ 1 ) + 2 e φ ( ρ 2 ) S p − 1 ϕ ( ξ ) φ ′ ( ρ 2 ) ( S S 1 + S 1 S 11 ) ) S 1 ≜ B 1 , 1 S 1 + 2 e φ ( ρ 2 ) S p − 1 ϕ ( ξ ) φ ′ ( ρ 2 ) S 1 2 S 11 ,

where

(3.26) B 1 , 1 S 1 = e φ ( ρ 2 ) S p − 2 ( ( p − 1 ) ϕ ( ξ ) S 1 2 + S S 1 ϕ 1 ) + 2 e φ ( ρ 2 ) S p − 1 ϕ ( ξ ) φ ′ ( ρ 2 ) S S 1 2 .

It is easy to see that

(3.27) ∣ B 1 , 1 S 1 ∣ ≤ c ( 1 + v ) ,

which means that

(3.28) B 1 , 1 S 1 ≥ − c ( 1 + v ) .

Noting that

(3.29) S 1 2 S 11 = α v 2 ,

we have

(3.30) 2 e φ ( ρ 2 ) S p − 1 ϕ ( ξ ) φ ′ ( ρ 2 ) S 1 2 S 11 = 2 α e φ ( ρ 2 ) φ ′ ( ρ 2 ) S p − 1 ϕ ( ξ ) v 2 .

Since φ is increasing, we can see that

(3.31) e φ ( ρ 2 ) S p − 1 ϕ ( ξ ) ≥ e φ ( 0 ) S p − 1 ϕ ( ξ ) ≥ δ 5 ,

where δ 5 = e φ ( 0 ) min ξ ∈ W { S p − 1 ( ξ ) ϕ ( ξ ) } > 0 . It follows from condition (A.3) and (3.30) that

(3.32) 2 e φ ( ρ 2 ) S p − 1 ϕ ( ξ ) φ ′ ( ρ 2 ) S 1 2 S 11 ≥ 2 α δ 0 δ 5 v 2 .

Therefore, putting (3.28) and (3.30) into (3.25), we obtain

(3.33) I 21 = B 1 S 1 ≥ 2 α δ 0 δ 5 v 2 − c v − c

for sufficiently large v ( ξ 3 ) .

We next deal with the term I 22 = Σ i F i i S 1 1 2 Q i i 1 S 1 − S 1 . It follows from (3.17) and (3.18) that

(3.34) I 22 = Σ i F i i S 1 1 2 Q i i 1 S 1 − S 1 = Σ i F i i 1 2 Q i i 11 S 1 2 + 1 2 Q i i 1 S 1 S 11 − S 1 2 ≥ c − c v 3 2

at the point ξ = ξ 3 .

Therefore, putting (3.33) and (3.34) into (3.23), we obtain

(3.35) I 2 ≥ 2 α δ 0 δ 5 v 2 − c v − v 3 2 − c ≥ α δ 0 δ 5 v 2 − c

for sufficiently large v ( ξ 3 ) .

We next estimate the term I 3 = − α v Σ i j F i j S i j . It is easy to see that

(3.36) I 3 = − α v Σ i j F i j S i j = − α v Σ i j F i i u i i + 1 2 Q i i 1 S 1 − S ≥ − c v 3 2 − c ,

at the point ξ = ξ 3 .

We next estimates the term I 4 = − 2 α Σ i j l F i j S l S l j S i . It follows from (3.17) and (3.18) that

(3.37) I 4 = − 2 α Σ i j l F i j S l S l j S i = − 2 α F 11 S 1 2 S 11 = − 2 α 2 F 11 v 2 .

Therefore, it follows from (3.19), (3.35), (3.36), and (3.37) that

(3.38) 0 ≥ Σ i = 1 4 I i ≥ α ( δ 0 δ 5 − α F 11 ) v 2 − c v 3 2 − c

at the point ξ = ξ 3 . Since δ 0 δ 5 > 0 and F 11 is bounded and positive, we choose α > 0 such that

(3.39) δ 0 δ 5 4 ≤ α F 11 ≤ δ 0 δ 5 2 .

Therefore,

(3.40) 0 ≥ ( δ 0 δ 5 ) 2 8 F 11 v 2 − c v 3 2 − c ≥ ( δ 0 δ 5 ) 2 16 max ξ ∈ W F 11 v 2 − c

for sufficiently large v ( ξ 3 ) , and thus, there exists a constant c , depends only on p , n , φ , ϕ , such that

(3.41) v 2 ≤ c

for sufficiently large v ( ξ 3 ) . This completes the proof of Lemma 3.2.

By the definition of u i j , we let

(3.42) G ( u i j ) = ( det u i j ) 1 n

and

(3.43) ψ = ( ϕ ( ξ ) S p − 1 e φ ( ρ 2 ) ) 1 n .

Then, equation (1.12) becomes

(3.44)□ G ( u i j ) = ψ ( ξ ) .

Lemma 3.3

For any fixed n ≥ 1 and p > n + 1 , we let S ∈ C ∩ C + ( W ) be a solution to (1.12) and u i j be the function defined in (3.9). Suppose that the conditions ( A . 1 ) – ( A . 3 ) hold. Then, there exists a positive constant c such that

(3.45) Δ u ≤ c .

Proof

The proof is also based on maximum principle. We let

(3.46) H = Σ i u i i .

Suppose that H achieves its maximum at the point ξ = ξ 4 . Without loss of generality, we may assume that ( H i j ) n × n is diagonal at the point ξ = ξ 4 . Therefore, at the point ξ = ξ 4 ,

(3.47) H j = 0

for any fixed j ∈ { 1 , 2 , … , n } , and ( H i j ) n × n is nonpositive at the point ξ = ξ 4 . For any fixed i , j , s , t ∈ { 1 , 2 , … , n } , we let

(3.48) G i j = ∂ G ∂ u i j , G i j , r s = ∂ 2 G ∂ u i j ∂ u r s .

Therefore, at the point ξ = ξ 4 ,

(3.49) 0 ≥ Σ i j G i j H i j = Σ i G i i H i i .

By the commutator identity, we have

(3.50) H i i = Δ u i i − n u i i + H .

Putting (3.50) into (3.49), we obtain

(3.51) 0 ≥ Σ i G i i Δ u i i − n Σ i G i i u i i + H Σ i G i i .

Taking the l th partial derivatives on both sides of (3.44) twice for any fixed l ∈ { 1 , 2 , … , n } , we have

(3.52) Σ i j G i j u i j l = ψ l , Σ i j s t G i j , r s u i j l u r s l + Σ i j G i j ( u i j ) l l = ψ l l

for any fixed l ∈ { 1 , 2 , … , n } . It follows from the concavity of G that

(3.53) Σ i j s t l G i j , r s u i j l u r s l ≤ 0 .

This implies that

(3.54) Σ i G i i Δ u i i ≥ Σ i j s t l G i j , r s u i j l u r s l + Σ i j G i j Δ u i j = Δ ψ

at the point ξ = ξ 4 .

It follows from Newton-MacLaurin inequality that

(3.55) Σ i G i i ≥ 1 ,

see [21]. Putting (3.54) and (3.55) into (3.51), we have

(3.56) 0 ≥ Δ ψ − n ψ + H Σ i G i i ≥ Δ ψ − n ψ + H ≥ Δ ψ − n ψ

at the point ξ = ξ 4 .

Now, we claim that

(3.57) Δ ψ ψ ≥ Σ i = 1 4 I i ≥ δ 0 Σ i j S i j 2 − c Σ i j S i j 2 − c

at the point ξ = ξ 4 , where δ 0 = min t > 0 φ ′ ( t ) > 0 .

Indeed, noting ρ 2 = ∣ ∇ S ∣ 2 + S 2 , for any fixed l ∈ { 1 , 2 , … , n } , taking l th partial derivatives on both sides of (3.43) twice, we have

(3.58) n ψ l ψ = ( log ϕ ) ′ + ( p − 1 ) S l S + 2 φ ′ ( ρ 2 ) ( Σ j S j S j l + S S l )

and

(3.59) n Δ ψ ψ − n ∣ ∇ ψ ∣ 2 ψ 2 = n Σ l ψ ψ l l − ψ l 2 ψ 2 = Σ l ( log ϕ ) ″ + ( p − 1 ) S S l l − S l 2 S 2 + 2 φ ′ ( ρ 2 ) ( Σ j S j l 2 + S j S j l l + S S l l + S l 2 ) + 4 φ ″ ( ρ 2 ) ( Σ j S j S j l + S S l ) 2 ≜ Σ i = 1 4 T i ,

where

(3.60) T 1 = n ( log ϕ ) ″ + 1 − p S 2 + 2 φ ′ ( ρ 2 ) + 4 φ ″ ( ρ 2 ) S 2 ∣ ∇ S ∣ 2 ,

(3.61) T 2 = p − 1 S + 2 φ ′ ( ρ 2 ) S Δ S + 8 φ ″ ( ρ 2 ) S Σ j l S j S l S j l = Σ j l p − 1 S + 2 φ ′ ( ρ 2 ) S δ j l + 8 φ ″ ( ρ 2 ) S S j S l S j l ,

(3.62) T 3 = 2 φ ′ ( ρ 2 ) Σ j l S j S j l l ,

and

(3.63) T 4 = 2 φ ′ ( ρ 2 ) Σ j , α S j α 2 + 4 φ ″ ( ρ 2 ) Σ j ( Σ i j S i S i j ) 2 .

Now, we claim that

(3.64) n Δ ψ ψ ≥ δ 0 Σ i j S i j 2 − c Σ i j S i j 2 − c

at the point ξ = ξ 4 .

We first obtain some estimates of T 1 . Noting that φ , ϕ ∈ C 4 , it follows from Lemmas 3.1 and 3.2 that

(3.65) ∣ T 1 ∣ ≤ c ,

and therefore,

(3.66) T 1 ≥ − c

at the point ξ = ξ 4 .

Moreover, it follows from Lemma 3.1 and Hölder inequality that

(3.67) Σ j α p − 1 S + 2 φ ′ ( ρ 2 ) S δ j α + 8 φ ″ ( ρ 2 ) S S j S α S j α ≤ c ∣ Σ j α S j α ∣ ≤ c Σ i j S i j 2 ,

which means that

(3.68) T 2 ≥ − c Σ i j S i j 2 .

It follows from Ricci identity that

(3.69) Σ i j S j S j i i = Σ i j S j S i i j + Σ i j R i j S i S j .

Therefore, we have

(3.70) ∣ 2 φ ′ ( ρ 2 ) Σ i j S j S j i i ∣ ≤ c ∣ Σ i j S j S i i j ∣ + c ∣ Σ i j R i j S i S j ∣ .

It follows from Lemma 3.2 that

(3.71) c ∣ Σ i j R i j S i S j ∣ ≤ c ∣ ∇ S ∣ 2 ≤ c .

It follows from (3.9) that

(3.72) Σ i j S j S i i j = Σ j S j ( Δ S ) j = Σ j S j ( Δ u ) j + Σ j S j 1 2 Q i j l S l − S j .

From (3.47), we have

(3.73) Σ j S j ( Δ u ) j = 0

at the point ξ = ξ 4 . Direct calculus shows that

(3.74) Σ j S j 1 2 Q i j l S l − S j = Σ j S j 1 2 Q i j l j S l − S j + Σ j S j 1 2 Q i j l S l j .

It follows from Lemma 3.2 and Hölder inequality that

(3.75) Σ j S j 1 2 Q i j l j S l − S j ≤ c

and

(3.76) Σ l j S j 1 2 Q i j l S l j ≤ c ∣ Σ l j S l j ∣ ≤ c n Σ l j S l j 2 .

Therefore,

(3.77) Σ j S j 1 2 Q i j l S l − S j ≤ c + c n Σ l j S l j 2 .

Putting (3.77) and (3.73) into (3.72), we have

(3.78) ∣ Σ i j S j S i i j ∣ = Σ j S j ( Δ u ) j + Σ j S j 1 2 Q i j l S l − S j ≤ c + c n Σ l j S l j 2 .

It follows from (3.78), (3.69), and (3.70) that

(3.79) T 3 = 2 φ ′ ( ρ 2 ) Σ i j S j S j i i ≥ − c − c n Σ l j S l j 2 .

Since φ ∈ C 2 is convex, we see that φ ″ ( ρ 2 ) ≥ 0 , and therefore,

(3.80) 4 φ ″ ( ρ 2 ) Σ j ( Σ i j S i S i j ) 2 ≥ 0 .

Combining (3.80) and (3.63), we obtain

(3.81) T 4 ≥ 2 φ ′ ( ρ 2 ) Σ j , α S j α 2 ≥ δ 0 Σ i j S i j 2 ,

where δ 0 = min t > 0 φ ′ ( t ) > 0 .

Equations (3.66), (3.68), (3.79), and (3.81) yield

(3.82) Δ ψ ψ ≥ Σ i = 1 4 T i ≥ δ 3 Σ i j S i j 2 − c Σ i j S i j 2 − c

at the point ξ = ξ 4 . This is the desired inequality (3.57).

By (3.56) and (3.57), we see

(3.83) δ 3 Σ i j S i j 2 − c Σ i j S i j 2 − c ≤ n

at the point ξ = ξ 4 . From (3.83), we can see that

(3.84) Σ i j S i j 2 ≤ c .

By the Hölder inequality, we have

(3.85) Δ S = ∣ Σ i S i i ∣ ≤ n Σ i S i i 2 ≤ n Σ i j S i j 2 ≤ c .

Noting that

(3.86) u i i = S i i − 1 2 Q i i j S j + S

we can conclude from equations (3.85) and (3.86) and Lemmas 3.1 and 3.2 that

(3.87) Δ u ≤ c

at the point ξ = ξ 4 . This completes the proof of Lemma 3.3.□

Now, we are in a position to prove Theorem 3.0.

Final proof of Theorem 3.0

It follows from (1.12) that equation (3.44) becomes

(3.88) ℱ ( u i j ) = 0 ,

provided ℱ ( u i j ) = G ( u i j ) − ψ . We let ℱ i j = ∂ ℱ ∂ u i j . It follows from Lemmas 3.1–3.3 that there exist positive constants λ and Λ , independent of S , such that

(3.89) 1 ≤ Λ λ < ∞ ,

and

(3.90) 0 < λ ζ 2 ≤ ℱ i j ζ i ζ j ≤ Λ ζ 2 ,

for any ζ = ( ζ 1 , ζ 2 , … , ζ n ) ∈ R n . That is, (3.88)

is elliptic uniformly.

Moreover, it is easy to see that G = ( det ) 1 n is concave with respect to ( u i j ) n × n , and therefore,

ℱ is concave with respect to ( u i j ) n × n .

Then, it follows from Theorem 17.14 of Gilbarg and Trudinger [18] that there exist τ 1 ∈ ( 0 , 1 ) and positive constant c such that

(3.91) ‖ u ‖ C 2 , τ 1 ( W ) ≤ c ,

(see [18], pp. 457–461). Therefore, there exist τ ∈ ( 0 , 1 ) and positive constant c such that

(3.92) ‖ S ‖ C 2 , τ ( W ) ≤ c .

This is the desired conclusion of Theorem 3.0.

4 Existence and uniqueness

This section devotes the proof of Theorem 1.1.

4.1 Part one: Uniqueness

This subsection devotes the proof of the uniqueness part of Theorem 1.1.

We let the set of the positive continuous function on W be C + ( W ) . For any ζ ∈ C , we let

(4.1) F ( S ) = det ( S i j − 1 2 Q i j l S l + δ i j S ) , J ( S ) = e − φ ( ρ 2 ) S 1 − p ,

(4.2) M ( S ) = F ( S ) J ( S ) ,

and

(4.3) M [ S ] ( ζ ) = d d ε M ( S + ε ζ ) ε = 0 .

Lemma 4.1

For any fixed n ≥ 1 and p > n + 1 . Suppose that S ∈ C ∩ C + ( W ) is a solution to (4.1). For any ζ ∈ C , we let M [ S ] ( ζ ) be the operator defined in (4.3) and

(4.4) M [ S ] ( ζ ) = a i j ζ S i j + b i ζ S i + C ζ S .

Suppose that the conditions (A.1)–(A.3) hold. Then, ( a i j ) n × n is positive b i is bounded, and C < 0 .

Proof

Taking logarithm on both sides of (4.2), we obtain

(4.5) M [ S ] ( ζ ) M ( S ) = 1 − p S ζ − 2 φ ′ ( ρ 2 ) ( S ζ + ∇ S ⋅ ∇ ζ ) + P i j B ( ζ ) ,

where ( P i j ) n × n is the inverse of the matrix S i j − 1 2 Q i j l S l + S δ i j n × n and

(4.6) B ( ζ ) = ζ i j − 1 2 Q i j l ζ l + ζ δ i j .

We let ζ = S v . Direct calculation shows that

(4.7) ζ i = S v i + S i v

and

(4.8) ζ i j = S v i j + ( S i v j + S j v i ) + S i j v .

Therefore, we obtain

(4.9) S ζ + ∇ S ⋅ ∇ ζ = ( S 2 + ∣ ∇ S ∣ 2 ) v + S ∇ S ⋅ ∇ v = ρ 2 v + S ∇ S ⋅ ∇ v ,

which implies that

(4.10) 1 − p S ζ − 2 φ ′ ( ρ 2 ) ( S ζ + ∇ S ⋅ ∇ ζ ) = ( 1 − p − 2 φ ′ ( ρ 2 ) ρ 2 ) v − 2 φ ′ ( ρ 2 ) S ∇ S ⋅ ∇ v .

It follows from (4.6) and (4.10) that

(4.11) B ( ζ ) = S v i j + ( S i v j + S j v i ) + S i j − 1 2 Q i j l S l + S δ i j v − 1 2 Q i j l S v l ,

and thus,

(4.12) P i j B ( ζ ) = S P i j v i j + 2 P i j S i v j + n v − 1 2 S Σ l P i j Q i j l v l

due to the symmetry of ( P i j ) n × n . Putting (4.10) and (4.12) into (4.5), we have

(4.13) G [ S ] ( v ) = M [ S ] ( v ) = S M ( S ) P i j v i j + ( 2 M ( S ) P i j S j − 1 2 S M ( S ) P t l Q t l i − 2 φ ′ ( ρ 2 ) M ( S ) S S i ) v i + ( n + 1 − p − 2 φ ′ ( ρ 2 ) ρ 2 ) M ( S ) v ≜ a i j v i j + b i v i + N v ,

where

(4.14) a i j = S M ( S ) P i j ,

(4.15) b i = 2 M ( S ) P i j S j − 1 2 S M ( S ) P t l Q t l i − 2 φ ′ ( ρ 2 ) M ( S ) S S i ,

and

(4.16) N = ( n + 1 − p − 2 φ ′ ( ρ 2 ) ρ 2 ) M ( S ) .

Since S ¯ , M ( S ¯ ) > 0 , ( P ¯ i j ) n × n is positive, we see that ( a i j ) n × n is positive. It follows from Lemma 3.2 that b i is bounded.

It follows from condtion (A.3) that

(4.17) φ ′ ( ρ 2 ) > 0 , φ ′ ( ρ 2 ) ρ 2 > 0

for any ξ ∈ W . Noting that M ( S ) is positive, we have

(4.18) − 2 φ ′ ( ρ 2 ) ρ 2 M ( S ) < 0 .

If p > n + 1 , we obtain C < 0 . This completes the proof of Lemma 4.1.□

Lemma 4.2

For any fixed n ≥ 1 and p > n + 1 . Suppose that S ∈ C ∩ C + ( W ) is a solution to (1.12). For any ζ ∈ C , we let M [ S ] ( ζ ) be the operator defined in (4.3) and

(4.19) M [ S ] ( ζ ) = 0 .

Suppose that the conditions (A.1)–(A.3) hold, then

(4.20) ζ ≡ 0 .

Proof

For any ζ ∈ C such that

(4.21) M [ S ] ( ζ ) = 0 ,

it follows from Lemma 4.1 and strong maximum principle of elliptic equations of second order that

(4.22) ζ ≡ 0 ,

see [18]. This is the desired conclusion of Lemma 4.2.□

Proposition 4.3

For any fixed n ≥ 1 and p > n + 1 , we let S 1 , S 2 ∈ C ∩ C + ( W ) be two solutions of (1.12). Suppose that the conditions (A.1)–(A.3) hold. Then,

(4.23) S 1 = S 2 .

Proof

Without loss of generality, we may assume that there exists

(4.24) t ≥ 1

such that

(4.25) t S 1 ( ξ ) ≥ S 2 ( ξ ) , t S 1 ( ξ 0 ) = S 2 ( ξ 0 ) ,

for any ξ ∈ W and some ξ 0 ∈ W . Since t ≥ 1 and φ is increasing, we obtain

(4.26) e φ ( ρ 2 ) ≤ e φ ( ( t ρ ) 2 ) .

For any solution S to (1.12), it is easy to see that

(4.27) F ( t S ) = t n F ( S ) = t n S p − 1 e φ ( ρ 2 ) ϕ ( ξ ) = t n + 1 − p ( t S ) p − 1 e φ ( ρ 2 ) ϕ ( ξ ) ≤ t n + 1 − p ( t S ) p − 1 e φ ( ( t ρ ) 2 ) ϕ ( ξ ) .

Therefore,

(4.28) F ( t S 1 ) J ( t S 1 ) ≤ t n + 1 − p ϕ ( ξ ) ≤ ϕ ( ξ )

due to the assumption that p > n + 1 and t ≥ 1 . Therefore,

(4.29) M ( t S 1 ) = F ( t S 1 ) J ( t S 1 ) ≤ ϕ ( ξ ) = M ( S 2 ) ,

which means that

(4.30) 0 ≥ M ( t S 1 ) − M ( S 2 ) = ∫ 0 1 d d ε M ( ( ε t S 1 ) + ( 1 − ε ) S 2 ) d ε ≜ a i j ( ξ ) t S 1 − S 2 S 2 i j + b i ( ξ ) t S 1 − S 2 S 2 i + C ( ξ ) t S 1 − S 2 S 2

for any ξ ∈ W . It follows from Lemma 4.1 that ( a i j ) n × n and − C are positive. Then, by strong maximum principle of elliptic equations of second order, we have

(4.31) t S 1 ( ξ ) − S 2 ( ξ ) S 2 ( ξ ) ≤ 0

for any ξ ∈ W , see [18]. This, together with (4.25) , implies that

(4.32) t S 1 ( ξ ) = S 2 ( ξ ) ,

and therefore,

(4.33) ρ 2 ( ξ ) = t ρ 1 ( ξ ) ≥ ρ 1 ( ξ )

for any ξ ∈ W . Therefore,

(4.34) F ( t S 1 ) = t n F ( S 1 ) = t n S 1 p − 1 e φ ( ρ 1 2 ) ϕ ( ξ ) = t n + 1 − p ( t S 1 ) p − 1 e φ ( ρ 1 2 ) ϕ ( ξ ) ≤ t n + 1 − p S 2 p − 1 e φ ( ρ 2 2 ) ϕ ( ξ ) = t n + 1 − p F ( S 2 ) = t n + 1 − p F ( t S 1 ) ,

which means that

(4.35) t ≤ 1

since p > n + 1 and the positivity of F . It follows from (4.24) and (4.34) that

(4.36) t = 1 .

By the arbitrariness of S 1 and S 2 , we obtain the desired equation (4.23) and this completes the proof of Proposition 4.3.□

4.2 Part two: Existence

This subsection devotes the proof of the existence part of Theorem 1.1.

Motivated by [19,21,23] and so on, we consider the following auxiliary problem with a parameter t ∈ [ 0 , 1 ] ,

(4.37) S 1 − p e − φ ( ρ 2 ) det ( S i j ( ξ ) − 1 2 Q i j l S l + δ i j S ( ξ ) ) = t ϕ ( ξ ) + ( 1 − t ) e − φ ( 1 ) ≜ f t , ∀ ξ ∈ W ,

where ρ 2 = ∣ ∇ S ∣ 2 + S 2 and 0 < ϕ ∈ C 2 ( W ) .

We let the set of the positive continuous function on W be C + ( W ) and

C = S ∈ C 2 , τ ( W ) : S i j − 1 2 Q i j l S l + δ i j S n × n is positive definite

(4.38) ℐ = { t ∈ [ 0 , 1 ] : S ∈ C ∩ C + ( W ) , (4.37) is solvable } .

Since f t ∈ C 2 ( W ) satisfying

0 < min { e − φ ( 1 ) , min ξ ∈ W ϕ ( ξ ) } ≤ f t ( ξ ) ≤ max { e − φ ( 1 ) , max ξ ∈ W ϕ ( ξ ) } < ∞ ∀ ξ ∈ W

for any t ∈ [ 0 , 1 ] , adopting some similar arguments in Section 3, we obtain

Lemma 4.4

For any fixed n ≥ 1 , p > n + 1 , and t ∈ [ 0 , 1 ] , we let S t ∈ C ∩ C + ( W ) be a solution of (4.37). Suppose that the conditions (A.1)–(A.3) hold. Then, there exists a constant c , independent of t , such that

0 < c − 1 ≤ ∣ S t ∣ C 2 , τ ( W ) ≤ c ,

for any t ∈ [ 0 , 1 ] and some τ ∈ ( 0 , 1 ) .

As a corollary of Lemma 4.4, we have

Corollary 4.5

For any fixed n ≥ 1 , p > n + 1 , and t ∈ [ 0 , 1 ] , we let ℐ is the set defined in (4.38). Suppose that the conditions (A.1)–(A.3) hold. Then, ℐ is closed.

Proof

It suffices to show that for any sequence { t j } j = 1 ∞ ⊆ ℐ satisfying

t j → t 0 ,

as j → ∞ for some t 0 ∈ [ 0 , 1 ] , we need to prove t 0 ∈ ℐ .

We let S j be a solution of problem (4.37) at t = t j . It follows from the conclusion of Lemma 4.4 that there exists a positive constant c , independent of j , such that

‖ S j ‖ C 2 , τ ( W ) ≤ c .

It follows from Ascoli-Arzela theorem that up to a subsequence, there exists a S 0 ∈ C 2 , τ ( W )

‖ S j − S 0 ‖ C 2 , τ ( W ) → 0

as j → ∞ . It is easy to see that

(4.39) S j 1 − p → S 0 1 − p , e − φ ( ρ j 2 ) → e − φ ( ρ 0 2 )

uniformly on W as j → ∞ , where ρ j 2 = S j 2 + ∣ ∇ S j ∣ 2 for any j ∈ { 0 , … } . Letting j → ∞ , we can see that ( t 0 , S 0 ) is a solution to the following problem:

(4.40) S 1 − p e − φ ( ρ 2 ) det S i j ( ξ ) − 1 2 Q i j l S l + δ i j S ( ξ ) = t ϕ ( ξ ) + ( 1 − t ) e − φ ( 1 ) ∀ ξ ∈ W .

Equation (4.40) implies that t 0 ∈ ℐ . This is the desired conclusion of Corollary 4.5.□

Lemma 4.6

For any fixed n ≥ 1 , p > n + 1 , and t ∈ [ 0 , 1 ] , we let ℐ is the set defined in (4.38). Suppose that the conditions (A.1)–(A.3) hold. Then, ℐ is open.

Proof

Suppose that there exists a t ¯ ∈ ℐ , it suffices to prove t ∈ ℐ for any t ∈ B δ ( t ¯ ) ∩ [ 0 , 1 ] . To achieve this goal, joint with implicit function theorem, we need to analyze the kernel of linearized equation associated with (4.37). We assume that S ¯ is a solution to (4.37) at t = t ¯ . For any ζ ∈ W , we let

(4.41) M ( S ) = e − φ ( ρ 2 ) S 1 − p det ( S i j − 1 2 Q i j l S l + δ i j S ) , f t = t ϕ ( ξ ) + ( 1 − t ) e − φ ( 1 ) ,

(4.42) G ( t , S ¯ ) = M ( S ¯ ) − f t , M [ S ¯ ] ( ζ ) = d d ε M ( S ¯ + ε ζ ) ε = 0 ,

and

(4.43) G [ S ¯ ] ( ζ ) = d d ε G ( S ¯ + ε ζ ) ε = 0 = d d ε M ( S ¯ + ε ζ ) ε = 0 .

By (4.37), we have

(4.44) M ( S ¯ ) = f t .

Taking logarithm on both sides of (4.44), since f t is independent of S ¯ , we obtain

(4.45) M ′ [ S ¯ ] ( ζ ) M ( S ¯ ) = 1 − p S ¯ ζ − 2 φ ′ ( ρ 2 ) ( S ¯ ζ + ∇ S ¯ ⋅ ∇ ζ ) + P ¯ i j B ( ζ ) ,

where ( P ¯ i j ) n × n is the inverse of the matrix S ¯ i j − 1 2 Q i j l S l + S ¯ δ i j n × n and

(4.46) B ( ζ ) = ζ i j − 1 2 Q i j l ζ l + ζ δ i j .

We let ζ = S ¯ v . Direct calculation shows that

(4.47) ζ i = S ¯ v i + S ¯ i v

and

(4.48) ζ i j = S ¯ v i j + ( S ¯ i v j + S ¯ j v i ) + S ¯ i j v .

Therefore, we obtain

(4.49) S ¯ ζ + ∇ S ¯ ⋅ ∇ ζ = ( S ¯ 2 + ∣ ∇ S ¯ ∣ 2 ) v + S ¯ ∇ S ¯ ⋅ ∇ v = ρ ¯ 2 v + S ¯ ∇ S ¯ ⋅ ∇ v ,

which implies that

(4.50) 1 − p S ¯ ζ − 2 φ ′ ( ρ ¯ 2 ) ( S ¯ ζ + ∇ S ¯ ⋅ ∇ ζ ) = ( 1 − p − 2 φ ′ ( ρ ¯ 2 ) ρ ¯ 2 ) v − 2 φ ′ ( ρ ¯ 2 ) S ¯ ∇ S ¯ ⋅ ∇ v .

It follows from (4.48) and (4.46) that

(4.51) B ( ζ ) = S ¯ v i j + ( S ¯ i v j + S ¯ j v i ) + S ¯ i j − 1 2 Q i j l S ¯ l + S ¯ δ i j v − 1 2 S Σ l Q i j l v l ,

and thus,

(4.52) P ¯ i j B ( ζ ) = S ¯ P ¯ i j v i j + 2 P ¯ i j S ¯ i v j + n v − 1 2 S ¯ Σ l P i j Q i j l v l

due to the symmetry of ( P ¯ i j ) n × n . Putting (4.50) and (4.52) into (4.45), we have

(4.53) G [ S ¯ ] ( v ) = M [ S ¯ ] ( v ) = S ¯ M ( S ¯ ) P ¯ i j v i j + M ( S ¯ ) ( 2 P ¯ i j S ¯ j − 1 2 S ¯ P t l Q t l i − 2 φ ′ ( ρ ¯ 2 ) S ¯ S ¯ i ) ∇ v + ( n + 1 − p − 2 φ ′ ( ρ ¯ 2 ) ρ ¯ 2 ) M t v ≜ a i j v i j + b i v i v l + N v ,

where

(4.54) a i j = S ¯ M ( S ¯ ) P ¯ i j , b i = M ( S ¯ ) 2 P ¯ i j S ¯ j − 1 2 S ¯ P t l Q t l i − 2 φ ′ ( ρ ¯ 2 ) S ¯ S ¯ i

and

(4.55) N = ( n + 1 − p − 2 φ ′ ( ρ ¯ 2 ) ρ ¯ 2 ) M ( S ¯ ) .

Since S ¯ , M ( S ¯ ) > 0 , ( P ¯ i j ) n × n is positive, we see that ( a i j ) n × n is positive. It follows from Lemma 4.4 that b i and S ¯ M ( S ¯ ) P i j Q i j l are bounded. It follows from condition (A.3) that

(4.56) φ ′ ( ρ ¯ ) > 0 ,

and therefore,

(4.57) φ ′ ( ρ ¯ ) ρ ¯ 2 > 0

for any ξ ∈ W . Noting that M ( S ¯ ) is positive, we have

(4.58) − 2 φ ′ ( ρ 2 ) ρ ¯ 2 M ( S ¯ ) < 0 .

If p > n + 1 , we obtain N < 0 . By strong maximum principle for elliptic equations of second order, we see that

(4.59) v ≡ 0

(see [18], pp. 35) and thus,

(4.60) ζ ≡ 0

since S ¯ > 0 . Then, by the standard Implicit Function Theorem, for any t ∈ B δ ( t ¯ ) ∩ [ 0 , 1 ] , there exists a S ∈ C 2 , τ ( W ) , such that G ( t , S ) = 0 . This means that t ∈ ℐ and completes the proof of Lemma 4.6.□

Final proof of Theorem 1.1

The proof of uniqueness part follows from Proposition 4.3. It suffices to prove the existence part. It is easy to see that S ≡ 1 is a solution of (4.37) at t = 0 . This means that ℐ is not-empty. This, together with Corollary 4.5 and Lemma 4.6, implies that ℐ = [ 0 , 1 ] . Taking t = 1 , we obtain the proof of existence part to Theorem 1.1.

Acknowledgments

The author would like to thank heartfeltly to anonymous referees and editors for their invaluable comments which are helpful to improve the quality of the revised version of our paper and would also like to thank heartfeltly to Professors Daomin Cao and Qiuyi Dai for useful guidance on the theory of nonlinear pdes and helpful comments on the first version of the present paper and Professors Yinghui Zhang, Rongli Huang and Qianzhong Ou for helpful suggestions on the revised version of the present paper.

Funding information: The work was supported by the Science and Technology Project of Guangxi (Guike AD21220114), China Postdoctoral Science Foundation (Grant: No.2021M690773) and the Key Laboratory of Mathematical and Statistical Model (Guangxi Normal University), the Education Department of Guangxi Zhuang Autonomous Region.
Conflict of interest: There is no conflict of interest in this work.
Data availability statement: Data availability is not applicable to this article since no new data were created or analyzed in this study.

References

[1] B. Andrews, Volume-preserving anisotropic mean curvature flow, Indiana Univ. Math. J. 50 (2001), 783–827. 10.1512/iumj.2001.50.1853Search in Google Scholar

[2] J. L. M. Barbosa, J. H. S. Lira, and V. I. Oliker, A priori estimates for starshaped compact hypersurfaces with prescribed mth curvature function in space forms, Nonlinear Problems in Mathematical Physics and Related Topics, I, Int. Math. Ser. (N. Y.), vol. 1, Kluwer/Plenum, New York, 2002, pp. 35–52. 10.1007/978-1-4615-0777-2_3Search in Google Scholar

[3] V. I. Bogachev, Gaussian Measures, American Mathematical Society, Providence, RI, 1998. 10.1090/surv/062Search in Google Scholar

[4] C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), 207–216. 10.1007/BF01425510Search in Google Scholar

[5] K. J. Böröczky, E. Lutwak, D. Yang, and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc. 26 (2013), 831–852. 10.1090/S0894-0347-2012-00741-3Search in Google Scholar

[6] H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), 366–389. 10.1016/0022-1236(76)90004-5Search in Google Scholar

[7] L. Caffarelli, Interior W2,p estimates for solutions of the Monge-Ampère equation, Ann. Math. 131 (1990), 135–150. 10.2307/1971510Search in Google Scholar

[8] L. Caffarelli, Some regularity properties of solutions to the Monge-Ampère equation, Comm. Pure Appl. Math. 44 (1991), 965–969. 10.1002/cpa.3160440809Search in Google Scholar

[9] S.-Y. Cheng and S.-T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), 495–516. 10.1002/cpa.3160290504Search in Google Scholar

[10] K.-S. Chou and X.-J. Wang, The Lp Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205 (2006), 33–83. 10.1016/j.aim.2005.07.004Search in Google Scholar

[11] A. Colesanti and I. Fragalà, The first variation of the total mass of log-concave functions and related inequalities, Adv. Math. 244 (2013), 708–749. 10.1016/j.aim.2013.05.015Search in Google Scholar

[12] N. Fang, S. Xing, and D. Ye, Geometry of log-concave functions: the Lp Asplund sum and the Lp Minkowski problem, Calc. Var. 61 (2022), 45, DOI: https://doi.org/10.1007/s00526-021-02155-7. 10.1007/s00526-021-02155-7Search in Google Scholar

[13] M. Fradelizi and M. Meyer, Some functional forms of Blaschke-Santaló inequality, Math. Z. 256 (2007), 379–395. 10.1007/s00209-006-0078-zSearch in Google Scholar

[14] J. Fröhlich, A. Knowles, B. Schlein, and V. Sohinger, Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions d≤3, Comm. Math. Phys. 356 (2017), 883–980. 10.1007/s00220-017-2994-7Search in Google Scholar

[15] M. E. Gage, Evolving plane curves by curvature in relative geometries, Duke Math. J. 72 (1993), 441–466. 10.1215/S0012-7094-93-07216-XSearch in Google Scholar

[16] R. J. Gardner and A. Zvavitch, Gaussian Brunn-Minkowski inequalities, Trans. Amer. Math. Soc. 362 (2010), 5333–5353. 10.1090/S0002-9947-2010-04891-3Search in Google Scholar

[17] I. M. Gel’fand and N. Ya Vilenkin, Generalized Functions, Vol. 4. Applications of Harmonic Analysis, Academic Press, New York-London, 1964. Search in Google Scholar

[18] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001. 10.1007/978-3-642-61798-0Search in Google Scholar

[19] B. Guan and P. Guan, Convex hypersurfaces of prescribed curvatures, Ann. Math. 156 (2002), no. 2, 655–673. 10.2307/3597202Search in Google Scholar

[20] P. Guan, J. Li, and Y. Li, Hypersurfaces of prescribed curvature measure, Duke Math. J. 161 (2012), 1927–1942. 10.1215/00127094-1645550Search in Google Scholar

[21] P. Guan and X. Ma, The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equation, Invent. Math. 151 (2003), 553–577. 10.1007/s00222-002-0259-2Search in Google Scholar

[22] Y. Huang, E. Lutwak, D. Yang, and G. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math. 216 (2016), 325–388. 10.1007/s11511-016-0140-6Search in Google Scholar

[23] Y. Huang and Y. Zhao, On the Lp dual Minkowski problem, Adv. Math. 332 (2018), 57–84. 10.1016/j.aim.2018.05.002Search in Google Scholar

[24] Y. Huang, D. Xi, and Y. Zhao, The Minkowski problem in the Gaussian probability space, Adv. Math. 385 (2021), DOI: https://doi.org/10.1016/j.aim.2021.107769. 10.1016/j.aim.2021.107769Search in Google Scholar

[25] Y. Huang, J. Liu, D. Xi, and Y. Zhao, Dual Curvature Measures for Log-concave Functions, arXiv:2210.02359. Search in Google Scholar

[26] B. Klartag and V. D. Milman, Geometry of log-concave functions and measures, Geom. Dedicata 112 (2005), 169–182. 10.1007/s10711-004-2462-3Search in Google Scholar

[27] H. Lewy, On differential geometry in the large, I. Minkowski’s problem, Trans. Amer. Math. Soc. 43 (1938), 258–270. 10.1090/S0002-9947-1938-1501942-3Search in Google Scholar

[28] J. Liu, The Lp -Gaussian Minkowski problem, Calc. Var. 61 (2022), 28, DOI: https://doi.org/10.1007/s00526-021-02141-z. 10.1007/s00526-021-02141-zSearch in Google Scholar

[29] E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131–150. 10.4310/jdg/1214454097Search in Google Scholar

[30] E. Lutwak and V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom. 41 (1995), 227–256. 10.4310/jdg/1214456011Search in Google Scholar

[31] H. Minkowski, Allgemeine Lehrsätzeüber die convexen Polyeder. Nachr. Ges. Wiss. Göttingen. 1897, pp. 198–219. Search in Google Scholar

[32] H. Minkowski. Volumen und Oberfläche, Math. Ann. 57 (1903), 447–495. 10.1007/BF01445180Search in Google Scholar

[33] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394. 10.1002/cpa.3160060303Search in Google Scholar

[34] V. I. Oliker, Hypersurfaces in Rn+1 with prescribed Gaussian curvature and related equations of Monge-Ampère type, Comm. Partial Differential Equations 9 (1984), 807–838. 10.1080/03605308408820348Search in Google Scholar

[35] A. V. Pogorelov, The Minkowski Multidimensional Problem, V.H. Winston & Sons, Washington, DC, 1978. Search in Google Scholar

[36] R. C. Reilly, The relative differential geometry of nonparametric hypersurfaces, Duke Math. J. 43 (1976), 705–721. 10.1215/S0012-7094-76-04355-6Search in Google Scholar

[37] L. Rotem, Surface area measures of log-concave functions, J. Anal. Math. 147 (2022), 373–400. 10.1007/s11854-022-0227-2Search in Google Scholar

[38] R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 2014. 10.1017/CBO9781139003858Search in Google Scholar

[39] G. Wang and C. Xia A Brunn-Minkowski inequality for a Finsler-Laplacian, Analysis 31 (2011), 103–115. 10.1524/anly.2011.1073Search in Google Scholar

[40] G. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations, 252 (2012), 1668–1700. 10.1016/j.jde.2011.08.001Search in Google Scholar

[41] C. Xia, On an anisotropic Minkowski problem, Indiana Univ. Math. J. 62 (2013), 1399–1430. 10.1512/iumj.2013.62.5083Search in Google Scholar

[42] C. Xia, Inverse anisotropic mean curvature flow and a Minkowski type inequality, Adv. Math. 315 (2017), 102–129. 10.1016/j.aim.2017.05.020Search in Google Scholar

[43] C. Zhou and C. Zhou, Moser-Trudinger inequality involving the anisotropic Dirichlet norm (∫ΩFn(∇u)dx)1n on W01,n(Ω), J. Funct. Anal. 276 (2019), 2901–2935. 10.1016/j.jfa.2018.12.001Search in Google Scholar

Received: 2022-12-26

Revised: 2023-04-24

Accepted: 2023-05-08

Published Online: 2023-06-13

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

Articles in the same Issue

https://doi.org/10.1515/ans-2022-0068

Keywords for this article

log-concave measure; anisotropy; L p Minkowski problem; Monge-Ampère equation; the continuous method

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