Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure

Qi-Rui Li; Caihong Yi

doi:10.1515/ans-2022-0033

Article Open Access

Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure

Qi-Rui Li and Caihong Yi

Published/Copyright: January 11, 2023

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From the journal Advanced Nonlinear Studies Volume 23 Issue 1

Abstract

In this article, we study the Musielak-Orlicz-Gauss image problem based on the Gauss curvature flow in Li et al. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. By the use of the topological method in Guang et al., a special initial condition is chosen such that the Gauss curvature flow converges to a solution of the Musielak-Orlicz-Gauss image problem.

Keywords: Gauss curvature flow; Monge-Ampère equation; Musielak-Orlicz-Gauss image problem; topological method

MSC 2010: 35J20; 35K96; 52A30

1 Introduction and overview of the main results

In this article, we continue our study of the Musielak-Orlicz-Gauss image problem [25], which is equivalent to finding solution u : S n → [ 0 , ∞ ) to the following Monge-Ampère equation:

(1.1) u ( u 2 + ∣ ∇ u ∣ 2 ) − n 2 G z ( u 2 + ∣ ∇ u ∣ 2 , ξ ) p λ ( ξ ) det ( ∇ 2 u + u I ) = f ( x ) ψ ( u , x ) , x ∈ S n ,

where G : ( 0 , ∞ ) × S n → R , G z = ∂ z G ( z , ξ ) , p λ and f are given functions.

The Brunn-Minkowski theory is the core of convex geometry and strongly influences in fully nonlinear partial differential equations. The Minkowski-type problems are an important part of the theory. The past few years have witnessed the great progress in the Minkowski-type problems, including the L p Minkowski problem [27], the Orlicz-Minkowski problem [17], the dual Minkowski [18], the L p dual Minkowski [29], and the dual Orlicz-Minkowski problems [11]. These Minkowski problems have been extensively studied, see, e.g., [1,3,6,7,9,10,12,16,20,22,26,32].

Recently, the Musielak-Orlicz function [15,30] was introduced into the Minkowski-type problem. The Musielak-Orlicz-Gauss problem, which aims to characterize the Musielak-Orlicz-Gauss image measure C ˜ Θ ( Ω , ⋅ ) for convex body Ω in R n + 1 , was posed in [21]. It naturally leads to a new generation of the Brunn-Minkowski theory, namely the Musielak-Orlicz-Brunn-Minkowski theory of convex bodies.

Let K be the set of all convex bodies in R n + 1 containing the origin, and K 0 ⊆ K be the set of all convex bodies with the origin in their interiors. Let C be the set of all Musielak-Orlicz function G : ( 0 , ∞ ) × S n → R such that both G and G z ( z , ξ ) = ∂ z G ( z , ξ ) are continuous on ( 0 , ∞ ) × S n . Let Θ = ( G , Ψ , λ ) be a given triple such that G ∈ C , Ψ ∈ C , and λ be a nonzero finite Borel measure on S n . The Musielak-Orlicz-Gauss problem asks (see [21]): under what conditions on the triple Θ and a nonzero finite Borel measure μ on S n do there exist a Ω ∈ K 0 and a constant τ ∈ R such that

(1.2) d μ = τ d C ˜ Θ ( Ω , ⋅ ) ?

Assume that λ is a nonzero finite Borel measure on S n and d λ ( ξ ) = p λ ( ξ ) d ξ with the function p λ : S n → ( 0 , ∞ ) being continuous. When the pregiven measure μ has a density f with respect to d ξ , (1.2) reduces to solving the following Monge-Ampère-type equation on S n :

(1.3) u ( u 2 + ∣ ∇ u ∣ 2 ) − n 2 G z ( u 2 + ∣ ∇ u ∣ 2 , ξ ) p λ ( ξ ) det ( ∇ 2 u + u I ) = γ f ( x ) ψ ( u , x ) ,

where ∇ and ∇ 2 are the gradient and Hessian operators with respect to an orthonormal frame on S n , γ > 0 is a constant, I is the identity matrix, and ξ = α Ω u ∗ ( x ) where α Ω u ∗ is the reverse radial Gauss map of Ω u – the convex body whose support function is u ( x ) for x ∈ S n .

Let C I and C d be the subclasses of C defined by

C I = { G ∈ C : G z = ∂ z G ( z , ξ ) > 0 on ( 0 , ∞ ) × S n } , C d = { G ∈ C : G z = ∂ z G ( z , ξ ) < 0 on ( 0 , ∞ ) × S n } .

The existence of solutions to the Musielak-Orlicz-Gauss image problem (1.3) was established:

by [21] under the condition G ∈ C d and Ψ ∈ C I ∪ C d ;
by [25] under the condition G ∈ C I and Ψ ∈ C I .

In [25], the authors attacked the problem by studying the following parabolic flow (1.4) with any given initial data ℳ 0 (a smooth, closed, uniformly convex hypersurface in R n + 1 enclosing the origin),

(1.4) ∂ ∂ t X ( x , t ) = ( − f ( ν ) ψ ( u , x ) r n G z ( r , ξ ) − 1 p λ − 1 ( ξ ) K + η ( t ) u ) ν , X ( x , 0 ) = X 0 ( x ) ,

where X ( ⋅ , t ) : S n → R n + 1 is the embedding that parameterizes a family of convex hypersurfaces ℳ t (in particular, X 0 is the parametrization of ℳ 0 ), K and ν denote the Gauss curvature and the unit outer normal of ℳ t at X ( x , t ) , ξ = α Ω t ∗ ( x ) , ψ : ( 0 , ∞ ) × S n → ( 0 , ∞ ) is defined as

(1.5) ψ ( z , x ) = − z Ψ z ( z , x )

and

(1.6) η ( t ) = ∫ S n f ψ ( u , x ) d x ∫ S n r G z ( r , ξ ) p λ ( ξ ) d ξ .

In this article, we study the Musielak-Orlicz-Gauss image problem (1.3) for the case G ∈ C I and Ψ ∈ C d . The main difficulties are to obtain the uniform estimates for the flow (1.4), namely the control of the shape of ℳ t . To handle this, we use the topological method developed in [13] to find a special initial condition such that the evolving hypersurfaces ℳ t satisfy

(1.7) B r ( 0 ) ⊂ Ω t ⊂ B R ( 0 ) ,

for some uniform constants R ≥ r > 0 independent of t , where Ω t is the convex body circumscribed by ℳ t . Our C 0 estimates (1.7) also need the following constraints on the functions G and Ψ :

Condition (A): G ( z , ξ ) is a positive and continuous function defined in ( 0 , + ∞ ) × S n such that lim z → 0 G ( z , ξ ) = 0 and
min ξ ∈ S n G z ( t , ξ ) ≥ α ⋅ t n + ε
for some positive constants α and ε when t is sufficiently large.
Condition (B): Ψ ( z , x ) and ψ ( z , x ) defined by (1.5) are positive and continuous functions defined in ( 0 , + ∞ ) × S n such that lim z → + ∞ Ψ ( z , x ) = 0 , and
min x ∈ S n ψ ( t , x ) ≥ β ⋅ t − n − 1 − ε
for some positive constants β and ε when t is sufficiently small.

The main result of this article is the following theorem.

Theorem 1.1

Let f , p λ ∈ C 1 , 1 ( S n ) be positive functions satisfying c 0 − 1 ≤ f , p λ ≤ c 0 for some constant c 0 > 1 . Suppose that G z ∈ C 1 , 1 ( ( 0 , ∞ ) × S n ) , Ψ z ∈ C 1 , 1 ( ( 0 , ∞ ) × S n ) , and G ∈ C I satisfying Condition (A) and Ψ ∈ C d satisfying Condition (B) . Then there is a uniformly convex and C 3 , α -smooth positive solution to (1.3), where α ∈ ( 0 , 1 ) .

We will show that (1.4) is a gradient flow to the following functional:

(1.8) J ( Ω ) = ∫ S n f Ψ ( u , x ) d x + V ˜ G , λ ( Ω t ) ,

where V ˜ G , λ ( Ω ) is the general dual volume of Ω ∈ K 0 with respect to λ defined by

(1.9) V ˜ G , λ ( Ω ) = ∫ S n G ( r Ω ( ξ ) , ξ ) d λ ( ξ ) .

Under conditions (A) and (B), we find that the functional J ( Ω ) becomes large if either the volume of Ω is sufficiently large or small, or the shape of Ω t is quite bad (see Proposition 3.1). This property is used to select the initial hypersurface by the topological argument of [13] for which the C 0 estimate (1.7) is valid. Once (1.7) is proved, we are able to show that flow (1.4) exists for all time. This together with the monotonicity of (1.8) implies that the flow (1.4) converges to a solution of (1.3).

We remark that in [13] the authors introduced their topological argument to resolve the L p Minkowski problem with super-critical exponents p < − n − 1 . Before that the problems in the sub-critical case p > − n − 1 have been extensively studied [2,4,6,7,8, 9,16,19,26,28,31], while the super-critical case remains widely open. Their approach was also applied to the L p dual Minkowski problem [14]. In this article, we further adopt the topological method of [13] to attack the Musielak-Orlicz-Gauss image problem.

This article is organized as follows. In Section 2, some properties of convex hypersurfaces are presented, and we show the preservation of V ˜ G , λ ( ⋅ ) along the flows (see Lemma 2.1) and the strict monotonicity of functionals (1.8) (see Lemma 2.2). In particular, we show a priori estimates and the long time existence of the flows (1.4) (see Theorems 2.3 and 2.4). Section 3 is dedicated to the proofs of Theorem 1.1. The functional J ( Ω ) will be very large if either the volume of Ω is sufficiently large or small, or the eccentricity of Ω is sufficiently large. We prove that the flow converges to a solution of (1.3) by using the topological method.

2 Preliminary and a priori estimates

For Ω ∈ K 0 , define its radial function r Ω : S n → ( 0 , ∞ ) and support function u Ω : S n → ( 0 , ∞ ) , respectively, by

(2.1) r Ω ( x ) = max { a ∈ R : a x ∈ Ω } and u Ω ( x ) = max { ⟨ x , y ⟩ , y ∈ Ω } , x ∈ S n ,

where ⟨ x , y ⟩ denotes the inner product in R n + 1 .

The polar dual convex body Ω ∗ of Ω :

Ω ∗ = { y ∈ R n + 1 : ⟨ y , z ⟩ ≤ 1 , ∀ z ∈ Ω } .

The Gauss map of ∂ Ω , denoted by ν Ω : ∂ Ω → S n , is defined as follows: for y ∈ ∂ Ω ,

ν Ω ( y ) = { x ∈ S n : ⟨ x , y ⟩ = u Ω ( x ) } .

Let ν Ω − 1 : S n → ∂ Ω be the reverse Gauss map such that

ν Ω − 1 ( x ) = { y ∈ ∂ Ω : ⟨ x , y ⟩ = u Ω ( x ) } , x ∈ S n .

Denote by α Ω : S n → S n the radial Gauss image of Ω . That is,

α Ω ( ξ ) = { x ∈ S n : x ∈ ν Ω ( r Ω ( ξ ) ξ ) } , ξ ∈ S n .

Define α Ω ∗ : S n → S n , the reverse radial Gauss image of Ω is as follows: for any Borel set E ⊆ S n ,

(2.2) α Ω ∗ ( E ) = { ξ ∈ S n : r Ω ( ξ ) ξ ∈ ν Ω − 1 ( E ) } .

We often omit the subscript Ω in r Ω , u Ω , ν Ω , ν Ω − 1 , α Ω , and α Ω ∗ if no confusion occurs.

Let ℳ be a smooth, closed, uniformly convex hypersurface in R n + 1 , enclosing the origin. The parametrization of ℳ is given by the inverse Gauss map X : S n → ℳ ⊆ R n + 1 . It follows from (2.1) and (2.2) that

(2.3) X ( x ) = r ( α ∗ ( x ) ) α ∗ ( x ) and u ( x ) = ⟨ x , X ( x ) ⟩ ,

where u is the support function of (the convex body circumscribed by) ℳ . It is well known that the Gauss curvature of ℳ is

(2.4) K = 1 det ( ∇ 2 u + u I ) ,

and the principal curvature radii of ℳ are the eigenvalues of the matrix b i j = ∇ i j u + u δ i j . Moreover, the following hold, see e.g. [24],

(2.5) r ξ = u x + ∇ u , r = u 2 + ∣ ∇ u ∣ 2 , and u = r 2 r 2 + ∣ ∇ r ∣ 2 .

Let u ( ⋅ , t ) and r ( ⋅ , t ) be the support and radial functions of ℳ t . Recall that (see, e.g., [5, Lemma 2.1])

(2.6) ∂ t r ( ξ , t ) r = ∂ t u ( x , t ) u .

By (2.3) and (2.4), the flow equation (1.4) for ℳ t can be reformulated by its support function u ( x , t ) as follows:

(2.7) ∂ t u ( x , t ) = − f ( x ) ψ ( u , x ) r n G z ( r , ξ ) − 1 p λ − 1 ( ξ ) K + η ( t ) u , u ( ⋅ , 0 ) = u 0 .

It is well known that J ( ξ ) , the determinant of the Jacobian of the radial Gauss image x = α Ω ( ξ ) for Ω ∈ K 0 , satisfies (see, e.g., [24])

(2.8) J ( ξ ) = r Ω n + 1 ( ξ ) K ( r Ω ( ξ ) ξ ) u Ω ( α Ω ( ξ ) ) .

It is clear that both (1.4) and (2.7) are parabolic Monge-Ampére types, their solutions exist for a short time. Therefore, the flow (1.4), as well as (2.7), have short-time solutions. Let X ( ⋅ , t ) be a smooth solution to the flow (1.4) with t ∈ [ 0 , T ) for some constant T > 0 . We now show that V ˜ G , λ ( ⋅ ) remains unchanged along the flow (1.4).

Lemma 2.1

Let G ∈ C I and Ψ ∈ C d . Let X ( ⋅ , t ) be a smooth solution to the flow (1.4) with t ∈ [ 0 , T ) , and ℳ t = X ( S n , t ) be a smooth, closed, and uniformly convex hypersurface. Suppose that the origin lies in the interior of the convex body Ω t enclosed by ℳ t for all t ∈ [ 0 , T ) . Then, for any t ∈ [ 0 , T ) , one has

(2.9) V ˜ G , λ ( Ω t ) = V ˜ G , λ ( Ω 0 ) .

Proof

It follows from (2.8) that, by letting x = α Ω ( ξ ) ,

∫ S n f ψ ( u , x ) d x = ∫ S n f ψ ( u , x ) u r n + 1 K d ξ .

This, together with (1.6), (2.6), and (2.8), yields that

d d t ∫ S n G ( r , ξ ) p λ ( ξ ) d ξ = ∫ S n G z ( r , ξ ) p λ ( ξ ) r t d ξ = ∫ S n G z ( r , ξ ) p λ ( ξ ) − f ψ ( u ) u r n + 1 G z − 1 ( r , ξ ) p λ − 1 K + r η ( t ) d ξ = − ∫ S n f ψ ( u , x ) u r n + 1 K d ξ + η ( t ) ∫ S n r G z ( r , ξ ) p λ ( ξ ) d ξ = − ∫ S n f ψ ( u , x ) u r n + 1 K d ξ + ∫ S n f ψ ( u , x ) d x = 0 .

In conclusion, V ˜ G , λ ( ⋅ ) remains unchanged along the flow (1.4), and in particular, (2.9) holds for any t ∈ [ 0 , T ) .□

Recall that Ψ z < 0 , the function ψ = − z Ψ z in (1.5). The lemma below shows that the functional J ( u ) is monotone along with the flow (1.4).

Lemma 2.2

Let G ∈ C I and Ψ ∈ C d . Let X ( ⋅ , t ) , ℳ t , and Ω t be as in Lemma 2.1. Then the functional J defined in (1.8) is nondecreasing along the flow (1.4). That is, d J ( u ( ⋅ , t ) ) d t ≥ 0 , with equality if and only if ℳ t satisfies the elliptic equation (1.3).

Proof

Let u ( ⋅ , t ) be the support function of ℳ t .

d J ( u ( ⋅ , t ) ) d t = ∫ S n f 2 ψ 2 ( u , x ) r n G z − 1 ( r , ξ ) p λ − 1 ( ξ ) u − 1 K d x − ∫ S n f ψ ( u , x ) d x 2 ∫ S n r G z ( r , ξ ) p λ ( ξ ) d ξ ≥ 0 .

Clearly, equality holds here if and only if equality holds for the Hölder inequality, namely, there exists a constant c ( t ) > 0 such that

(2.10) f ψ ( u , x ) u r n G z − 1 ( r , ξ ) p λ − 1 ( ξ ) ( det ( ∇ 2 u + u I ) ) − 1 = c ( t ) .

Moreover, it can be proved by (1.6) and (2.10) that c ( t ) = η ( t ) as follows:

η ( t ) = ∫ S n f ψ ( u , x ) d x ∫ S n r G z ( r , ξ ) p λ ( ξ ) d ξ = c ( t ) ∫ S n r − n G z ( r , ξ ) p λ ( ξ ) u K − 1 d x ∫ S n r G z ( r , ξ ) p λ ( ξ ) d ξ = c ( t ) .

This concludes the proof.□

Theorem 2.3

Suppose that f , p λ , G z , and Ψ z are both positive and C 1 , 1 -smooth. Let u ( ⋅ , t ) be a positive, smooth, and uniformly convex solution to flow (2.7), t ∈ [ 0 , T ) . Assume that

(2.11) 1 / C 0 ≤ u ( x , t ) ≤ C 0

for all ( x , t ) ∈ S n × [ 0 , T ) . Then

(2.12) C − 1 I ≤ ( ∇ 2 u + u I ) ( x , t ) ≤ C I , ∀ ( x , t ) ∈ S n × [ 0 , T ) ,

where C is a positive constant depending only on n , C 0 , ∣ ∣ f ∣ ∣ C 1 , 1 ( S n ) , ∣ ∣ p λ ∣ ∣ C 1 , 1 ( S n ) , ∣ ∣ ψ ∣ ∣ C 1 , 1 ( ( 0 , ∞ ) × S n ) , ∣ ∣ G z ∣ ∣ C 1 , 1 ( ( 0 , ∞ ) × S n ) , and the initial hypersurface, but is independent of T.

Proof

It follows from (2.5) that

(2.13) max x ∈ S n ∣ ∇ u ( x , t ) ∣ ≤ max x ∈ S n r ( x , t ) = max x ∈ S n u ( x , t ) ≤ C 0 .

It is direct to yield the bound of η defined by (1.6):

(2.14) 1 / C 1 ≤ η ( t ) ≤ C 1 , ∀ t ∈ [ 0 , T ) ,

where the constant C > 0 (abuse of notations) depends only on f , p λ , G , Ψ , and Ω 0 but is independent of t ∈ [ 0 , T ) .

In [25], we obtained the priori estimates for a more general equation of the form

(2.15) ∂ u ∂ t ( x , t ) = − Φ ( x , u , ∇ u ) ( det ( ∇ 2 u + u I ) ) − 1 + u η ( t ) on S n × [ 0 , T ) .

It follows from (2.11), (2.13), and (2.14) that (2.12) holds. This is a direct consequence of Lemmas 5.1 and 5.2 in [25] by letting

Φ ( x , u , ∇ u ) = f ( x ) ψ ( u , x ) ( u 2 + ∣ ∇ u ∣ 2 ) n 2 p λ − 1 u x + ∇ u u 2 + ∣ ∇ u ∣ 2 G z − 1 u 2 + ∣ ∇ u ∣ 2 , u x + ∇ u u 2 + ∣ ∇ u ∣ 2 . □

In view of (2.12), the flow (2.7) is uniformly parabolic and then ∣ ∂ t u ∣ L ∞ ( S n × [ 0 , T ) ) ≤ C (abuse of notation C ). It follows from the results of Krylov and Safonov [23] that the Hölder continuity estimates for ∇ 2 u and ∂ t u can be obtained. Hence, the standard theory of the linear uniformly parabolic equations can be used to obtain the higher-order derivative estimates, which further implies the long time existence of a positive, smooth, and uniformly convex solution to the flow (2.7). Moreover, we have the following theorem.

Theorem 2.4

Assume the conditions in Theorem 2.3. Let T max be the maximal time such that the solution u ( ⋅ , t ) exists on [ 0 , T max ) . Then T max = ∞ and u satisfies

(2.16) 1 / C ≤ u ( x , t ) ≤ C for all ( x , t ) ∈ S n × [ 0 , ∞ ) ,

(2.17) 1 / C ≤ ∣ η ( t ) ∣ ≤ C for all t ∈ [ 0 , ∞ ) ,

(2.18) ∇ 2 u + u I ≥ 1 / C ε on S n × [ 0 , ∞ ) ,

(2.19) ∣ u ∣ C x , t k , l ( S n × [ 0 , ∞ ) ) ≤ C k , l ,

where C and C k , l are constants depending on ε , G , Ψ , f , p λ , and Ω 0 .

3 Proof of Theorem 1.1

3.1 An estimate for the functional (1.8)

For any convex body Ω in R n + 1 , let E ( Ω ) denote John’s minimum ellipsoid of Ω . We have

1 n + 1 E ( Ω ) ⊂ Ω ⊂ E ( Ω ) .

Let a 1 ( Ω ) ≤ a 2 ( Ω ) ≤ … ≤ a n + 1 ( Ω ) be the lengths of semi-axes of E ( Ω ) . Denote e ℳ = e Ω = a n + 1 a 1 the eccentricity of ℳ ≔ ∂ Ω . We show an estimate for the functional (1.8) as follows.

Proposition 3.1

Let G and Ψ satisfy conditions stated in Theorem 1.1. Suppose that f , p λ satisfy c 0 − 1 ≤ f , p λ ≤ c 0 for some constant c 0 > 1 . For any given constant A > J ( B 1 ( 0 ) ) , if one of the quantities e Ω , Vol ( Ω ) , [ Vol ( Ω ) ] − 1 , and [ dist ( O , ∂ Ω ) ] − 1 is sufficiently large, then J ( Ω ) > A .

Proof

We divide the proof into three steps.

Step 1: If e Ω ≥ e for a large constant e > 1 , we have J ( Ω ) > A .

By a proper rotation of coordinates, we assume that E = E ( Ω ) is given by

(3.1) E − ζ E = z ∈ R n + 1 : ∑ i = 1 n + 1 z i 2 a i 2 ≤ 1 ,

where ζ E = ( ζ 1 , … , ζ n + 1 ) is the center of E , and a 1 ≤ … ≤ a n + 1 .

First, we construct inequalities about the quantity ∫ S n f Ψ ( u , x ) d x . Let x 0 ∈ S n be the point such that u ( x 0 ) = min S n u = d , where u is the support function of Ω and d = dist ( O , ∂ Ω ) . We choose i # and switch e i # and − e i # if necessary such that

x 0 ⋅ e i # = max { ∣ x 0 ⋅ e i ∣ : 1 ≤ i ≤ n + 1 } .

This implies that x 0 ⋅ e i # ≥ c n , where c n denotes a constant which depends only on n but may change from line by line.

Let w ( x ) = u ( x ) + u ( − x ) , x ∈ S n , be the width of Ω in x . Since 1 n + 1 E ( Ω ) ⊂ Ω ⊂ E ( Ω ) , we have

d ≤ min S n w ≤ c n a i #

and

2 a i n + 1 ≤ w ( e i ) ≤ 2 a i , ∀ i = 1 , … , n + 1 .

By switching e i and − e i if necessary, we assume that u ( e i ) ≤ c n a i for all i = 1 , … , n + 1 .

Now we consider the cone V in R n + 1 with the vertex p 0 = r ∗ ( x 0 ) x 0 and the base

C ≕ convex hull of { O , r ∗ ( e k ) e k } k ≠ i # ⊂ Ω ∗ ,

where r ∗ is the radial function of the polar dual body Ω ∗ of Ω .

We consider the following subset of V :

V ′ = υ ∈ V : υ i # ≥ r ∗ ( x 0 ) 2 x 0 ⋅ e i # .

Recall that r ∗ = 1 / u . Hence, we have

(3.2) r ∗ ( x 0 ) = 1 u ( x 0 ) = 1 d and r ∗ ( e k ) ≥ c n a k , for all k ≥ 1 .

By using (1.5) and (3.2), we obtain that

(3.3) ∫ S n f Ψ ( u , x ) d x ≥ C 0 ∫ S n Ψ ( 1 / r ∗ , x ) d x ≥ C 0 ∫ S n ∫ 0 r ∗ ( x ) ∂ Ψ ( z , x ) ∂ z ⋅ ( − 1 / τ 2 ) d τ d x ≥ C 0 ∫ Ω ∗ ψ ( 1 / ∣ υ ∣ , υ / ∣ υ ∣ ) 1 / ∣ υ ∣ ∣ υ ∣ − n − 2 d υ ≥ C 0 ∫ V ψ ( 1 / ∣ υ ∣ , υ / ∣ υ ∣ ) 1 / ∣ υ ∣ ∣ υ ∣ − n − 2 d υ ≥ C 0 ∫ V ′ ψ ( 1 / ∣ υ ∣ , υ / ∣ υ ∣ ) ∣ υ ∣ − n − 1 d υ ≥ C 0 min ( z , x ) ∈ [ d , 2 d / c n ] × S n ψ ( z , x ) ⋅ d n + 1 Vol ( V ′ ) ≥ C 0 min ( z , x ) ∈ [ d , 2 d / c n ] × S n ψ ( z , x ) ⋅ d n + 1 c n d Vol ( C ) ≥ C 0 min ( z , x ) ∈ [ d , 2 d / c n ] × S n ψ ( z , x ) ⋅ d n ⋅ a i # ∏ i = 1 n + 1 a i − 1 ≥ C 0 min ( z , x ) ∈ [ d , 2 d / c n ] × S n ψ ( z , x ) ⋅ d n + 1 ∏ i = 1 n + 1 a i − 1 ,

where C 0 denotes a constant which depends only on n but may change from line by line.

Next, we construct inequalities about the general volume V ˜ G , λ ( Ω ) . Recall that (3.1), ζ E = ( ζ 1 , … , ζ n + 1 ) . We can further assume that ζ n + 1 ≥ 0 . Since 1 n + 1 E ⊂ Ω , we have

(3.4) u ( e n + 1 ) ≥ ζ n + 1 + 1 n + 1 a n + 1 .

Hence, there exists a point p 0 ∈ Ω such that

(3.5) p 0 ⋅ e n + 1 = u ( e n + 1 ) ≥ ζ n + 1 + 1 n + 1 a n + 1 .

Consider the hyperplane L which is orthogonal to e n + 1 and passes through ζ E :

L = { υ ∈ R n + 1 : ( υ − ζ E ) ⋅ e n + 1 = 0 } .

Let P = L ∩ 1 n + 1 E be the intersection of L with the ellipsoid 1 n + 1 E , and V be the cone in R n + 1 with base P and the vertex p 0 . Clearly V ⊂ Ω . Let us consider the following subset of V :

V ′ = υ ∈ V : υ n + 1 − ζ n + 1 ≥ 1 2 ( p 0 ⋅ e n + 1 − ζ n + 1 ) .

This together with (3.4) implies that

(3.6) ∣ υ ∣ ≥ a n + 1 2 ( n + 1 ) , ∀ υ ∈ V ′ .

It is easy to see that

(3.7) Vol ( V ′ ) ≥ c ¯ n a n + 1 ∏ i = 1 n a i ,

for some positive constant c ¯ n which depends only on n . Since Ω contains the origin and Ω ⊂ E , we have

(3.8) ∣ υ ∣ ≤ c ¯ n a n + 1 , ∀ υ ∈ Ω .

Combining (3.6), (3.7), and (3.8), we can obtain that

(3.9) V ˜ G , λ ( Ω ) = ∫ S n G ( r Ω ( ξ ) , ξ ) p λ ( ξ ) d ξ ≥ C ¯ 0 ∫ S n G ( r Ω ( ξ ) , ξ ) d ξ ≥ C ¯ 0 ∫ S n ∫ 0 r ( ξ ) ∂ G ( z , ξ ) ∂ z d z d ξ ≥ C ¯ 0 ∫ Ω G z ( ∣ υ ∣ , υ / ∣ υ ∣ ) ∣ υ ∣ − n d υ ≥ C ¯ 0 ∫ V ′ G z ( ∣ υ ∣ , υ / ∣ υ ∣ ) ∣ υ ∣ − n d υ ≥ C ¯ 0 min ( z , ξ ) ∈ a n + 1 2 ( n + 1 ) , a n + 1 × S n G z ( z , ξ ) ⋅ a n + 1 − n ⋅ Vol ( V ′ ) ≥ C ¯ 0 min ( z , ξ ) ∈ a n + 1 2 ( n + 1 ) , a n + 1 × S n G z ( z , ξ ) ⋅ a n + 1 − n ⋅ a n + 1 ∏ i = 1 n a i ≥ C ¯ 0 min ( z , ξ ) ∈ a n + 1 2 ( n + 1 ) , a n + 1 × S n G z ( z , ξ ) ⋅ a n + 1 − n + 1 ∏ i = 1 n a i ,

where C ¯ 0 denotes a constant which depends only on n but may change from line by line.

From (3.3) and (3.9), there exists a positive constant C depending only on f , g , C 0 , and C ¯ 0 such that

(3.10) [ J ( Ω ) ] 2 ≥ C min ( z , x ) ∈ [ d , 2 d / c n ] × S n ψ ( z , x ) ⋅ d n + 1 min ( z , ξ ) ∈ a n + 1 2 ( n + 1 ) , a n + 1 × S n G z ( z , ξ ) ⋅ a n + 1 − n .

Recall that the functions G ( t , x ) and Ψ ( t , x ) satisfy the conditions (A) and (B), respectively, we conclude that:

When t is sufficiently large, min x ∈ S n G z ( t , x ) ≥ α ⋅ t n + ε for some positive constants α and ε , then there exists a large positive constant M such that

(3.11) min x ∈ S n G z ( t , x ) ≥ α ⋅ t n + ε , as t ≥ M .

When t is sufficiently small, min x ∈ S n ψ ( t , x ) ≥ β ⋅ t − n − 1 − ε for some positive constants β and ε , then there exists a small positive constant δ such that

(3.12) min x ∈ S n ψ ( t , x ) ≥ β ⋅ t − n − 1 − ε , as t ≤ δ .

Since V ˜ G , λ ( Ω t ) = V ˜ G , λ ( Ω 0 ) , G z ( z , x ) > 0 on ( 0 , ∞ ) × S n . It is easy to obtain that

(3.13) 2 d / c n ≤ C 1 and a n + 1 2 ( n + 1 ) ≥ C 2 ,

for some positive constants C 1 and C 2 depending only on the initial hypersurface ∂ Ω 0 .

Combining (3.11), (3.12), and (3.13), inequality (3.10) is mainly discussed in the following two cases.

Case 1: C 1 < δ for the constant δ in (3.12) and the constant C 1 in (3.13). In this case, by (3.13), we have 2 d / c n ≤ C 1 < δ , and then either M ≥ a n + 1 2 ( n + 1 ) ≥ C 2 or a n + 1 2 ( n + 1 ) ≥ max { C 2 , M } for the constant M in (3.11) and the constant C 2 in (3.13).

If 2 d / c n ≤ C 1 < δ and M ≥ a n + 1 2 ( n + 1 ) ≥ C 2 , by (3.12), (3.10) becomes

(3.14) J 2 ≥ C min ( z , x ) ∈ [ d , 2 d / c n ] × S n ψ ( z , x ) ⋅ d n + 1 ⋅ min ( z , ξ ) ∈ [ a n + 1 2 ( n + 1 ) , a n + 1 ] × S n G z ( z , ξ ) ⋅ a n + 1 − n ≥ C ˆ β d − n − 1 − ε d n + 1 ⋅ min ( z , ξ ) ∈ [ C 2 , 2 ( n + 1 ) M ] × S n G z ( z , ξ ) ⋅ a n + 1 − n ≥ C ˆ a n + 1 e Ω − ε ⋅ min ( z , ξ ) ∈ [ C 2 , 2 ( n + 1 ) M ] × S n G z ( z , ξ ) ⋅ a n + 1 − n ≥ C ˆ M e Ω − ε M − n ,

where C ˆ denotes a constant but may change from line by line. Since a n + 1 d ≥ a n + 1 a 1 = e Ω is sufficiently large, (3.14) implies that J ( Ω ) ≥ A .

If 2 d / c n ≤ C 1 < δ and a n + 1 2 ( n + 1 ) ≥ max { C 2 , M } , by (3.11) and (3.12), (3.10) becomes

(3.15) J 2 ≥ C min ( z , x ) ∈ [ d , 2 d / c n ] × S n ψ ( z , x ) ⋅ d n + 1 ⋅ min ( z , ξ ) ∈ [ a n + 1 2 ( n + 1 ) , a n + 1 ] × S n G z ( z , ξ ) ⋅ a n + 1 − n ≥ C ¯ β d − n − 1 − ε d n + 1 ⋅ α a n + 1 n + ε ⋅ a n + 1 − n ≥ C ¯ a n + 1 e Ω − ε a n + 1 ε ≥ C ¯ e Ω ε ,

where C ¯ denotes a constant but may change from line by line. Since e Ω is sufficiently large, (3.15) implies that J ( Ω ) ≥ A .

Case 2: C 1 ≥ δ for the constant δ in (3.12) and the constant C 1 in (3.13). In this case, by (3.13), we have either δ ≤ 2 d / c n ≤ C 1 or 2 d / c n < δ , and a n + 1 2 ( n + 1 ) ≥ C 2 . We will discuss this case as follows.

If δ ≤ 2 d / c n ≤ C 1 , since a n + 1 d ≥ a n + 1 a 1 = e Ω is sufficiently large, hence a n + 1 2 ( n + 1 ) ≥ max { C 2 , M } . By (3.11), (3.10) becomes

(3.16) J 2 ≥ C min ( z , x ) ∈ [ d , 2 d / c n ] × S n ψ ( z , x ) ⋅ d n + 1 ⋅ min ( z , ξ ) ∈ [ a n + 1 2 ( n + 1 ) , a n + 1 ] × S n G z ( z , ξ ) ⋅ a n + 1 − n ≥ C ˜ min ( z , x ) ∈ [ δ c n / 2 , C 1 ] × S n ψ ( z , x ) d n + 1 ⋅ α a n + 1 n + ε a n + 1 − n ≥ C ˜ ⋅ d n + 1 ⋅ ( e Ω d ) ε ≥ C ˜ ⋅ δ n + 1 + ε ⋅ e Ω ε ,

where C ˜ denotes a constant but may change from line by line. Since e Ω is sufficiently large, (3.16) implies that J ( Ω ) ≥ A .

If d / c n < δ , J 2 ≥ A can be obtained by using the same argument in Case 1.

Step 2: If either Vol ( Ω ) ≤ v 0 and Vol ( Ω ) ≥ v 0 − 1 for a small constant v 0 , then J ( Ω ) > A .

(3.17) J ( u ) ≥ ∫ S n f Ψ ( u , x ) d x ≥ C f min x ∈ S n Ψ ( a n + 1 , x ) ≥ C f min x ∈ S n Ψ C a 1 a n + 1 1 [ Vol ( Ω ) ] 1 n + 1 − 1 , x

and

(3.18) J ( u ) ≥ ∫ S n G ( r , ξ ) p λ ( ξ ) d ξ ≥ C p λ min ξ ∈ S n G ( d , ξ ) ≥ C p λ min ξ ∈ S n G C d a n + 1 [ Vol ( Ω ) ] 1 n + 1 , ξ .

Recall that G is increasing in its first variable, and Ψ is decreasing in its first variable. If either d / a n + 1 or a 1 / a n + 1 is sufficiently close to 0, then J ( Ω ) > A by the argument in Step 1. Hence, we assume that d / a n + 1 or a 1 / a n + 1 are away from 0. By (3.17) and (3.18), if either [ Vol ( Ω ) ] − 1 or Vol ( Ω ) is large, then J ( Ω ) > A .

Step 3: If dist ( O , ∂ Ω ) ≤ d 0 for a small constant d 0 > 0 , then J ( Ω ) > A .

Assume that a j ≤ C A d for all 1 ≤ j ≤ n + 1 for some C A ≥ 1 . Otherwise, a j d is sufficiently large for some j , so then a n + 1 d is sufficiently large. By the argument in Step 1, we have J ( Ω ) > A .

Under the above assumption, if d is sufficiently small, then Vol ( Ω ) becomes very small. By the argument in Step 2, we have J ( Ω ) > A .

This completes the proof.□

Remark 3.1

Let ℳ t , t ∈ [ 0 , T max ) , be a solution to the flow (1.4). By Proposition 3.1, if J ( ℳ t ) < A for a constant A independent of t , then there exist positive constants e 0 , v 0 , d 0 depending on A , but independent of t , such that

(3.19) e ℳ t ≤ e 0 , v 0 ≤ Vol ( Ω t ) ≤ v 0 − 1 , and B d 0 ( 0 ) ⊂ Ω t ,

where Ω t is the convex body enclosed by ℳ t . Note that (3.19) implies (2.11). Hence, the a priori estimate (2.12) holds, and the long-time existence of solution can be obtained by Theorem 2.4. Therefore, all we need is to establish the condition J ( ℳ t ) < A for some constant A .

3.2 A modified flow of (1.4)

We introduce a modified flow of (1.4) such that for any initial condition, the solution exists for all time t ≥ 0 . It is more convenient to work with a flow that exists for all t ≥ 0 . Let us fix a constant

(3.20) A 0 = 2 max x ∈ S n Ψ 1 2 ( n + 1 ) , x ∣ ∣ f ∣ ∣ L 1 ( S n ) + max x ∈ S n G ( 2 , ξ ) ∣ ∣ p λ ∣ ∣ L 1 ( S n ) .

If the minimum ellipsoid of Ω is B 1 ( 0 ) , then 1 n + 1 B 1 ( 0 ) ⊂ Ω ⊂ B 1 ( 0 ) and hence

(3.21) J ( Ω ) ≤ 1 2 A 0 .

Denote by Cl ( N ) the convex body enclosed by N . For a closed, smooth, and uniformly convex hypersurface N such that Ω 0 = Cl ( N ) ∈ K 0 , we define a family of hypersurfaces ℳ ¯ N ( t ) with initial hypersurface N as follows:

If J ( ℳ N ( t ) ) < A 0 for all time t ≥ 0 , let ℳ ¯ N ( t ) = ℳ N ( t ) for all time t ≥ 0 , where ℳ N ( t ) is the solution to (1.4).
If J ( N ) < A 0 , and J ( ℳ N ( t ) ) < A 0 , and J ( ℳ N ( t ) ) reaches A 0 at the first time t 0 > 0 , we define
ℳ ¯ N ( t ) = ℳ N ( t ) , if 0 ≤ t < t 0 , ℳ N ( t 0 ) , if t ≥ t 0 .
If J ( N ) ≥ A 0 , we let ℳ ¯ N ( t ) ≡ N for all t ≥ 0 . That is, the solution is stationary.

For convenience, we call ℳ ¯ N ( t ) a modified flow of (1.4). Moreover, we have the following properties.

ℳ ¯ N ( t ) is defined for all time t ≥ 0 , and by Lemma 2.2, J ( ℳ ¯ N ( t ) ) is nondecreasing. In particular, we have J ( ℳ ¯ N ( t ) ) ≤ max { A 0 , J ( N ) } , ∀ t ≥ 0 .
If either dist ( O , N ) is very small, or Vol ( Ω 0 ) is sufficiently large or small, or e Ω 0 is sufficiently large, by Proposition 3.1, we have ℳ ¯ N ( t ) ≡ N , ∀ t ≥ 0 .

3.3 Homology of a class of ellipsoids

Here we recall the homology of a class of ellipsoids A I introduced in [13], such that an ellipsoid E with J ( E ) < A 0 is contained in A I . By Proposition 3.1, we have

Corollary 3.2

For the constant A 0 given by (3.20), there exist sufficiently small constants d ¯ and v ¯ , and sufficiently large constant e ¯ , such that for any Ω ∈ K 0 ,

If dist ( O , ∂ Ω ) ≤ d ¯ , then J ( Ω ) > A 0 .
If e Ω ≥ e ¯ , then J ( Ω ) > A 0 .
If Vol ( Ω ) ≤ v ¯ or Vol ( Ω ) ≥ 1 / ( n + 1 ) n + 1 v ¯ − 1 , then J ( Ω ) > A 0 .

Let K be the metric space consisting of nonempty, compact, and convex sets in R n + 1 , equipped with the Hausdorff distance. Denote by K ¯ 0 the closure of K 0 in K .

Fix the constants d ¯ , v ¯ , e ¯ in Corollary 3.2. Let A I be the set of ellipsoids E ∈ K ¯ 0 such that v ¯ ≤ Vol ( E ) ≤ 1 / v ¯ , and e E ≤ e ¯ . Denote by A the following subset of A I

A = { E ∈ A I : Vol ( E ) = ω n , and either e E = e ¯ or dist ( O , ∂ Ω ) = 0 } .

Here, ω n = ∣ B 1 ( 0 ) ∣ is the volume of B 1 ( 0 ) , and e E is the eccentricity of E .

We also denote by E I the set of ellipsoids in A I centered at the origin, and by E the set of ellipsoids in A centered at the origin. These sets are all metric spaces by equipping the Hausdorff distance.

It was proved in [13] that E I is contractible and so the homology H k ( E I ) = 0 for all k ≥ 1 . Moreover, A I is homeomorphic to E I × B 1 ( 0 ) . Hence, A I is contractible and the homology

(3.22) H k ( A I ) = 0 for all k ≥ 1 .

Denote

(3.23) P = { E ∈ A I : either Vol ( E ) = v ¯ , or Vol ( E ) = 1 / v ¯ , or e E = e ¯ , or O ∈ ∂ E } .

It is the boundary of A I if we regard A I as a set in the topological space of all ellipsoids. Moreover, there is a retraction Ψ from A I \ { B 1 } to P . Namely, Ψ : A I \ { B 1 } → P is continuous and Ψ ∣ P = i d . The following two theorems were also proved in [13].

Proposition 3.3

We have the following results.

H k + 1 ( P ) = H k ( A ) for all k ≥ 1 .
There is a long exact sequence
⋯ → H k + 1 ( A ) → H k ( E × S n ) → H k ( E ) ⊕ H k ( S n ) → H k ( A ) → ⋯

Proposition 3.4

Let n ∗ = n ( n + 1 ) 2 . The homology group H n ∗ + n − 1 ( E ) = Z .

3.4 Selection of a good initial condition

In this subsection, we use Propositions 3.3 and 3.4 to select a special initial condition in I such that the solution to the Gauss curvature flow (1.4) satisfies the uniform estimate. The idea is similar to that in [13]. For any ellipsoid N such that Cl ( N ) ∈ A I , let ℳ ¯ N ( t ) be the solution to the modified flow. We have the following properties:

If Cl ( N ) is close to P in Hausdorff distance or in P , we have J ( N ) ≥ A 0 and so ℳ ¯ N ( t ) ≡ N for all t (see Corollary 3.2).
If Cl ( N ) is close to B 1 ( 0 ) in Hausdorff distance, then J ( N ) < A 0 .
By our definition of the modified flow, J ( ℳ ¯ N ( t ) ) < max { A 0 , J ( N ) } for all t , then by Remark 3.1, if M ¯ N ( t ) is not identical to ℳ ¯ N ( 0 ) = N , then

(3.24) e ℳ ¯ N ( t ) ≤ e ¯ , v ¯ ≤ Vol ( ℳ ¯ N ( t ) ) ≤ 1 / v ¯ , and B d ¯ ( 0 ) ⊂ Cl ( ℳ ¯ N ( t ) ) , ∀ t ≥ 0 .

Lemma 3.5

For every t > 0 , there exists N = N t with Cl ( N ) ∈ A I , such that the minimum ellipsoid of ℳ ¯ N ( t ) is unit ball B 1 ( 0 ) .

Proof

Suppose by contradiction that there exists t ′ > 0 such that, for any Ω ∈ A I , E N ( t ′ ) ≠ B 1 ( 0 ) , where N = ∂ Ω and E N ( t ′ ) is the minimum ellipsoid of Ω N ( t ′ ) ≔ Cl ( ℳ ¯ N ( t ′ ) ) .

By Corollary 3.2, E N ( t ′ ) ∈ A I . Hence, we can define a continuous map T : A I → P by

Ω ∈ A I ↦ E N ( t ′ ) ∈ A I \ { B 1 } ↦ Ψ ( E N ( t ′ ) ) ∈ P ,

where Ψ is the retraction after (3.23), and B 1 = B 1 ( 0 ) for short. Note that when Ω ∈ P , we have J ( Ω ) ≥ A 0 and thus E N ( t ′ ) = E N ( 0 ) = Ω . This implies that T ∣ P = i d P . Hence, T is a retraction from A I to P , and so there is an injection from H ∗ ( P ) to H ∗ ( A I ) . By (3.22), we then have

H k ( P ) = 0 for all k ≥ 1 .

It follows from Proposition 3.3 (ii) that

H k ( E × S n ) = H k ( E ) ⊕ H k ( S n ) for all k ≥ 1 .

Computing the left-hand side by the Künneth formula and using the fact H k ( S n ) = Z if k = 0 or k = n , and H k ( S n ) = 0 otherwise, we further obtain

H k ( E ) ⊕ H k − n ( E ) = H k ( E ) ⊕ H k ( S n ) .

However, this contradicts Proposition 3.4 by taking k = n ∗ + 2 n − 1 in the above.□

In the following, we prove the convergence of the flow (1.4) with a specially chosen initial condition. Take a sequence t k → ∞ and let N k = N t k be the initial data from Lemma 3.5. By our choice of A 0 (see (3.20) and (3.21)), Lemma 3.5 implies that

(3.25) J ( ℳ ¯ N k ( t k ) ) ≤ 1 2 A 0 .

Hence, by the monotonicity of the functional J , we have

ℳ ¯ N k ( t ) = ℳ N k ( t ) , ∀ t ≤ t k .

Since Cl ( N k ) ∈ A I and B d ¯ ( 0 ) ⊂ Cl ( N k ) , by Blaschke’s selection theorem, there is a subsequence of N k which converges in Hausdorff distance to a limit N ∗ such that Cl ( N ∗ ) ∈ A I and B d ¯ ⊂ Cl ( N ∗ ) .

Next we show that the flow (1.4) starting from N ∗ satisfies J ( ℳ N ∗ ( t ) ) < A 0 for all t .

Lemma 3.6

For any t ≥ 0 , we have

J ( ℳ ¯ N ∗ ( t ) ) ≤ 3 4 A 0 .

Hence,

ℳ ¯ N ∗ ( t ) = ℳ N ∗ ( t ) , ∀ t > 0 .

Proof

For any given t > 0 , since N k → N ∗ and t k → ∞ , when k is sufficiently large such that t k > t , we have

J ( ℳ ¯ N ∗ ( t ) ) − J ( ℳ N k ( t ) ) ≤ 1 4 A 0 .

By the monotonicity of the functional J

J ( ℳ ¯ N k ( t ) ) ≤ J ( ℳ N k ( t k ) ) .

Combining the aforementioned two inequalities with (3.25), we obtain that

J ( ℳ ¯ N ∗ ( t ) ) = J ( ℳ ¯ N ∗ ( t ) ) − J ( ℳ N k ( t ) ) + J ( ℳ N K ( t ) ) ≤ J ( ℳ ¯ N ∗ ( t ) ) − J ( ℳ N k ( t ) ) + J ( ℳ N K ( t k ) ) ≤ 1 4 A 0 + 1 2 A 0 = 3 4 A 0 . □

3.5 Convergence of the flow and existence of solutions to (1.3)

Let Ω N ∗ ( t ) = Cl ( ℳ N ∗ ( t ) ) and u ( ⋅ , t ) be its support function. By Lemma 3.6, ℳ N ∗ ( t ) satisfies (3.24). Hence,

d ¯ ≤ u ( x , t ) ≤ C , ∀ ( x , t ) ∈ S n × [ 0 , ∞ ) ,

where C = ( n + 1 ) / ( v ¯ ω n − 1 d ¯ n ) . Hence, condition (2.11) holds, and we obtain the existence of solutions to (1.3) as follows.

Proof of Theorem 1.1

Denote ℳ ( t ) = ℳ N ∗ ( t ) and J ( t ) = J ( ℳ ( t ) ) . By Lemmas 2.2 and 3.6,

J ( t ) < A 0 and J ′ ( t ) ≥ 0 , ∀ t ≥ 0 .

Therefore,

∫ 0 ∞ J ′ ( t ) d t ≤ limsup T → ∞ J ( T ) − J ( 0 ) ≤ A 0 .

This implies that there exists a sequence t i → ∞ such that

J ′ ( t i ) = ∫ S n f 2 ψ 2 ( u , x ) r n G z − 1 ( r , ξ ) p λ − 1 ( ξ ) u − 1 K d x − ∫ S n f ψ ( u , x ) d x 2 ∫ S n r G z ( r , ξ ) p λ ( ξ ) d ξ t = t i → 0 .

Passing to a subsequence, we obtain by the priori estimates that u ( ⋅ , t i ) → u ∞ in C 3 , α ( S n ) -topology and u ∞ solves (1.3), where

γ = lim t i → ∞ 1 η ( t i ) = ∫ S n r ∞ ( ξ ) G z ( r ∞ ( ξ ) , ξ ) p λ ( ξ ) d ξ ∫ S n f ( x ) ψ ( u ∞ , x ) d x

with r ∞ = r Ω ∞ the radial function of the convex body Ω ∞ ∈ K 0 whose support function is u ∞ .□

Funding information: The research of Li has been supported by NSFC (No. 12031017), Zhejiang Provincial NSFC (No. LR23A010002) and by the Fundamental Research Funds for the Central Universities (No. 226-2022-00157). The research of Yi has been supported by Scientific Research Foundation for Scholars of HZNU (No. 4085C50220204091) and by the Zhejiang Provincial NSFC (No. LQ23A010005).
Conflict of interest: Authors state no conflict of interest.

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Received: 2022-07-14

Accepted: 2022-08-28

Published Online: 2023-01-11

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

Articles in the same Issue

https://doi.org/10.1515/ans-2022-0033

Keywords for this article

Gauss curvature flow; Monge-Ampère equation; Musielak-Orlicz-Gauss image problem; topological method

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