Generalized Gaussian curvature flows related to the Orlicz Gaussian Minkowski problem

Yannan Liu; Yuxin Peng

doi:10.1515/ans-2023-0188

Article Open Access

Generalized Gaussian curvature flows related to the Orlicz Gaussian Minkowski problem

Yannan Liu and Yuxin Peng

Published/Copyright: May 22, 2025

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From the journal Advanced Nonlinear Studies Volume 25 Issue 3

Abstract

In this paper, we investigate two anisotropic Gaussian curvature flows. Through establishing the long-time existence and congergence for these two flows, we derive the existence results for the Orlicz-Gaussian Minkowski problem in both origin-symmetric and general convex body settings.

Keywords: Monge-Ampère equation; Orlicz-Minkowski problem; Gauss curvature flow; existence of solutions

2010 Mathematics Subject Classification: 35K96; 52A20

1 Introduction

Let M ₀ be a smooth, closed, strictly convex hypersurface in the Euclidean space R n , which encloses the origin, and is given by a smooth embedding X 0 : S n − 1 → R n . Consider a family of closed hypersurfaces M t with M t = X ( S n − 1 , t ) , where X : S n − 1 × [ 0 , T ) → R n is a smooth mapping satisfying the following initial value problem:

(1) ∂ X ∂ t ( x , t ) = − ( 2 π ) n f ( ν ) K ( x , t ) 〈 X , ν 〉 e | X | 2 2 φ ( 〈 X , ν 〉 ) ν + X , X ( x , 0 ) = X 0 ( x ) .

In the whole paper, f is a positive smooth function on the unit sphere S n − 1 , ν is the unit outer normal vector of the hypersurface M _t at the point X(x, t), K ( x , t ) is the Gauss curvature of M _t at X(x, t), ⟨⋅, ⋅⟩ is the standard inner product in R n , φ is a positive smooth function defined in (0, + ∞), and T is the maximal time for which the solution exists.

Curvature flows, as an efficient method, provide alternative methods to solve elliptic PDEs arising from convex geometry, see e.g. [1]–[9].

If the flow (1) exists for all time and converges to some hypersurface, the support function h of the limiting hypersurface satisfies

(2) 1 ( 2 π ) n φ ( h ) e − | ∇ h | 2 + h 2 2 det ( ∇ 2 h + h I ) = f on S n − 1 .

Here h is a function defined on S n − 1 , ∇² h = (∇_ij h) is the Hessian matrix of covariant derivatives of h with respect to an orthonormal frame on S n − 1 , and I is the unit matrix of order n − 1. Eq. (2) is a Monge-Ampère type equation.

When φ(s) = 1, Eq. (2) is the Gaussian Minkowski problem, which was recently proposed by Huang, Xi & Zhao in [10]. For a convex body K in R n with origin O ∈ K, the Gaussian volume is defined by

γ n ( K ) = 1 ( 2 π ) n ∫ K e − | x | 2 2 d x .

In [10], the authors proved that there exists a unique even solution for Gaussian Minkowski problem, when f is even with | f | L 1 < 1 ( 2 π ) , and γ n ( K ) > 1 2 . From the volume form, we can see the limit of the Gaussian area density 1 ( 2 π ) n e − r 2 2 r n − 1 , is 0, whenever r → 0 or r → ∞. This is because that the Gaussian probability space thins out exponentially as you move away from the origin. Consequently, and hence both larger and smaller convex bodies in R n can have relatively small Gaussian surface area.

Similar to the study of Minkowski problem in Lebesgue space, it is natural to consider the L _p case for Gaussian Minkowski problem. When φ(s) = s ^1−p, Eq. (2) is just the L _p Gaussian Minkowski problem,

1 ( 2 π ) n h 1 − p e − | ∇ h | 2 + h 2 2 det ( ∇ 2 h + h I ) = f on S n − 1 ,

which was first studied in [11]. Recently, more generalizations of Gaussian Minkowski problem have been investigated, and one can refer to [12]–[14]. The planar Gaussian Minkowski problem was studied in [15]. The L _p and the dual L _p Minkowski problems in Lebesgue space have been studied extensively, one can see [16]–[21].

In this paper, we are concerned with the Orlicz case of Gaussian Minkowski problem. The Orlicz Minkowski problem in Lebesgue space is a fundamental issue in the Orlicz-Brunn-Minkowski theory in convex geometry, which has attracted great attention from many scholars, see for example, [9], [22]–[31]. The Orlicz Gaussian Minkowski problem is a generalization of the Gaussian Minkowski problem, and it asks what are the necessary and sufficient conditions for a Borel measure on the unit sphere S n − 1 to be the Orlicz Gaussian surface area measure of a convex body in R n .

We first study the long-time existence and convergence for the flow (1), and establish existence of smooth solution for the Orlicz Gaussian Minkowski problem under some conditions.

Theorem 1.

Suppose φ ∈ C ^∞(0, + ∞) satisfies

(3) lim sup s → + ∞ φ ( s ) s n − 1 e − s 2 2 < lim inf s → 0 + φ ( s ) s n − 1 .

Then, for all f ∈ C ∞ ( S n − 1 ) satisfying

(4) 1 ( 2 π ) n lim sup s → + ∞ φ ( s ) s n − 1 e − s 2 2 < f < 1 ( 2 π ) n lim inf s → 0 + φ ( s ) s n − 1 ,

the flow (1) has a smooth solution M _t for all time t > 0, and a subsequence of M _t converges in C ^∞ to a positive, smooth, uniformly convex solution to Eq. (2).

The similar condition (4) was used by the first author and her collaborator in [8] to prove the existence of the dual Orlicz Minkowski problem. We also consider the normalized Orlicz Gaussian Minkowski problem:

(5) c φ ( h ) e − | ∇ h | 2 + h 2 2 det ( ∇ 2 h + h I ) = f on S n − 1 ,

where c is a positive constant.

To obtain the existence of solution for Eq. (5), we construct flow

(6) ∂ X ∂ t ( x , t ) = − f ( ν ) K ( x , t ) 〈 X , ν 〉 e | X | 2 2 φ ( 〈 X , ν 〉 ) η ( t ) ν + X , X ( x , 0 ) = X 0 ( x ) .

η is given by

(7) η ( t ) = ∫ S n − 1 e − ρ 2 2 ρ ( u , t ) n d u ∫ S n − 1 f ( x ) h ( x , t ) / φ ( h ) d x ,

where ρ(⋅, t) and h(⋅, t) are the radial function and support function of the convex hypersurface M _t respectively. We will see η(t) in (6) is used to keep M _t normalized in a certain sense.

To get the long-time existence of the flow (6), we need to impose some constraints on φ.

(A) φ is a continuous and positive function defined in (0, + ∞) such that ϕ ( s ) = ∫ 0 s 1 / φ ( τ ) d τ exists for every s > 0 and is unbounded as s → +∞;

(B) φ is a continuous and positive function defined in (0, + ∞) such that for every s > 0, ϕ ( s ) = ∫ s + ∞ 1 / φ ( τ ) d τ exists, and for s near 0, ϕ(s) ≤ Ns ^p for some positive constant N and some number p ∈ (−n, 0).

One can easily see that the special case φ(s) = s ^1−p satisfies (A) when p > 0, and (B) when −n < p < 0. In fact, these two assumptions have been used in [22], [24]–[26], [32] to prove existence results of Olicz-Minkowsi problem by variational methods, and two more assumptions on φ were needed: lim s → 0 + φ ( s ) = 0 and φ is monotone increasing. The authors used the two assumptions in [9] to obtain the existence of solution for Olicz-Minkowsi problem without the monotone condition on φ.

When f is even, namely f(−x) = f(x) for any x ∈ S n − 1 , we obtain the following long-time existence and convergence of the flow (6).

Theorem 2.

Assume M ₀ is a smooth, closed, origin-symmetric, uniformly convex hypersurface in R n . If f is a smooth and even function on S n − 1 , and φ ∈ C ^∞(0, + ∞) satisfies (A) or (B), then the flow (6) has a unique smooth solution M _t for all time t > 0. Moreover, when t → ∞, a subsequence of M _t converges in C ^∞ to a smooth, closed, origin-symmetric, uniformly convex hypersurface, whose support function is a smooth even solution to Eq. (5) for some positive constant c.

As applications, we have

Corollary 1.

Assume φ and f are smooth and positive functions on S n − 1 satisfying (3) and (4) respectively, then there exists a smooth solution to Eq. (2).

Corollary 2.

Assume f is a smooth and even function on S n − 1 . If φ is a smooth function satisfying (A) or (B), then there exists a smooth even solution to Eq. (5) for some positive constant c.

This paper is organized as follows. In Section 2, we give some basic knowledge about the flow (1). In Section 3, the long-time existence of the flow (1) will be proved. First, we derive uniform positive upper and lower bounds for support functions of M t . Then, the bounds of principal curvatures are derived via proper auxiliary functions and delicate computations. So the long-time existence follows by standard arguments. In Section 4, by considering some geometric functionals, we complete the proofs of Theorems 1 and 2.

2 Preliminaries

Let R n be the n-dimensional Euclidean space, and S n − 1 be the unit sphere in R n . Assume M is a smooth closed uniformly convex hypersurface in R n . We may assume that M encloses the origin. The support function h of M is defined as

h ( x ) ≔ max y ∈ M ⟨ y , x ⟩ , ∀ x ∈ S n − 1 ,

where ⟨⋅, ⋅⟩ is the standard inner product in R n . And the radial function ρ of M is given by

ρ ( u ) ≔ max λ > 0 : λ u ∈ M , ∀ u ∈ S n − 1 .

Note that ρ(u)u ∈ M.

Denote the Gauss map of M by ν _M. Then M can be parametrized by the inverse Gauss map X : S n − 1 → M with X ( x ) = ν M − 1 ( x ) . The support function h of M can be computed by

(8) h ( x ) = ⟨ x , X ( x ) ⟩ , x ∈ S n − 1 .

Note that x is just the unit outer normal of M at X(x). Let e _ij be the standard metric of the sphere S n − 1 , and ∇ be the corresponding connection on S n − 1 . Differentiating (8), we have

∇ i h = 〈 ∇ i x , X ( x ) 〉 + 〈 x , ∇ i X ( x ) 〉 .

Since ∇_i X(x) is tangent to M at X(x), we have

∇ i h = ⟨ ∇ i x , X ( x ) ⟩ .

It follows that

(9) X ( x ) = ∇ h + h x .

By differentiating (8) twice, the second fundamental form A _ij of M can be computed in terms of the support function, see for example [33],

(10) A i j = ∇ i j h + h e i j ,

where ∇_ij = ∇_i∇_j denotes the second order covariant derivative with respect to e _ij. The induced metric matix g _ij of M can be derived by Weingarten’s formula,

(11) e i j = ⟨ ∇ i x , ∇ j x ⟩ = A i k A l j g k l .

The principal radii of curvature are the eigenvalues of the matrix b _ij = A ^ik g _jk. When considering a smooth local orthonormal frame on S n − 1 , by virtue of (10) and (11), we have

(12) b i j = A i j = ∇ i j h + h δ i j .

We will use b ^ij to denote the inverse matrix of b _ij. The Gauss curvature of X(x) ∈ M is given by

K ( x ) = [ det ( ∇ i j h + h δ i j ) ] − 1 .

From the evolution equations of X(x, t) in flow (1) and (6), we derive the evolution equations of the corresponding support functions h(x, t):

(13) ∂ h ∂ t ( x , t ) = − ( 2 π ) n f ( x ) e | ∇ h | 2 + h 2 2 K h / φ ( h ) + h ( x , t ) , x ∈ S n − 1

and

(14) ∂ h ∂ t ( x , t ) = − η ( t ) f ( x ) e | ∇ h | 2 + h 2 2 K h / φ ( h ) + h ( x , t ) , x ∈ S n − 1 .

Denote the radial function of M _t by ρ(u, t). From (9), u and x are related by

(15) ρ ( u ) u = ∇ h ( x ) + h ( x ) x .

The evolution equations of the radial function under flows (1) and (6) then follow from (13) and (14)

(16) ∂ ρ ∂ t ( u , t ) = − ( 2 π ) n f ( x ) e ρ 2 2 K ρ / φ ( h ) + ρ ( u , t ) ,

and

(17) ∂ ρ ∂ t ( u , t ) = − η ( t ) f ( x ) e ρ 2 2 K ρ / φ ( h ) + ρ ( u , t ) ,

where K denotes the Gauss curvature at ρ(u, t)u ∈ M _t and f takes value at the unit normal vector x(u, t).

From

γ n ( t ) = ∫ S n − 1 d u ∫ 0 ρ ( u , t ) e − r 2 2 r ( u , t ) n − 1 d r ,

we have the following computations:

∂ t γ n ( t ) = ∫ S n − 1 e − ρ 2 2 ρ ( u , t ) n − 1 ∂ t ρ d u , = ∫ S n − 1 e − ρ 2 2 ρ n d u − η ( t ) ∫ S n − 1 ρ n f ( x ) K / φ ( h ) d u , = ∫ S n − 1 e − ρ 2 2 ρ n d u − η ( t ) ∫ S n − 1 f ( x ) h / φ ( h ) d x .

By the expression of η(t) in (7), we have

(18) ∂ t γ n ( t ) ≡ 0 ,

namely the Gaussian volume of the convex body bounded by M _t remains unchanged.

3 Long-time existence of the flow

In this section, we will give a priori estimates about support functions and obtain the long-time existence of flows (1) and (6) under assumptions of Theorems 1 and 2.

In the following of this paper, we always assume that M ₀ is a smooth, closed, uniformly convex hypersurface in R n , f is a smooth, positive and function on S n − 1 , and h : S n − 1 × [ 0 , T ) → R is a smooth solution to the evolution equation (14) with the initial h(⋅, 0) the support function of M ₀. Here T is the maximal time for which the solution exists. Let M _t be the convex hypersurface determined by h(⋅, t), and ρ(⋅, t) be the corresponding radial function.

We first prove h(⋅, t) has uniform positive upper and lower bounds for t ∈ [0, T) under the flow (1).

Lemma 1.

Let h be a smooth solution of (13) on S n − 1 × [ 0 , T ) , and suppose f, φ are smooth functions satisfying (3) and (4). Then

1 C ≤ h ( x , t ) ≤ C ,

where C is a positive constant depending on max S n − 1 h ( x , 0 ) , min S n − 1 h ( x , 0 ) , max S n − 1 f ( x ) , min S n − 1 f ( x ) , lim sup s → + ∞ φ ( s ) s n − 1 e − s 2 2 and lim inf s → 0 + φ ( s ) s n − 1 .

Proof.

Suppose max S n − 1 h ( x , t ) is attained at x 0 ∈ S n − 1 . Then at x ⁰, we have

∇ h = 0 , ∇ 2 h ≤ 0 , ρ = h ,

and

∇ 2 h + h I ≤ h I .

So at x ⁰,

∂ h ∂ t = − ( 2 π ) n f ( x ) K ( x ) h φ ( h ) e ρ 2 2 + h ≤ − ( 2 π ) n f ( x ) h 2 − n φ ( h ) e ρ 2 2 + h = ( 2 π ) n h 2 − n φ ( h ) e ρ 2 2 1 ( 2 π ) n h n − 1 φ ( h ) e − ρ 2 2 − f ( x )

Let A ̄ = lim sup s → + ∞ φ ( s ) s n − 1 e − s 2 2 . By (4), ε = 1 2 ( ( 2 π ) n min S n − 1 f ( x ) − A ̄ ) is positive and there exists a positive constant C ₁ > 0 such that

h n − 1 φ ( h ) e − h 2 2 < A ̄ + ε

for h > C ₁. It follows that

1 ( 2 π ) n h n − 1 φ ( h ) e − ρ 2 2 − f ( x ) < 1 ( 2 π ) n A ̄ + 1 ( 2 π ) n ε − min S n − 1 f ( x ) < 0 ,

from which we have

∂ h ∂ t < 0

at maximal points. So

h ≤ max C 1 , max S n − 1 h ( x , 0 ) .

Similarly, we can estimate min S n − 1 h ( x , t ) . Suppose min S n − 1 h ( x , t ) is attained at x ¹, then at this point,

∇ h = 0 , ∇ 2 h ≥ 0 , ρ = h ,

and

∇ 2 h + h I ≥ h I .

And at x ¹,

∂ h ∂ t ≥ − ( 2 π ) n f ( x ) h 1 − n h φ ( h ) e ρ 2 2 + h = ( 2 π ) n h 2 − n φ ( h ) e ρ 2 2 1 ( 2 π ) n h n − 1 φ ( h ) e − ρ 2 2 − f ( x ) .

Let A ̲ = lim inf s → 0 + φ ( s ) s n − 1 . Noticing that lim s → 0 e − ρ 2 2 = 1 , then by (4), ε = 1 2 ( A ̲ − ( 2 π ) n max S n − 1 f ( x ) ) is positive and there exists some positive constant C ₂ > 0 such that

h n − 1 φ ( h ) e − ρ 2 2 > A ̲ − ε

for h < C ₂. It follows by (4)

h n − 1 φ ( h ) e − ρ 2 2 − f ( x ) > A ̲ − ε − max S n − 1 f ( x ) > 0 ,

which shows that

∂ h ∂ t > 0

at minimal points. Hence

h ≥ min C 2 , min S n − 1 h ( x , 0 ) .

The proof of the lemma is completed. □

Now, we prove h(⋅, t) has the uniform positive upper and lower bounds of for t ∈ [0, T) under the flow (6).

Lemma 2.

Under the assumptions of Theorem 2, there exists a positive constant C independent of t, such that for every t ∈ [0, T)

1 / C ≤ ρ ( ⋅ , t ) ≤ C on S n − 1 .

It means that

1 / C ≤ h ( ⋅ , t ) ≤ C on S n − 1 .

Proof.

Let

(19) J ( t ) = ∫ S n − 1 ϕ ( h ( x , t ) ) f ( x ) d x , t ≥ 0 .

First suppose φ satisfies (A). We claim that in this case J(t) is non-increasing. In fact, recalling (14), we have

J ′ ( t ) = ∫ S n − 1 ϕ ′ ( h ( x , t ) ) ∂ t h f ( x ) d x = ∫ S n − 1 − η ( t ) f ( x ) e ρ 2 2 K h / φ ( h ) + h f ( x ) / φ ( h ) d x = ∫ S n − 1 f ( x ) h / φ ( h ) d x − η ( t ) ∫ S n − 1 e ρ 2 2 f ( x ) 2 K h / φ ( h ) 2 d x .

By the definition of η(t) in (7), there is

η ( t ) = ∫ S n − 1 e − ρ 2 2 h / K d x ∫ S n − 1 f ( x ) h / φ ( h ) d x .

Hence

(20) J ′ ( t ) ∫ S n − 1 f ( x ) h / φ ( h ) d x = ∫ S n − 1 f ( x ) h / φ ( h ) d x 2 − ∫ S n − 1 e − ρ 2 2 h / K d x ⋅ ∫ S n − 1 e ρ 2 2 f ( x ) 2 K h / φ ( h ) 2 d x = ∫ S n − 1 h / K e − ρ 2 2 ⋅ f φ ( h ) K h e ρ 2 2 d x 2 − ∫ S n − 1 e − ρ 2 2 h / K d x ⋅ ∫ S n − 1 e ρ 2 2 f ( x ) 2 K h / φ ( h ) 2 d x ≤ 0 ,

where the last inequality is due to the Hölder’s inequality. Therefore, J(t) is non-increasing.

For each t, write

R t = max u ∈ S n − 1 ρ ( u , t ) = ρ ( u t , t )

for some u t ∈ S n − 1 . Since M _t is origin-symmetric, we have by the definition of support function that

h ( x , t ) ≥ R t | ⟨ x , u t ⟩ | , ∀ x ∈ S n − 1 .

Since J(t) is non-increasing, it follows

J ( 0 ) ≥ J ( t ) = ∫ S n − 1 ϕ ( h ( x , t ) ) f ( x ) d x ≥ f min ∫ S n − 1 ϕ ( h ( x , t ) ) d x ≥ f min ∫ S n − 1 ϕ ( R t | ⟨ x , u t ⟩ | ) d x = f min ∫ S n − 1 ϕ ( R t | x 1 | ) d x .

Denote S 1 = x ∈ S n − 1 : | x 1 | ≥ 1 / 2 , then

J ( 0 ) ≥ f min ∫ S n − 1 ϕ ( R t / 2 ) d x = f min ϕ ( R t / 2 ) | S 1 | ,

which implies that ϕ(R _t/2) is uniformly bounded from above. By assumption (A), ϕ(s) is strictly increasing and tends to +∞ as s → +∞. Thus R _t is uniformly bounded from above.

Recalling γ _n(t) ≡ γ _n(0) by (18), one can easily obtain the uniform positive lower bound of ρ(⋅, t). In fact, by the concept of minimum ellipsoid of a convex body, there exists a positive constant C _n depending only on n, such that

γ n ( t ) ≤ V o l n ( t ) ≤ C n R t n − 1 ⋅ min u ∈ S n − 1 ρ ( u , t ) .

Thus the uniform positive lower bound of ρ(⋅, t) follows from their uniform upper bound.

When φ satisfies (B), we see that ϕ′(s) = −1/φ(s). From (19), we have

J ′ ( t ) = ∫ S n − 1 ϕ ′ ( h ( x , t ) ) ∂ t h f ( x ) d x = − ∫ S n − 1 [ − η ( t ) f ( x ) K h / φ ( h ) + h ] f ( x ) / φ ( h ) d x = η ( t ) ∫ S n − 1 f ( x ) 2 K h / φ ( h ) 2 d x − ∫ S n − 1 f ( x ) h / φ ( h ) d x ≥ 0 ,

which shows that J(t) is non-decreasing.

Let a be a positive number to be determined. Write

S t = x ∈ S n − 1 : h ( x , t ) ≥ a .

Since ϕ is non-increasing,

∫ S t ϕ ( h ( x , t ) ) f ( x ) d x ≤ ∫ S t ϕ ( a ) f ( x ) d x ≤ ϕ ( a ) f L 1 ( S n − 1 ) .

By lim_s→+∞ ϕ(s) = 0, one can take a sufficiently large a, such that

∫ S t ϕ ( h ( x , t ) ) f ( x ) d x < J ( 0 ) / 2 , ∀ t ∈ [ 0 , T ) .

Note that a depends only on f and φ.

Recalling assumption (B), when s near 0, ϕ(s) ≤ Ns ^p for some positive constant N and some number p ∈ (−n, 0). Since ϕ is smooth in (0, + ∞), one can easily see that there exists a positive number N ̃ such that

ϕ ( s ) ≤ N ̃ s p , ∀ s ∈ ( 0 , a ) .

Now we can estimate J(t) as follows:

J ( t ) = ∫ S n − 1 \ S t + ∫ S t ϕ ( h ( x , t ) ) f ( x ) d x ≤ N ̃ ∫ S n − 1 \ S t h ( x , t ) p f ( x ) d x + J ( 0 ) / 2 ≤ N ̃ f max ∫ S n − 1 h ( x , t ) p d x + J ( 0 ) / 2 ,

which together with J(t) ≥ J(0) implies that

∫ S n − 1 h ( x , t ) p d x ≥ J ( 0 ) 2 N ̃ f max .

Noticing that h(⋅, t) is even, p ∈ (−n, 0) and γ _n(t) ≡ γ _n(0), by a simple computation, one can see that h(⋅, t) has uniform positive upper and lower bounds. For the more details can also see [14]. □

By the equality ρ ² = h ² + |∇h|², we can easily obtain the gradient estimates from the previous lemmas.

Corollary 3.

Under the assumptions of Theorem 1 or 2, we have

| ∇ h ( x , t ) | ≤ C , ∀ ( x , t ) ∈ S n − 1 × [ 0 , T ) ,

where C is a positive constant depending only on constants in Lemmas 1 or 2.

To obtain the long-time existence of the flow (1) and (6), we further need to establish uniform upper and lower bounds for the principal curvature. In this paper, we only give details of proof for flow (6), and for the flow (1), it is almost the same.

In the rest of this section, we take a local orthonormal frame {e ₁, …, e _n−1} on S n − 1 such that the standard metric on S n − 1 is {δ _ij}. Double indices always mean to sum from 1 to n − 1. For convenience, we also write

F = η ( t ) e ρ 2 2 f ( x ) K ( x ) h / φ ( h ) .

By Lemma 2, for any t ∈ [0, T), h(⋅, t) always ranges within a bounded interval I′ = [1/C, C], where C is the constant in these two lemmas.

To obtain the C ² estimate of h, we first derive the upper bound of the Gaussian curvature for the flow (6).

Lemma 3.

Under the assumptions of Theorem 2, we have

K ( x , t ) ≤ C , ∀ ( x , t ) ∈ S n − 1 × [ 0 , T ) ,

where C is a positive constant independent of t.

Proof.

Consider the following auxiliary function:

Q ( x , t ) = 1 h − ε 0 ( F − h ) = − h t h − ε 0 ,

where ɛ ₀ is a positive constant satisfying

ε 0 < min S n − 1 × [ 0 , T ) h ( x , t ) .

Recalling that F = η ( t ) f ( x ) e ρ 2 2 K ( x ) h / φ ( h ) and that h has uniform positive upper and lower bounds, the upper bound of K ( x , t ) follows from that of Q(x, t). Hence we only need to derive the upper bound of Q(x, t).

It is easy to obtain the evolution equation of Q(x, t) and F:

∂ Q ∂ t − F b i j ∇ i j Q = 1 h − ε 0 F t − F b i j F i j + Q + Q 2 + ( Q + 1 ) F b i j h i j h − ε 0 + ∇ i Q F b i j h j h − ε 0 + ∇ j Q F b i j h i h − ε 0

and

∂ F ∂ t − F b i j ∇ i j F = − F ( n − 1 ) + F 2 b i j δ i j + K ( x , t ) f ( x ) ∂ ∂ t h φ ( h ) e ρ 2 2 η ( t ) .

At a spatial maximal point of Q(x, t), if we take an orthonormal frame such that b _ij is diagonal, we have

∂ Q ∂ t − b i i F ∇ i i Q ≤ 1 h − ε 0 F t − b i i F ∇ i i F + Q + Q 2 + F b i i h i i h − ε 0 + Q F b i i h i i h − ε 0 ≤ F Q 1 − h h − ε 0 ∑ i b i i + C 1 Q + C 2 Q 2 + 1 h − ε 0 K ( x , t ) f ( x ) ∂ ∂ t h φ ( h ) e ρ 2 2 η ( t ) .

Since

∂ η ( t ) ∂ t = 1 ∫ S n − 1 h φ ( h ) f ( x ) d x ∫ S n − 1 e − ρ 2 2 − ρ ρ t + n ρ n − 1 ρ t d u − ∫ S n − 1 ρ n d u ∫ S n − 1 h φ ( h ) f ( x ) d x 2 ∫ S n − 1 f ( x ) 1 φ ( h ) − φ ′ ( h ) h φ 2 h t d x = 1 ∫ S n − 1 h φ ( h ) f ( x ) d x ∫ S n − 1 f ( x ) h φ ( h ) η ( t ) ρ 2 − n − e − ρ 2 2 h ρ n − 2 K d x − ∫ S n − 1 ρ n d u ∫ S n − 1 h φ ( h ) f ( x ) d x 2 ∫ S n − 1 f ( x ) 1 φ ( h ) − φ ′ ( h ) h φ 2 h t d x ≤ 1 ∫ S n − 1 h φ ( h ) f ( x ) d x ∫ S n − 1 f ( x ) h φ ( h ) η ( t ) ρ 2 − n d x − ∫ S n − 1 ρ n d u ∫ S n − 1 h φ ( h ) f ( x ) d x 2 ∫ S n − 1 f ( x ) 1 φ ( h ) − φ ′ ( h ) h φ 2 h t d x ≤ C 3 + C 4 Q ,

where C ₃ and C ₄ are positive constants depending on f C ( S n − 1 ) , ‖ φ ‖ C 1 ( I ′ ) and ‖ h ‖ C 1 ( S n − 1 × [ 0 , T ) ) . In the above inequality we have used the fact that

0 = ∂ t γ n ( t ) = ∫ S n − 1 e − ρ 2 2 ρ n − 1 ρ t d u .

Now we have

∂ ∂ t h φ ( h ) e ρ 2 2 η ( t ) = e ρ 2 2 h φ ( h ) ∂ η ( t ) ∂ t + 1 φ ( h ) − φ ′ ( h ) h φ 2 η ( t ) h t + ρ 2 h h t ≤ C 5 + C 6 Q .

For Q large enough, there is

1 / C 0 K ≤ Q ≤ C 0 K ,

and

∑ i b i i ≥ ( n − 1 ) K 1 n − 1 .

Hence, for large Q, we obtain

∂ Q ∂ t ≤ C 1 Q 2 C 2 − ε 0 Q 1 n − 1 < 0 .

Then the upper bound of K ( x , t ) follows. □

Now we can estimate lower bounds of principal curvatures κ _i(x, t) of M _t for i = 1, …, n − 1.

Lemma 4.

Under the assumptions of Theorem 2, we have

κ i ≥ C , ∀ ( x , t ) ∈ S n − 1 × [ 0 , T ) ,

where C is a positive constant independent of t.

Proof.

Consider the auxiliary function

w ( x , t ) = log λ max ( b i j ) − A ⁡ log ⁡ h + B | ∇ h | 2 ,

where A, B are positive constants to be determined, and λ _max is the maximal eigenvalue of b _ij.

For any fixed t, we assume that max S n − 1 w ( x , t ) is attained at q ∈ S n − 1 . At q, we take an orthogonal frame such that b _ij(q, t) is diagonal and λ _max(q, t) = b ₁₁(q, t). Now we can write w(x, t) as

w ( x , t ) = log b 11 − A ⁡ log ⁡ h + B | ∇ h | 2 .

We first compute the evolution equation of w as follows

(21) ∂ w ∂ t − F b i i ∇ i i w = b 11 ∂ b 11 ∂ t − F b i i ∇ i i b 11 + F b i i ( b 11 ) 2 ( ∇ i b 11 ) 2 − A h ∂ h ∂ t − F b i i ∇ i i h − F b i i h i 2 h 2 + 2 B h k ∂ h k ∂ t − F b i i ∇ i i h k − 2 B F b i i h i i 2 .

Let

M = log f ( x ) e ρ 2 2 η ( t ) h / φ ( h ) ,

then

log ⁡ F = log K + M .

Differentiating the above equation, we have that

∇ k F F = 1 K ∂ K ∂ b i j ∇ k b i j + ∇ k M = − b i j ∇ k b i j + M k ,

and

∇ k l F F − ∇ k F ∇ l F F 2 = − b i j ∇ k l b i j + b i i b j j ∇ k b i j ∇ l b i j + ∇ l k M .

Recalling the evolution equation of h, we have

(22) ∂ h ∂ t − F b i i ∇ i i h = − F n + h + F h ∑ i b i i ,

and

(23) ∂ h k ∂ t − F b i i ∇ i i h k = − M k F + h k + F h k ∑ i b i i .

We also have

∂ h k l ∂ t = − ∇ k l F + h k l = − ∇ k F ∇ l F F + F b i j ∇ k l b i j − F b i i b j j ∇ k b i j ∇ l b i j − F ∇ l k M + h k l .

By the Gauss equation, see e.g. [33] for details,

∇ k l h i j = ∇ i j h k l + 2 δ k l h i j − 2 δ i j h k l + δ k j h i l − δ l i h k j ,

we get

∂ b k l ∂ t = F b i j ∇ i j b k l + δ k l F ( n − 2 ) − F b i j δ i j b k l + F b i k h i l − F b j l h k j + b k l − ∇ k F ∇ l F F − F b i i b j j ∇ k b i j ∇ l b i j − F ∇ l k M .

When k = l = 1, the above equation is

(24) ∂ b 11 ∂ t = F b i i ∇ i i ( b 11 ) + F ( n − 2 ) − F ∑ i b i i b 11 + b 11 − F 1 2 F − F b i i b j j ∇ 1 ( b i j ) 2 − F M 11 .

Inserting (22), (23) and (24) into (21), we obtain

∂ w ∂ t − F b i i ∇ i i w ≤ b 11 F ( n − 2 ) + 1 − A − b 11 F M 11 + A F n h − A F ∑ i b i i − 2 B F h k M k + 2 B | ∇ h | 2 + 2 B F | ∇ h | 2 ∑ i b i i − 2 B F ∑ i b i i + 4 B F h ( n − 1 ) = b 11 F ( n − 2 ) − b 11 F M 11 − 2 B F h k M k − ( A − 2 B | ∇ h | 2 ) F ∑ i b i i − ( A − 1 − 2 B | ∇ h | 2 ) − 2 B F ∑ i b i i + 4 B F h ( n − 1 ) + A F n h .

If we let A satisfy

A ≥ 2 B max S n − 1 × [ 0 , T ) | ∇ h | 2 + 1 ,

then

(25) ∂ w ∂ t − F b i i ∇ i i w ≤ b 11 F ( n − 2 ) − b 11 F M 11 − 2 B F h k M k − 2 B F ∑ i b i i + 4 B F h ( n − 1 ) + A F n h .

Now, we estimate −b ¹¹ FM ₁₁ − 2BFh _k M _k. Since

M = log η ( t ) f ( x ) h / φ ( h ) e ρ 2 2 = log ⁡ f ( x ) + log ⁡ h ( x ) − log ⁡ φ ( h ) + log ⁡ η ( t ) + 1 2 ρ 2 ,

then

∇ k M = f k f + h k h + ρ ρ k − φ ′ φ h k = f k f + h k h + h h k + h k i h i − φ ′ φ h k ,

where we have used the relation ρ ² = h ² + |∇h|². And

∇ 11 M = f 11 f − f 1 2 f 2 + h 11 h − h 1 2 h 2 − φ ″ h 1 2 + φ ′ h 11 φ + ( φ ′ ) 2 h 1 2 φ 2 + h 1 2 + h h 11 + h i h 11i + h 1 i 2 .

Therefore, we obtain

− 2 B h k M k = − 2 B h k f k f + h k h + h k h + h h k + h k i h i − φ ′ φ h k ≤ 2 B | ∇ f ‖ ∇ h | f + | ∇ h | 2 h + h | ∇ h | 2 + | ∇ h | 2 | φ ′ | φ − 2 B h k i h i h k ≤ c 1 B − 2 B h k i h i h k ,

where c ₁ is a positive constant depending on upper and lower bounds of f, φ(h) and h, and upper bounds of their first order derivatives. We also have

− b 11 M 11 = − b 11 f 11 f − f 1 2 f 2 − h 1 2 h 2 − φ ″ h 1 2 φ + ( φ ′ ) 2 h 1 2 φ 2 + h 1 2 + b 11 φ ′ φ + h − 1 h ( b 11 − h ) − b 11 h 1 i 2 − b 11 h i h 11i ≤ c 2 b 11 + c 3 − b 11 h i b 11i ,

where c ₂, c ₃ are positive constants depending on ‖ φ ‖ C 2 ( I ′ ) , ‖ f ‖ C 2 ( S n − 1 ) , ‖ h ‖ C 1 ( S n − 1 × [ 0 , T ) ) , and lower bounds of φ(h), f and h.

From the definition of w,

b 11 ∇ l b 11 = ∇ l w + A h h l − 2 B b l k h k + 2 B h h l .

At point q, we have

− b 11 M 11 ≤ c 4 B + c 5 b 11 + c 6 + 2 B h k 2 b k k ,

where c ₄, c ₅ and c ₆ are the following c _i are all positive constants independent of t.

Thus, we have proved that

− b 11 F M 11 − 2 B F h k M k ≤ F c 7 B + c 8 b 11 + c 9 .

Now, from (25) we have

∂ w ∂ t − F b i i ∇ i i w ≤ F c 10 B + c 11 b 11 + c 12 − 2 B F ∑ i b i i + 4 B F h ( n − 1 ) + A F n h .

If we take B = 1, then for b _ii large enough, there is

∂ w ∂ t − F b i i ∇ i i w ≤ F c 10 + c 11 b 11 + c 12 − 2 F ∑ i b i i + 4 F h ( n − 1 ) + A F n h < 0 ,

which implies that

∂ w ∂ t < 0 .

Therefore w has a uniform upper bound, and so does λ _max(b _ij). The conclusion of this lemma then follows. □

Combining Lemma 3, we see that the principal curvatures of M _t has uniform positive upper and lower bounds. This together with Lemma 2 and Corollary 3 implies that the evolution equation (14) is uniformly parabolic on any finite time interval. Thus, the standard parabolic theory show that the smooth solution of (14) exists for all time. And by these estimates again, a subsequence of M _t converges in C ^∞ to a positive, smooth, uniformly convex hypersurface M _∞ in R n .

4 Convergence of the flow

In this section, we will complete the proofs of Theorems 1 and 2.

Proof of Theorem 1: To obtain the convergence, we define the functional

L ( M ) = ∫ S n − 1 ( 2 π ) n ϕ ( h ) f ( x ) d x − ∫ S n − 1 G ̃ ( ρ , u ) d u ,

where

G ̃ ( r , u ) = ∫ − ∞ r e − s 2 2 s n − 1 d s , ϕ ( t ) = ∫ 0 t 1 φ ( s ) d s .

First, we show that the functional L(M _t) is non-increasing along the flow (1).

We begin from the equation

∂ L ∂ t = ∫ S n − 1 ( 2 π ) n f ( x ) φ ( h ) ∂ h ∂ t d x − ∫ S n − 1 e − ρ 2 2 ρ n − 1 ∂ ρ ∂ t d u .

Since

1 ρ ( u , t ) ∂ ρ ∂ t = 1 h ( x , t ) ∂ h ∂ t ,

then by the fact

ρ n d u = h ( x ) K ( x ) d x ,

we have

∂ L ∂ t = ∫ S n − 1 ( 2 π ) n f ( x ) φ ( h ) − e − ρ 2 2 K ( x ) ∂ h ∂ t d x = − ∫ S n − 1 e − ρ 2 2 K ( x ) h ( 2 π ) n f ( x ) φ ( h ) K ( x ) e ρ 2 2 − 1 2 d x ≤ 0 .

The equality holds if and only if

( 2 π ) n f ( x ) φ ( h ) K ( x ) e ρ 2 2 = 1 .

From Lemma 1, Corollary 3 and the definition of L(M _t), there exists a positive constant C which is independent of t, such that

(26) | L ( M t ) | ≤ C , ∀ t > 0 .

It is easy to see

(27) L ( M 0 ) − L ( M t ) = ∫ 0 t d t ∫ S n − 1 e − | X t | 2 2 K ( x ) h ( x , t ) ( 2 π ) n f ( x ) φ ( h ) K ( x ) e | X t | 2 2 − 1 2 d x .

Combining (26) and (27), and recalling Lemmas 3 and 4, we have that

∫ 0 ∞ d t ∫ S n − 1 ( 2 π ) n f ( x ) φ ( h ) K ( x ) e | X t | 2 2 − 1 2 d x < ∞ .

This implies that there exists a subsequence of times t _j → ∞ such that

∫ S n − 1 ( 2 π ) n f ( x ) φ ( h ) K ( x ) e | X t j | 2 2 − 1 2 d x → 0 as t j → ∞ .

Passing to the limit, we obtain

∫ S n − 1 ( 2 π ) n f ( x ) φ ( h ∞ ) K ( x ) e | ∇ ̄ h ∞ | 2 2 − 1 2 d x = 0 ,

where h _∞ is the support function of M _∞, and X ∞ = ∇ ̄ h ∞ . This implies that

( 2 π ) n f ( x ) φ ( h ∞ ) K ( x ) e | ∇ ̄ h ∞ | 2 2 = 1 , ∀ x ∈ S n − 1 ,

which is just Eq. (2). The proof of Theorem 1 is now completed.

Proof of Theorem 2:

Let h ̃ be the support function of M _∞. We need to prove that h ̃ is a solution to the equation (5) for some positive constant c.

Recall the functional

J ( t ) = ∫ S n − 1 ϕ ( h ( x , t ) ) f ( x ) d x , t ≥ 0 .

By the assumptions on ϕ, and Lemma 2, there exists a positive constant C independent of t, such that

(28) J ( t ) ≤ C , ∀ t ≥ 0 .

We also note that, by proof of Lemma 2, J(t) is non-increasing when φ satisfies (A), and non-decreasing when φ satisfies (B).

We now begin the proof with the assumption (A). Recalling J′(t) ≤ 0 for any t > 0. From

∫ 0 t [ − J ′ ( t ) ] d t = J ( 0 ) − J ( t ) ≤ J ( 0 ) ,

we have

∫ 0 ∞ [ − J ′ ( t ) ] d t ≤ J ( 0 ) ,

This implies that there exists a subsequence of times t _j → ∞ such that

− J ′ ( t j ) → 0 as t j → ∞ .

Recalling (20):

J ′ ( t j ) ∫ S n − 1 f ( x ) h / φ ( h ) d x = ∫ S n − 1 h / K e − ρ 2 2 ⋅ f φ ( h ) K h e ρ 2 2 d x 2 − ∫ S n − 1 e − ρ 2 2 h / K d x ⋅ ∫ S n − 1 e ρ 2 2 f ( x ) 2 K h / φ ( h ) 2 d x

Since h and K have uniform positive upper and lower bounds, by passing to the limit, we obtain

∫ S n − 1 e − ρ ̃ 2 2 h ̃ / K ̃ ⋅ f φ ( h ̃ ) K ̃ e ρ ̃ 2 2 h ̃ d x 2 = ∫ S n − 1 e − ρ ̃ 2 2 h ̃ / K ̃ d x ⋅ ∫ S n − 1 f 2 K ̃ e ρ ̃ 2 2 h ̃ / φ ( h ̃ ) 2 d x ,

where K ̃ is the Gauss curvature of M _∞. By the equality condition for the Hölder’s inequality, there exists a constant c ≥ 0 such that

c e − ρ ̃ 2 2 φ ( h ̃ ) / K ̃ = f on S n − 1 ,

which is just Eq. (5). Note h ̃ and K ̃ have positive upper and lower bounds, c should be positive.

For the proof with the assumption (B). Recalling J′(t) ≥ 0 for any t > 0. By estimate (28),

∫ 0 t J ′ ( t ) d t = J ( t ) − J ( 0 ) ≤ J ( t ) ≤ C ,

which leads to

∫ 0 ∞ J ′ ( t ) d t ≤ C .

This implies that there exists a subsequence of times t _j → ∞ such that

J ′ ( t j ) → 0 as t j → ∞ .

Now using almost the same arguments as above, one can prove h ̃ solves Eq. (5) for some positive constant c. The proof of Theorem 2 is completed.

Corresponding author: Yannan Liu, School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, 100048, P.R. China, E-mail: liuyn@th.btbu.edu.cn

Funding source: the National Natural Science Foundation of China

Award Identifier / Grant number: 12141103

Funding source: the National Natural Science Foundation of China

Award Identifier / Grant number: 12071017

Acknowledgments

The authors are grateful to the anonymous referees for their careful reading and constructive suggestions. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 12141103 and 12071017).

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: Yannan Liu (Corresponding author): Methodology, Funding acquisition, Review & Editing. Yuxing Peng: Validation, Original Draft.
Use of Large Language Models, AI and Machine Learning Tools: LanguageTool to detect spelling mistakes.
Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this paper.
Research funding: None declared.
Data availability: Not applicable.

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Received: 2024-11-16

Revised: 2025-04-20

Accepted: 2025-04-20

Published Online: 2025-05-22

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

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https://doi.org/10.1515/ans-2023-0188

Keywords for this article

Monge-Ampère equation; Orlicz-Minkowski problem; Gauss curvature flow; existence of solutions

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