Rotationally symmetric solutions to the L p -Minkowski problem ✩

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Journal of Differential Equations

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Rotationally symmetric solutions to the L _p -Minkowski problem ^✩

Jian Lu

^a

, Xu-Jia Wang

^b

^,∗

aDepartment of Mathematics, Tsinghua University, Beijing 100084, China

bCentre for Mathematics and Its Applications, Australian National University, Canberra, ACT 0200, Australia

a r t i c l e i n f o a b s t r a c t

Article history:

Received 12 September 2012 Available online 22 October 2012 Keywords:

Monge–Ampère equation Minkowski problem A priori estimates Existence of solutions

In this paper we study theLp-Minkowski problem for p= −ⁿ−^1, which corresponds to the critical exponent in the Blaschke–Santalo inequality. We first obtain volume estimates for general solutions, then establish a priori estimates for rotationally symmetric solu- tions by using a Kazdan–Warner type obstruction. Finally we give sufficient conditions for the existence of rotationally symmetric so- lutions by a blow-up analysis. We also include an existence result for the Lp-Minkowski problem which corresponds to the super- critical case of the Blaschke–Santalo inequality.

©2012 Elsevier Inc. All rights reserved.

1. Introduction

Let f be a positive function on the unit sphere Sⁿ. In this paper we are concerned with the solvability of the equation

det

∇

^2H

+

^{H I}

=

^f

Hⁿ⁺² on Sⁿ

,

(1.1)

where His the support function of a bounded convex body K in the Euclidean spaceR^{n+1,I is the} unit matrix,∇^2H=

(

∇i jH

)

is the covariant derivatives of H with respect to an orthonormal frame onSⁿ.

✩ The first author was supported by the Natural Science Foundation of China (Grant No. 11131005) and the Doctoral Programme Foundation of Institution of Higher Education of China. The second author was supported by the Australian Research Council.

*

Corresponding author.

E-mail addresses:lj-tshu04@163.com(J. Lu),Xu-Jia.Wang@anu.edu.au(X.-J. Wang).

0022-0396/$ – see front matter ©2012 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.jde.2012.10.008

Eq. (1.1) is the Lp-Minkowski problem of Lutwak [16] with p= −ⁿ−1. It is called the cen- troaffine Minkowski problem in[9], and is of particular interest due to its invariance under projective transformations on Sⁿ. This equation also arises in a number of applications. It describes self-similar solutions to the anisotropic curve shortening flow [3,10]. The associated parabolic equation also re- ceived considerable interest in image processing [2]. Eq. (1.1) corresponds to the critical case of the Blaschke–Santalo inequality [18], and its existence of solution is a rather complicated problem. The situation is similar, in some aspects, to the Nirenberg problem and the prescribing scalar curvature problem on the sphere, which involve critical exponents of the Sobolev inequalities and have been extensively studied [6,8,12,14,15]. For Eq. (1.1), it is known that when f is constant, all ellipsoids centered at the origin are solutions to (1.1) [5]. So one cannot obtain a priori estimates for solutions without additional assumptions on f. Similarly to the prescribing scalar curvature problem, there exist obstructions for the existence of solutions, such as a Kazdan–Warner type one in[9].

Eq. (1.1) has been studied in a number of papers, see[1,7,11,13,20]for the casen=^{1, and[9,16,17]}

forn

>

1. Whenn=1, (1.1) is a nonlinear ordinary differential equation, which arises in the inves- tigation of self-similar solutions to the anisotropic curve shortening flow [3,10]. Sufficient conditions for the existence of solutions have been found in [1,7,11,13,20] by different methods. In this paper we study then-dimensional case of Eq. (1.1) forn1, especially when f is a rotationally symmetric function.

First we have the following volume estimates.

Theorem 1.1.There exist positive constants Cn

,

C˜n, depending only on n, such that for any solution H to Eq.(1.1), we have

Cn

f_min

|

^K

|

C

˜

n

fmax

,

(1.2)

where fmin=infSⁿ f , fmax=sup_Sⁿ f , and

|

^K

| =

¹ n

+

¹

Sⁿ

Hdet

∇

^2H

+

^{H I}

is the volume of the corresponding convex body K .

Next we consider a priori estimates and existence of rotationally symmetric solutions, that is, solutions which are rotationally symmetric with respect to the x_n+1-axis in Rⁿ⁺¹. In the spherical coordinates, a rotationally symmetric function f on Sⁿ can be regarded as a function on[⁰

, π

]^{, such} that f

(θ )

= ^f

(

x1

, . . . ,

xn+¹

)

withxn+¹=^cos

θ

. In particular f

(

0

)

is the value of f at the north pole and f

( π )

is the value of f at the south pole. Using the superscriptto denote _d^d_θ, we introduce the following two quantities associated with f,

ni

(

f

) =

−

_π^f

(

^π₂

),

n

²

,

0

(

f

(θ ) −

^f

(

^π₂

))

tan

θ

d

θ,

n

=

¹

,

and

pi

(

f

) =

π 0

f

(θ )

cot

θ

d

θ.

Note that by the rotational symmetry, we have f

(

0

)

= ^f

( π )

=^0.

Theorem 1.2.Assume that f ∈^C2

(

Sⁿ

)

when n=^{2and f} ∈^C6

(

Sⁿ

)

when n=2, that f is positive, rotationally symmetric, and that f

(

^π₂

)

=^{0, ni}

(

f

)

=^{0and pi}

(

f

)

=0. Then there exist positive constants C

,

C depending˜ only on n and f , such that for any rotationally symmetric solution H to Eq.(1.1), we have

C

^H

C

˜ .

(1.3)

By the above a priori estimate, we then have the following existence result.

Theorem 1.3.Under assumptions of Theorem1.2, if ni

(

f

) <

0and pi

(

f

) >

0, then Eq.(1.1)admits a rotation- ally symmetric solution.

The proof of the a priori estimates (1.3) is inspired by[1,13], which treats the one dimensional case of the above problem, and by [6], which treats prescribing scalar curvature problem on the sphere. For this approach, we need the rotational symmetry to conclude the uniqueness of solutions in a limiting procedure. For the prescribing scalar curvature problem, the corresponding uniqueness is a consequence of the Liouville theorem.

With the a priori estimates, one can study the existence of solutions by the topological degree theory, as was in[1,13]for the one dimensional case. In this paper we choose a different approach to the existence, namely by a blow-up analysis. However, additional conditions are needed in this approach, just as in the approach by the degree method [1,13]. The blow-up analysis is of some interest itself, as it may apply to the non-rotationally symmetric case as well. We plan to explore this approach further in a subsequent work. In this paper we use the Kazdan–Warner type obstruction to establish the a priori estimates (1.3) and will restrict ourselves to the rotationally symmetric case only. Note also that even in the casen=1, our conditions are different from those in[1,13,20].

The paper is organized as follows. In Section2, we recall an obstruction for the existence of solu- tions in[9]and prove Theorem1.1. Then we prove the a priori estimates, Theorem 1.2, in Section3, and the existence Theorem1.3in Section4. In Section 5, we prove an existence result for the rotation- ally symmetric solutions toL_p-Minkowski problem, in the super-critical case of the Blaschke–Santalo inequality.

The first author would like to thank his supervisor, Professor Huaiyu Jian, for many discussions.

2. A necessary condition and volume estimates

In this section we recall a necessary condition introduced in [9] and give an upper and lower bounds for volume estimates.

Let B be an arbitrarily given

(

n+¹

)

×

(

n+¹

)

matrix. The matrix generates a projective vector field

ξ

, given by

ξ(

x

) =

Bx

−

x^TBx

x

,

x

∈

Sⁿ

.

(2.1)

It was proved that a solution to Eq. (1.1) must satisfy the following necessary condition.

Proposition 2.1.Let H be a C³-solution to Eq.(1.1). Then for the projective vector field

ξ

given by(2.1), we have

Sⁿ

∇

ξf

Hⁿ⁺¹

=

⁰

.

(2.2)

This proposition was proved in[9]using the gnomonic projection. Here we prove it by the moving frame method. The idea of the proof is essentially the same. First we prove the following integral identities on Sⁿ.

Lemma 2.2.For any C³-function u on Sⁿ, and any tangent vector field

ξ

of form(2.1), we have

Sⁿ

udet

∇

^2u

+

^{u I}

div

ξ = (

n

+

¹

)

Sⁿ

det

∇

^2u

+

^{u I}

∇

ξu

,

(2.3)

Sⁿ

u

∇

ξdet

∇

²u

+

u I

= −(

n

+

2

)

Sⁿ

det

∇

²u

+

u I

∇

ξu

.

(2.4)

Proposition2.1follows readily from this lemma. Indeed, if His a solution to Eq. (1.1), then

Sⁿ

∇

ξf Hⁿ⁺¹

=

Sⁿ

∇

ξ

(

Hⁿ⁺²det

(∇

²H

+

H I

))

Hⁿ⁺¹

=

Sⁿ

(

n

+

²

)

det

∇

^2H

+

^{H I}

∇

ξH

+

^H

∇

ξdet

∇

^2H

+

^{H I}

=

⁰

.

In the following we will denote by u,i= ∇iuandu,i j= ∇jiuthe covariant derivatives of u, in an orthonormal frame on Sⁿ. We also denote as usual that

δ

i j=1 ifi= jand

δ

i j=0 ifi=j.

Proof of Lemma2.2. For simplicity we denote

(

u_{,i j}+^u

δ

i j

)

by

(

A_{i j}

)

, and by A^{i j} the cofactor of A_{i j}. One easily sees that Ai j,kis a symmetric tensor and A^{i j}_,j=0. Hence the following integration by parts holds for any smooth functions

ϕ , ψ

onSⁿ,

A^{i j}

ϕ

_{,i j}

ψ = −

A^{i j}

ϕ

_,i

ψ

_,j

=

A^{i j}

ϕ ψ

_{,i j}

.

(2.5)

All integrals in the proof of Lemma 2.2 is over the unit sphere Sⁿ.

Write

ξ

=

ξ

^ke_k, then

ξ

^k=^xT_,kBx. Calculating its covariant derivatives, we get

ξ

_,^k_i

=

^xT_,k^Bx,i

−

^xTBx

δ

ki

=

x^TBx_,i

,k

,

(2.6)

ξ

_,^k_{i j}

= − ξ

^k

δ

i j

− ξ

^j

δ

_ki

−

^xTBx,i

δ

_jk

−

^xTBx,j

δ

_ki

.

(2.7) Hence

n

det

(

A_{i j}

)∇

ξu

=

A^{i j}

∇

^2u

+

^{u I}

ξ

^ku_,k

=

A^{i j}u

ξ

^ku_,k

,i j

+

^{Ai ju}

δ

i j

ξ

^ku_,k

=

u A^{i j}

ξ

_,^k_{i j}u_,k

+ ξ

_,^k_iu_,kj

+ ξ

_,^k_ju_,ki

+ ξ

^ku_,ki j

+

u A^{i j}

δ

i j

ξ

^ku_,k

=

u A^{i j}u_,k

ξ

_,^k_{i j}

+ ξ

^k

δ

i j

+

2u A^{i j}

ξ

_,^k_iu_,kj

+

u A^{i j}

ξ

^k

(

Aki,j

−

u_,j

δ

ki

).

Taking into account of (2.7) and the symmetry of Ai j,k, the above equality reads

n

det

(

A_{i j}

) ∇

ξu

=

−

^{u Ai ju},k

ξ

^j

δ

_ki

+

^xTBx,i

δ

_jk

+

^xTBx,j

δ

_ki

+

2u A^{i j}

ξ

_,^k_iu_,kj

+

^{u Ai j}

ξ

^kA_{i j,k}

−

^{u Ai ju},j

ξ

ⁱ

=

−

^{2u Ai ju},j

ξ

ⁱ

−

^{2u Ai ju},jx^TBx_,i

+

^{2u Ai j}

ξ

_,^k_iu_,kj

+

^u

ξ

^k

(

detA_{i j}

)

,k

=

−

^{Ai j}

u²

,j

ξ

ⁱ

+

^xTBx,i

+

^{2u Ai j}

ξ

_,^k_iu_,kj

+

u

∇

ξdet

(

Ai j

).

(2.8)

By virtue of (2.5) and (2.6), the first integral can be simplified as follows,

u²A^{i j}

ξ

ⁱ

+

x^TBx_,i

,j

+

2u A^{i j}

ξ

_,^k_iu_,kj

=

u²A^{i j}

ξ

_,ⁱ_j

+ ξ

_,^j_i

+

2u A^{i j}

ξ

_,^k_iu_,kj

=

2u A^{i j}

u

ξ

_,^j_i

+ ξ

_,^k_iu_,kj

=

2u A^{i j}

ξ

_,^k_iA_kj

=

2udet

(

A_{i j}

)δ

_kⁱ

ξ

_,^k_i

=

²

udet

(

A_{i j}

)

div

ξ.

Hence (2.8) becomes

n

det

(

A_{i j}

) ∇

ξu

=

²

udet

(

A_{i j}

)

div

ξ +

u

∇

ξdet

(

A_{i j}

).

(2.9)

On the other hand, the divergence theorem gives

det

(

Ai j

) ∇

ξu

+

u

∇

ξdet

(

Ai j

) +

udet

(

Ai j

)

div

ξ =

⁰

.

(2.10) Now the lemma follows immediately from (2.9) and (2.10). 2

An important property of Eq. (1.1) is its invariance under the projective transformation group SL

(

n+¹

)

. More precisely, let H be a solution to this equation and K the associated convex body inRⁿ⁺¹. Then after making a unimodular linear transformation A^T∈^SL

(

n+¹

)

, the convex body K is changed toKA with support functionHA. We have

H_A

(

x

) = |

^Ax

| ·

^H

Ax

|

^Ax

| ,

x

∈

^Sn

.

(2.11)

Indeed, by the definition of support function,

HA

(

x

) =

^sup

p_A∈^KA

p^T_Ax

=

^sup

p∈K

A^Tp

T

x

=

^sup

p∈^Kp^TAx

= |

^Ax

| ·

^H

Ax

|

^Ax

| .

One can also verify, see[9], thatHAsolves the equation

det

∇

^2HA

+

^HAI

=

^fA

Hⁿ_A⁺²

,

f_A

(

x

) =

^f

Ax

|

^Ax

| .

(2.12)

To understand formula (2.11), it is helpful to consider the corresponding convex body. The support function is the distance from the origin to the tangent plane, and (2.11) is the formula which tells how the distance changes under linear transformation.

It is known that for any non-degenerate convex bodyK, there is a unique ellipsoidE which attains the minimum volume among all ellipsoids containing K [19]. This ellipsoid E is called theminimum ellipsoidofK [19], which satisfies

1

n

+

^1E

⊂

^K

⊂

^E

,

(2.13)

where

α

E= {

α (

x−^x0

)

+^x0|^x∈^E}^andx0 is the center ofE. We sayK isnormalizedifE is a ball.

Next we consider the volume estimate for the solution H. Let K be the convex body with support function H. Recall that the volume ofK is given by

|

^K

| =

¹ n

+

¹

Sⁿ

Hdet

∇

^2H

+

^{H I}

=

¹ n

+

¹

Sⁿ

f Hⁿ⁺¹

.

Lemma 2.3.There exists a positive constant Cndepending only on n, such that for any solution H to Eq.(1.1) we have

H_min

·

^Hn_max

· |

^K

|

^Cnf_min

.

(2.14) Proof. By extendingHtoRⁿ⁺¹such that it is homogeneous of degree one and by the convexity of H, one sees that|∇^H|^Hmax:=^supSⁿH. Hence for any fixed point x₀∈^{Sn, we have}

H

(

x

)

^H

(

x₀

) +

^Hmax

|

^x

−

^x0

| ∀

^x

∈

^Sn

,

(2.15) where| · |means the standard metric inR^n+1.

Direct computation shows

Sⁿ

1 Hⁿ⁺¹

Sⁿ

1

(

H

(

x0

) +

^Hmax

|

^x

−

^x0

| )

ⁿ⁺¹

=

π 0

σ

nsinⁿ⁻¹

θ

(

H

(

x0

) +

Hmax2 sin^θ₂

)

^n+1d

θ

π

2 0

C_n

θ

ⁿ⁻¹

(

H

(

x₀

) +

^Hmax

θ )

^n+1d

θ

^Cn

H

(

x₀

)

Hⁿ_max

π

2 0

tⁿ⁻¹

(

1

+

^t

)

^n+1dt

,

(2.16)

where the spherical coordinate system with respect tox0 is used and

σ

n is the area of unit sphere inRⁿ. Thus we have

H

(

x0

)

Hⁿ_max

Sⁿ

1 Hⁿ⁺¹

^Cn for a different constantCn. Sincex0is any given point, we obtain

H_min

·

^Hnmax

Sⁿ

1

Hⁿ⁺¹

^Cn

.

(2.17)

Therefore

H_min

·

^Hnmax

· |

^K

| =

^Hmin

·

^Hnmax

·

¹ n

+

¹

Sⁿ

f

Hⁿ⁺¹

^Cnf_min

. 2

Proof of Theorem1.1. As the estimates (1.2) are invariant under unimodular linear transformation, we only need to prove it for normalizedH. Let R be the radius of the minimum ellipsoid ofH (actually a ball), then

ω

n+¹

R n

+

1

n+¹

|

K

| ω

n+¹Rⁿ⁺¹

,

(2.18)

where

ω

n+1is the volume of unit ball inR^n+1. Noting that

H_min

·

^Hnmax

^Hnmax⁺¹

(

2R

)

ⁿ⁺¹

(

2n

+

²

)

ⁿ⁺¹

ω

n+1

|

^K

| ,

by virtue of Lemma2.3, one immediately gets the first inequality.

On the other hand,

|

^K

| =

¹ n

+

¹

Sⁿ

Hdet

∇

^2H

+

^{H I}

¹

n

+

¹

Sⁿ

Hⁿ⁺²det

∇

^2H

+

^{H I}

1 n+2

Sⁿ

det

∇

^2H

+

^{H I}

n+1 n+2

=

¹ n

+

¹

Sⁿ

f

1 n+2

Sⁿ

det

∇

^2H

+

^{H I}

n+1 n+2

.

The last integral is equal to the area of the convex hypersurface

∂

K with support functionH, namely

Sⁿ

det

∇

^2H

+

^{H I}

n+1 n+2

=

^area

(

H

)

ⁿⁿ⁺⁺¹²

σ

n+¹Rⁿ

ⁿ⁺¹

n+2

.

Hence we obtain

|

^K

|

^Cn

(

fmax

)

ⁿ⁺¹²

Rⁿ

ⁿ_n⁺₊¹₂

.

Namely

R

^Cnf

1 2n+2 max

,

which together with (2.18) leads to the second inequality of (1.2). 2

We note that when n=1, similar volume estimates were obtained in [1] for centro-symmetric solutions. For normalized solution we then have

Corollary 2.4.There exist positive constants Cn

,

C˜ndepending only on n, such that for any normalized solu- tion H to Eq.(1.1),

Cnf_minf⁻

2n+1 2n+2

max

^H

C

˜

nf

1 2n+2 max

.

3. A priori estimates

From now on we only consider rotationally symmetric solutions to Eq. (1.1). In this case, f must also be rotationally symmetric, and the obstruction (2.2) can be written as

Proposition 3.1.Let H be a rotationally symmetric C³-solution to Eq.(1.1). Then

π 0

f

(θ )

sinⁿ

θ

cos

θ

Hⁿ⁺¹

(θ )

^d

θ =

⁰

.

(3.1)

Proof. Whenn=1, (3.1) can be proved directly by integration by parts[1,13].

Let

ξ

be the vector field given by (2.1) with B=

(

bαβ

)

. Then

∇

ξf

=

f

(θ )

x^T_,θBx

=

f

(θ )

cot

θ

x^T

−

v

(θ )

Bx

,

wherev

(θ )

=

(

0

, . . . ,

0

,

csc

θ )

. Therefore

Sⁿ

∇

ξf Hⁿ⁺¹

=

π 0

f

(θ )

d

θ

Hⁿ⁺¹

(θ )

S_θ

x^T_,θBx

,

whereS_θ= {^x∈^{Sn: x}n+1=^cos

θ}

. Direct computation shows

Sθ

x_,θ^TBx

=

cot

θ

Sθ

x^TBx

−

v

(θ )

B

Sθ

x

=

^cot

θ

α b_αα

S_θ

x^αx^αd

σ −

^bn+1,n+1cot

θ

S_θ

d

σ

=

n

α=1b_αα

n

−

^bn+1,n+1 cot

θ

sin²

θ

S_θ

d

σ

=

^trB

− (

n

+

1

)

bn+¹,n+¹

n

· σ

nsinⁿ

θ

cos

θ.

Hence

Sⁿ

∇

ξf

Hⁿ⁺¹

=

^trB

− (

n

+

1

)

bn+¹,n+¹

n

· σ

n

π 0

f

(θ )

sinⁿ

θ

cos

θ

Hⁿ⁺¹

(θ )

^d

θ.

Thus (3.1) holds. 2

For a rotationally symmetric solution Hto (1.1), one can choose matrix

A

=

^diag

aⁿ⁺¹¹

, . . . ,

aⁿ⁺¹¹

,

a⁻ⁿ⁺ⁿ¹

(3.2)

such that HA is normalized, where a

>

0, and HA, fA are defined in (2.11) and (2.12). To prove Theorem 1.2, we have two cases to consider, that is eithera→ ∞^ora→^0+.

From (2.12), we can write fAas

f_A

(θ ) =

^f

γ

a

(θ )

,

(3.3)

where

γ

a

(θ ) =

arccos

cos

θ

ia

(θ ) ,

ia

(θ ) =

a²sin²

θ +

^cos2

θ .

(3.4)

In fact,

Ax

=

aⁿ⁺¹¹x1

, . . . ,

aⁿ⁺¹¹xn

,

a⁻ⁿ⁺ⁿ¹xn+¹

T

,

|

^Ax

| =

^a−n+n1

a²

(

x₁

)

²

+ · · · + (

x_n

)

²

+ (

x_n+1

)

²

.

(3.5) Therefore in the spherical coordinates we have

Ax

|

^Ax

| =

· ,

^cos

θ

i_a

(θ )

T

,

which implies

fA

(θ ) =

f

·,

^cos

θ

ia

(θ ) =

f

γ

a

(θ ) .

First we prove two auxiliary lemmas.

Lemma 3.2.Let

ϕ

a∈^C[⁰

, π

_]be a sequence of uniformly bounded functions. If

ϕ

aconverges to a constant

ϕ

_∞

>

0locally uniformly in

(

0

, π )

as a→ +∞^{, then}

π

0

ϕ

a

(θ )

f

γ

a

(θ )

−

^f

( π /

2

)

a³sinⁿ

θ

cos

θ

i²_a

(θ )

^d

θ

=

⎧ ⎨

⎩

Cn

ϕ

_∞

(

ni

(

f

) +

^o

(

1

)),

n

³

, ϕ

_∞loga²

(

ni

(

f

) +

^o

(

1

)),

n

=

²

, ϕ

_∞a

(

ni

(

f

) +

^o

(

1

)),

n

=

¹

,

(3.6)

where Cn=_π

0 sinⁿ⁻³

θ

cos²

θ

d

θ

.

Proof. Let

Λ

adenote the integral on the left hand side of (3.6).

Ifn3, we write

Λ

aas

Λ

a

=

π 0

ϕ

a

(θ ) ·

f

γ

a

(θ )

−

^f

( π /

2

)

atan

θ ·

^a2sinn−1

θ

cos²

θ

i²_a

(θ )

^d

θ.

Whena→ +∞, one easily verifies that

f

γ

_a

(θ )

−

^f

( π /

2

)

atan

θ

^sup

[0,π]

f

,

f

γ

_a

(θ )

−

^f

( π /

2

)

atan

θ → −

^f

( π /

2

),

where the convergence is uniform on any closed interval of

(

0

, π )

. By the bounded convergence theorem, we obtain

a→+∞lim

Λ

a

=

π 0

ϕ

_∞

·

−

^f

( π /

2

)

·

^sinn−3

θ

cos²

θ

d

θ = −

^Cn

ϕ

_∞f

( π /

2

).

Namely

Λ

a

=

^Cn

ϕ

_∞

_{· −}

f

( π /

2

) +

^o

(

1

) .

If n=2, we shall use Taylor expansion to evaluate

Λ

a. Denote ˜f

(

t

)

= ^f

(

arccost

)

. Then ˜f ∈ C³[−¹

,

1]^{if f}∈^C6

(

S²

)

¹and

f

γ

a

(θ )

= − ˜

f

cos

θ

ia

(θ ) ·

^asin

θ

ia

(θ )

= −

˜

_f

(

0

) + ˜

^f

(

0

)

^cos

θ

i_a

(θ ) +

^O

(

1

)

^cos

2

θ

i²_a

(θ ) ·

^asin

θ

i_a

(θ ) .

Hence

Λ

a

= −

π 0

ϕ

a

(θ ) ˜

_f

(

0

)

^a

4sin³

θ

cos²

θ

i_a⁴

(θ )

^d

θ −

π 0

O

(

1

)

^a

4sin³

θ

cos³

θ

i⁵_a

(θ )

^d

θ

−

π 0

ϕ

a

(θ )

˜

_f

(

0

)

^asin

θ

i_a

(θ ) +

^f

( π /

2

)

^a

3sin²

θ

cos

θ

i²_a

(θ )

^d

θ.

Noting ˜f

(

0

)

= −^f

(

^π₂

)

and ˜f

(

0

)

= ^f

(

^π₂

)

, one sees that, asa→ +∞^,

Λ

a

= −

ϕ

_∞

+

^o

(

1

)

f

( π /

2

)

1

+

^o

(

1

)

loga²

+

^O

(

1

)

−

π 0

ϕ

_a

(θ )

f

( π /

2

)

1

−

^asin

θ

i_a

(θ )

a³sin²

θ

cos

θ

i²_a

(θ )

^d

θ

= ϕ

_∞

·

−

^f

( π /

2

) +

^o

(

1

)

loga²

−

^a

π 0

O

(

1

)

1

−

^asin

θ

i_a

(θ )

^cos

θ

d

θ

= ϕ

_∞

_{· −}

f

( π /

2

) +

^o

(

1

)

loga²

+

^O

(

1

)

= ϕ

_∞

_{· −}

f

( π /

2

) +

^o

(

1

)

loga²

.

1 We note that ˜f∈^C2is not sufficient. For example, let ˜f(t)=¹+^t−(1−^t2)^3/2. Then˜f isC²but notC³. Observing that

˜f(0)=^{0 and}˜f(0)=1, one can compute

π

2

0

˜f _cosθ

ia(θ ) acosθ

ia(θ ) =π+¹+^{2 log 2} 2 +^o(1),

but

π

2

0

˜f(0)+ ˜^f(0)^cosθ

i_a(θ )+^o(1)^cosθ i_a(θ )

acosθ i_a(θ ) =π

2+^o(1).

They are not equal. The reason is that theo(1)above is not really small nearθ=^0.

Ifn=1, applying the variable substitution

θ

=

γ

_a−1

(

t

)

(more details of this substitution is given below), we find that

Λ

a

=

π 0

ϕ

a

γ

_a−1

(

t

)

f

(

t

) −

f

( π /

2

)

a³sintcost sin²t

+

^a2cos2tdt

=

^a

π 0

ϕ

a

γ

_a−1

(

t

)

f

(

t

) −

^f

( π /

2

)

tant

·

^a2cos2t sin²t

+

^a2cos2tdt

.

Noting that

f

(

t

) −

^f

( π /

2

)

tant

^sup

[0,π]

f

,

we have

a→+∞lim a⁻¹

Λ

a

=

π 0

ϕ

_∞

·

f

(

t

) −

^f

( π /

2

)

tant dt

.

Hence

Λ

a

= ϕ

_∞a

·

π

0

f

(

t

) −

^f

( π /

2

)

tant dt

+

^o

(

1

)

.

This lemma is proved. 2

Lemma 3.3.Let

ϕ

a be a sequence of continuous, uniformly bounded functions on[⁰

, π

]. Assume that

ϕ

a converges a.e. to a function

ϕ

0

>

0as a→^{0+. Then}

π 0

ϕ

a

(θ )

f

γ

a

(θ )

sinⁿ

θ

cos

θ

i_a²

(θ )

^d

θ = ϕ

0

( π /

2

) ·

pi

(

f

) +

^o

(

1

)

.

(3.7)

Proof. Let

Λ

adenote the integral on the left hand side of (3.7). Consider the variable substitution

θ = γ

_a−1

(

t

) =

^arccos

acost

j_a

(

t

) ,

where

j_a

(

t

) =

sin²t

+

^a2cos2t

.

(3.8)

Direct computation shows

cos

θ =

^acost j_a

(

t

) ,

sin

θ =

^sint j_a

(

t

) ,

i_a

(θ ) =

^a

j_a

(

t

) ,

d

θ =

^a

j²_a

(

t

)

^dt

.

Then we find that

Λ

a

=

π 0

ϕ

a

γ

_a−1

(

t

)

f

(

t

)

sint j_a

(

t

)

n

·

^acost j_a

(

t

) ·

j_a

(

t

)

a

2

·

^a j_a²

(

t

)

^dt

=

π 0

ϕ

_a

γ

_a−1

(

t

)

f

(

t

)

^sin

ntcost jⁿ_a⁺¹

(

t

)

^dt

=

π 0

ϕ

a

γ

_a−1

(

t

)

·

^f

(

t

)

cott

·

^sin

n+¹t jⁿ_a⁺¹

(

t

)

^dt

.

Observing that

f

(

t

)

cott

^sup

[⁰,π]

f

, ϕ

a

γ

_a−1

(

t

)

→ ϕ

0

( π /

2

)

a.e.

,

we obtain by the bounded convergence theorem that

lim

a→⁰⁺

Λ

_a

=

π 0

ϕ

₀

( π /

2

) ·

^f

(

t

)

cott dt

.

Hence

Λ

a

= ϕ

0

( π /

2

) ·

π

0

f

(

t

)

cott dt

+

^o

(

1

)

. 2

Now we use Lemmas 3.2 and 3.3 to obtain the a priori estimates (1.3).

Proof of Theorem1.2. By Theorem 1.1, we only need to obtain a uniform positive lower bound for rotationally symmetric solutions. Suppose to the contrary that there exists a sequence of rotationally symmetric solutions H_k to Eq. (1.1) such that min_SⁿH_k→^{0+ ask}→ ∞^{. For each}k, there exists a matrix

A_k

=

^diag

a

1 n+1 k

, . . . ,

a

1 n+1 k

,

a⁻

n n+1 k

,

(3.9)

such that HA_k, given by (2.11), is a normalized rotationally symmetric solution to (2.12). We have eithera_k→ ∞^orak→^0+.