A PROPER AFFINE SPHERE THEOREM RELATED TO HOMOGENEOUS FUNCTIONS

ZHAO Lei-na

(College of Mathematics and Statistics;College of Transportation,Chongqing Jiaotong University,Chongqing 400074,China)

A PROPER AFFINE SPHERE THEOREM RELATED TO HOMOGENEOUS FUNCTIONS

ZHAO Lei-na

(College of Mathematics and Statistics;College of Transportation,Chongqing Jiaotong University,Chongqing 400074,China)

In this paper,we focus on the affine sphere theorem related to homogeneous function.Based on Hopf maximum principle,we obtain that the affine sphere theorem does hold for given elementary symmetric curvature problems under concavity conditions.In particular,it gives a new proof of Deicke’s theorem on homogeneous functions.

affine sphere theorem;homogeneous functions

1 Main theorems

LetLbe a positive function of classC4(Rn/{0})with homogeneous of degree one.Introducing a matrixgof elements

Deicke[4]showed that the matrixgis positive and the following theorem,a short and elegant proof was presented in Brickell[1].

Theorem 1.1Let detgbe a constant on Rn/{0}.Thengis a constant matrix on Rn/{0}.

Theorem 1.1 is very important in affine geometry[10,11,13]and Finsler geometry[4].There are lots of papers introducing the history and progress of these problems,for example[7].A laplacian operator and Hopf maximum principle is the key point of Deicke[4]’s proof.However,our method depends on the concavity of the fully nonlinear operator,we give a new method to prove more generalized operator than Theorem 1.1,for considering operatorF(g),which including the operator of determinant.

Theorem 1.2LetF(g)be a constant on Rn/{0},F(g)be concave with respect to matrixg,and the matrixbe positive semi-de finite.Thengis a constant matrix on Rn/{0}.

In fact

(1)IfF(g)=logdetg,Theorem 1.2 is just Theorem 1.1.

(2)An interesting example of Theorem 1.2 is,whereSk(g)is the elementary symmetric polynomial of eigenvalues ofg.The concavity ofF(g)was from Ca ff arelli-Nirenberg-Spruck[3].A similar Liouville problem for theS2equation was obtained in[2].

It is easy to see that the method of Brickell[1]does not apply to our Theorem 1.2.

On the other hand,there are some remarkable results for homogeneous solution to partial differential equations.Han-Nadirashvili-Yuan[6]proved that any homogeneous order 1 solution to nondivergence linear elliptic equations in R3must be linear,and Nadirashvili-Yuan[8]proved that any homogeneous degree other than 2 solution to fully nonlinear elliptic equations must be“harmonic”.In fact,our methods can also be used to deal with the following hessian type equations

More recently,Nadirashvili-Vlǎdut?[9]obtained the following theorem.

Theorem 1.3Letube a homogeneous order 2 real analytic function in R4/{0}.Ifuis a solution of the uniformly elliptic equationF(D2u)=0 in R4/{0},thenuis a quadratic polynomial.

However,our theorem say that above theorem holds providedFwith some concavity/convexity property.Pingali[12]can show for 3-dimension,there is concave operatorGformFwithout some concavity/convexity property,for example

forλ1≤ λ2≤ λ3are eigenvalues of hessian matrixD2u.Then

has a uniformly positive gradient and is concave ifλ1＞3.That is to say,using our methods,there is a simple proof of Theorem 1.3 if one can construct a concave operator with respect toFin Theorem 1.3.

2 Proof of Theorem 1.2

Here we firstly list the Hopf maximum principle to be used in our proof,see for example[5].

Lemma 2.1Letube aC2function which satisfies the differential inequality

in an open domain ?,where the symmetric matrixaijis locally uniformly positive de finite in ? and the coefficientsaij,biare locally bounded.Ifutakes a maximum valueMin ?thenu≡M.

Proof of Theorem 1.2Di ff erentiating this equation twice with respect tox

one has

The concavity ofF(g)with respect togsays that the matrixis positive semi-de finite.In particular,

We firstly consider(2.2)as an inequality in unit sphereSn?1,

that is to say using Hopf maximum principle of Lemma 2.1 and taking ?=Sn?1,it shows thatgkkis constant onSn?1,and it is so on Rn/{0}becausegkkis positively homogeneous of degree zero.Then,owing to the matrixFijgijklbe positive semi-de finite

Using Hopf maximum principle again andgklis positively homogeneous of degree zero,then the matrixgis constant matrix.We complete the proof of Theorem 1.2.

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齊次函數的一個仿射球定理

趙磊娜
(重慶交通大學數學與統計學院;交通運輸學院,重慶 400074)

本文研究了相關齊次函數的仿射球定理.利用Hopf極大值原理,對任意給定的帶凹性條件的初等對稱曲率問題,獲得了此類仿射球定理.特別地,這也給出了Deicke齊次函數定理的一個新證明.

仿射球定理;齊次函數

O175.25

35B50;35J15

A

0255-7797(2017)06-1173-04

date:2017-01-08Accepted date:2017-04-25

Supported by the Science and Technology Research program of Chongqing Municipal Education Commission(KJ1705136).

Biography:Zhao Leina(1981–),female,born at Qingdao,Shandong,lecture,major in partial differential and its applications.