帶梯度項(xiàng)的廣義復(fù)Monge-Ampère型方程的梯度估計(jì)

袁 日 榮

(廈門大學(xué)數(shù)學(xué)科學(xué)學(xué)院,福建 廈門 361005)

(1)

(2)

若c0=1且其他的常數(shù)為零,則方程(1)就是著名的復(fù)Monge-Ampère型方程.帶梯度項(xiàng)的復(fù)Monge-Ampère型方程的一個(gè)特殊且重要的例子是Sasakian度量空間的測地線方程[1].本研究的目的之一是將他們的估計(jì)推廣到更一般的情形.當(dāng)c0=c1=…=cn-3=0時(shí),筆者得到了方程(1)的C2,α-估計(jì)[2].

對(duì)于標(biāo)準(zhǔn)的方程而言,即χ為流形M上一個(gè)光滑的實(shí) (1,1)-形式,這類形如式(1) 的方程的研究可追溯到文獻(xiàn)[3-6]，這些工作研究了復(fù) Monge-Ampère 方程.從那以后,這類方程引起了很多有趣且重要的研究,可參考文獻(xiàn)[7-14]及其引用的文獻(xiàn).

在陳述本文中主要結(jié)果之前,需要介紹一些符號(hào).在局部坐標(biāo)(z1,…,zn)下,記

(3)

同時(shí)需假設(shè)χ滿足如下結(jié)構(gòu)條件：

(4)

本文中的主要結(jié)果可如下表述:

(5)

那么對(duì)于方程(1)滿足χu>0的解u∈C3(M),存在一個(gè)依賴|ψ|C0,1(M)及其他已知信息的常數(shù)C,使得

(6)

注1定理 1推廣了Guan等[1]的梯度估計(jì).條件(5)可看成一種錐條件[13],有興趣的讀者可參考文獻(xiàn) [9-10，12，22].

1 預(yù)備知識(shí)

記λ=λ(χu)為χu關(guān)于K?hler形式ω的特征根.令σk為一個(gè)初等對(duì)稱函數(shù),其定義為

(7)

方程(1)可等價(jià)地寫成

(8)

本文中主要結(jié)果的關(guān)鍵是下面的引理1.Fang等[9]首先對(duì)反σk方程證明這個(gè)引理1.Guan[15-16]和Székelyhidi[22]將它推廣到更一般的Hessian方程.

引理1假設(shè)條件(5)成立,那么存在兩個(gè)正常數(shù)R0,ε>0,使得當(dāng)|χu|≥R0時(shí),有

(9)

(10)

(11)

2 定理的證明

定理1的證明考慮如下閘函數(shù)

φ=Aeη,

其中A和B為兩個(gè)待定的正常數(shù).

(12)

由假設(shè)可知,在p點(diǎn)處有

(13)

約定下文中的計(jì)算均在p點(diǎn)處進(jìn)行.通過計(jì)算可知

(14)

由此

(15)

對(duì)方程 (8) 進(jìn)行求導(dǎo)可得

(16)

現(xiàn)在,利用前文中的假設(shè) (4) 得到

(17)

式中的L為方程 (8) 的線性化算子

v∈C2(M).

(18)

簡單計(jì)算可得

φi=φηi,

(19)

使用 Cauchy-Schwarz 不等式和假設(shè) (4),得到

(20)

和

(21)

再由式(13),(15)～(21),有

(22)

(23)

下面,依據(jù)引理1來進(jìn)行討論.假設(shè) |λ|≤R0,其中R0,ε均為引理 1中的常數(shù),那么存在常數(shù)K1>0,使得

是平凡的.

若|λ|≥R0,則由式(4)和引理 1可知

因此,

(24)

不失一般性,可假設(shè)λ1≥…≥λn.如果A,B?1,并且

那么

(25)

為證明式(25),僅需要驗(yàn)證存在某個(gè)正數(shù)δ>0,使得λn≥δ.

綜上可知，如果λi≤λj，則

fjλj≤fiλi,

因而,

(26)

由Cauchy-Schwarz 不等式

(27)

由式(22),(25),(27) 和引理1可得到定理1梯度估計(jì)的證明.

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