某些偏微分算子的非解析亚椭圆性  

Nonanalytic Hypoellipticity for Certain Partial Differential Operators

作　　者：马玉兰[1] 钟承奎[1] 

机构地区：[1]兰州大学数学系

出　　处：《兰州大学学报（自然科学版）》1998年第3期15-19,共5页Journal of Lanzhou University(Natural Sciences)

摘　　要：讨论由向量场的平方和构成的某些二阶偏微分算子的非解析亚椭圆性．所用方法是构造这些算子的奇异解后，将这些偏微分算子的解析亚椭圆性的讨论转化成某些特殊的常微分方程的研究．该构造过程是非常清晰和直接的．而且它涉及到那些带有依赖大参数位势的广义调和振子的第一特征值和第一特征函数的研究，最终获得了如下两个结果：１）令非负整数ｌ，ｋ满足ｌ＞ｋ且ｌ＝２ｋ＋１，则算子Ｐ＝（ｘ）２＋（ｘｋｙ）２＋（ｘｌｔ）２在原点ｏ∈Ｒ３不是解析亚椭圆的．２）对非负整数ｋｉ≥１，ｉ＝１，２，３，…，ｎ，算子Ｑ＝（ｘ）２＋（ｙ）２＋ｘ２ｋ１（ｔ１）２＋…＋ｘ２ｋｎ（ｔｎ）２其中（ｘ，ｙ，ｔ１，…，ｔｎ）∈Ｒｎ＋２，则Ｑ在原点ｏ∈Ｒｎ＋２不是解析亚椭圆的．The non-analytic hypoellipticity for certain second order partial differential operators expressed as sums of squares of complex vector fields is discussed here. By the method of constructing singular solutions to these operators, the discussion of the above partial differential operators is transformed into a study of some special ordinary differential equations. This construction is quite explicit and it is involved with the study of the first eigenvalue and the first eigenfunction of a generalized harmonic oscillator with a potential depending on a large parameter. At last, the following results are obtained. 1) Let non-negative integers l,k, satisfy l>k and l=2k+1, the operatorP=( x) 2+(x k y) 2+(x t t) 2is not analytic hypoelliptic at o∈R 3. 2) For all non-negative integers k i≥1,i=1,2,…,n,the operatorQ=( x) 2+( y) 2+x 2k 1 ( t 1 ) 2+…+x 2k n ( t n ) 2,(x,y,t 1,…,t n)∈R n+2 is not analytic hypoelliptic at o∈R n+2 .

关 键 词：解析亚椭圆性 特征值 特征函数 偏微分算子 

分 类 号：O175.3[理学—数学]