四阶椭圆型方程组主次特征值之比的下界

Lower Bound of the Ratio of First Eigenvalue to Second One for Elliptic Differential Equations with Fourth Order

对四阶椭圆型方程组的广义低阶特征值进行估计,利用选定的试验函数与主特征向量的正交条件、向量与矩阵的运算、分部积分法和不等式估计等技巧,证明了这类问题中的主特征向量、试验函数与主特征值间的关系,获得了关于主次特征值之比的一个下界估计不等式,还发现此界与空间的维数有关,但与所论区域的几何度量无关,并将结论推广至一般情形.

Generalized lower order eigenvalue estimate for elliptic differential equations with fourth order is considered.The techniques used are the property of orthogonality between selected trail functions and the first eigenvalue,operation of vector and matrix,integration by parts and inequality estimate etc.The relationship existed among the first eigenvector,trail functions and the first eigenvalue in this kind of problem are proved.The estimated inequality of the lower bound of the ratio of the first eigenvalue to second one is obtained.This bound depends on the space dimension,but not on the measure of the domain in which the problem is concerned.The conclusion is to be generalized to the general case.