高阶椭圆算子Dirichlet本征值的下界

Lower Bounds for Dirichlet Eigenvalues of Higher-order Elliptic Operators

设Ω是 Rm( m≥ 2 )中的一个有界区域 ,其边界足够光滑 ,考察 2 p( p≥ 1 )阶椭圆算子 ( - 1 ) p ∑|α|=|β|=p α( Aαβ β)在 Dirichlet边界条件下的本征值问题 ,给出了其本征值的一个下界 ,该下界除与维数 m有关外仅依赖于区域Ω的体积 .

Let Ω be a bounded domain in R m(m≥2) with sufficiently smooth boundary. Under the Dirichlet boundary condition, the eigenvalue problem of elliptic operators (-1) p∑|α|=|β|=p α(A αβ  β) with order 2p(p≥1) is discussed. This paper is to give a lower bound for the eigenvalue. Our bound depends on the volume of the domain Ω, except for the dimension m.