First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains
First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains
复制标题
流形或域上椭圆算子的第一特征值和格林函数比较
DOI：
10.1007/bf02843153
复制
发表时间：
1997
期刊：
影响因子：
--
通讯作者：
A. Ancona
中科院分区：
文献类型：
--
作者：
A. Ancona
给定一个完备的黎曼流形M（或RN中的一个区域U）和两个二阶椭圆算子L1，L2在M（或U）中的条件，主要是在无穷大附近（或U）。（近φ U），这意味着它们的绿色函数在大小上是等价的。对于具有给定参考点O的完备流形的情形，条件如下：L1和L2是弱强制的且局部良好的，在[0，∞]上存在可积的且非增的正函数，使得在每个球B（x，1）<$M中L1和L2之间的“距离”（待定义）小于<$（d（x，O））。同时证明了这类椭圆算子谱底的连续性。概括进行了讨论。应用到域的情况下导致迪尼型准则的Lipschitz域（或更一般地说，Hölder型域）。
Given a complete Riemannian manifoldM (or a regionU inRN) and two second-order elliptic operators L1, L2in M (resp.U, conditions, mainly in terms of proximity near infinity (resp. near ∂U) between these operators, are found which imply that their Green’s functions are equivalent in size. For the case of a complete manifold with a given reference pointO the conditions are as follows:L1 andL2 are weakly coercive and locally well-behaved, there is an integrable and nonincreasing positive function Ф on [0, ∞[ such that the “distance” (to be defined) betweenL1 andL2 in each ballB(x, 1 ) ⊂M is less than Ф(d(x, O)). At the same time a continuity property of the bottom of the spectrum of such elliptic operators is proved. Generalizations are discussed. Applications to the domain case lead to Dini-type criteria for Lipschitz domains (or, more generally, Hölder-type domains).