Maximum and anti-maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations
Maximum and anti-maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations
复制标题
椭圆方程正解的微扰理论的极大值和反极大值原理以及本征函数估计
DOI：
10.1007/s002080050307
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发表时间：
1999
影响因子：
1.4
通讯作者：
Y. Pinchover
中科院分区：
文献类型：
--
作者：
Y. Pinchover
抽象。本文讨论了二阶椭圆型偏微分方程扰动理论的一些新结果，这些结果与方程的正性有关。我们继续研究一些不同的“小”扰动的概念，并讨论它们的关系比较绿色的功能，完善的最大和反最大原则，基态，和本征函数的衰减。
特别地，我们证明了，如果V是定义在区域上的次临界薛定谔算子H的半小扰动的正函数，
$\Omega\subset \mathbb{R}^d$，以及
$\{\phi_k\}_{k\geq 0}$是方程的（Dirichlet）本征函数
$Hu=\lambda Vu$，则对于任何
$k\geq 0$，函数
$\phi_k/\phi_0$是有界的，并且有一个直到Martin边界的连续扩张
$（\Omega，H）$，其中
$\phi_0 $是具有主本征值的H的基态
$\mada_0 $.
Abstract. In this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations. We continue the study of some different notions of “small” perturbations and discuss their relations to comparisons of Green's functions, refined maximum and anti-maximum principles, ground state, and the decay of eigenfunctions.
In particular, we show that if V is a positive function which is a semismall perturbation of a subcritical Schrödinger operator H defined on a domain
$\Omega\subset \mathbb{R}^d$, and
$\{\phi_k\}_{k\geq 0}$ are the (Dirichlet) eigenfunctions of the equation
$Hu=\lambda Vu$, then for any
$k\geq 0$, the function
$\phi_k/\phi_0$ is bounded and has a continuous extension up to the Martin boundary of the pair
$(\Omega, H)$, where
$\phi_0$ is the ground state of H with a principal eigenvalue
$\lambda_0$.