Mathematical Sciences: Semilinear Elliptic Equations and Systems

数学科学：半线性椭圆方程和系统

• 批准号: 8914778
• 负责人: Henrik Egnell
• 金额: $ 3.28万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1991-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8914778&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Semilinear; Elliptic; Equations

Of the five projects outlined in this mathematical research plan, four will focus on semilinear elliptic partial differential equations while the fifth concerns eigenvalue problems for second order ordinary differential equations. The first tasks involve studies of existence and properties of positive solutions of semilinear equations motivated by attempts to prove large deviation results for weighted occupation time of the critical branching Brownian field. These probabilistic questions have been the subject of recent investigations by several researchers. Standard techniques do not apply in this context because of a lack of certain compactness conditions. When the semilinear equations were first introduced, some power of the unknown function appeared as an isolated element of the equation. In the present work, the power of the unknown is multiplied by a continuous function (with zero boundary data). The goal is to obtain conditions on the multiplier which will ensure the existence of positive solutions- especially solutions which are not radial. Related work will seek to quantify the number of nonradial positive solutions of classes of equations in which the number of radial solutions has been established. In addition, work will be done on the problem of maximizing and minimizing the higher order eigenvalues of the time independent Schrodinger equation. The variation takes place as the potential is allowed to change within a class of potentials. The first order case has been resolved for potentials in the classical Lebesgue spaces. Initial efforts will concentrate on the one-dimensional equation.

在这项数学研究中概述的五个项目中， 计划，四将重点放在半线性椭圆偏 微分方程，而第五个问题的特征值 二阶常微分方程的解。 第一个任务涉及存在和性质的研究 半线性方程的正解 试图证明加权占用的大偏差结果 临界分支布朗场的时间 这些 概率问题一直是最近 几位研究人员的调查。 标准技术确实 不适用于这种情况，因为缺乏某些 紧性条件 当半线性方程第一次 介绍时，未知函数的某些功能出现为 方程的孤立元素。 在目前的工作中， 未知数乘以一个连续函数（零 边界数据）。 目标是获得关于 乘数，这将确保存在的积极解决方案- 特别是非径向的解。 相关的工作将试图量化非径向的数量 正解的类方程，其中的数目 径向解决方案已经建立。 此外，工作将 在最大化和最小化高阶的问题上做的 与时间无关的薛定谔方程的特征值。 的 当电位允许变化时， 在一个类的潜力。 第一个案例是 经典勒贝格空间中的势。 最初的努力将集中在一维方程。