Mathematical Sciences: Semilinear Elliptic Equations and Systems

在这个数学研究计划中概述的五个项目中，四个将专注于半线性椭圆型偏微分方程，而第五个将关注二阶常微分方程本征值问题。第一个任务是研究半线性方程正解的存在性和性质，其动机是证明临界分枝布朗场的加权占用时间的大偏差结果。这些概率问题是最近几位研究人员调查的主题。由于缺乏某些紧凑性条件，标准技术不适用于此上下文。当半线性方程首次被引入时，未知函数的一些幂作为方程的孤立元素出现。在本工作中，未知数的功率乘以一个连续函数(边界数据为零)。目标是获得关于乘子的条件，以确保正解的存在--特别是非径向的解。相关工作将寻求量化已确定径向解数的一类方程的非径向正解的数目。此外，还将对与时间无关的薛定谔方程的高阶特征值的最大化和最小化问题进行研究。当允许势在一类势中变化时，这种变化就会发生。解决了经典勒贝格空间中位势的一阶情形。最初的努力将集中在一维方程上。

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.