Mathematical Sciences: LP Regularity for Nonelliptic Differential Equations

数学科学：非椭圆微分方程的 LP 正则性

基本信息

批准号：
9203904
负责人：
Hart Smith
金额：
$ 4.14万
依托单位：
University of Washington
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1992
资助国家：
美国
起止时间：
1992-07-01 至 1994-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9203904&HistoricalAwards=false
关键词：
Mathematical Sciences LP Regularity Nonelliptic

项目摘要

This award supports mathematical research concerned with nonlinear differential equations which are nonelliptic. The goals are to establish regularity results about solutions belonging to classes of functions in various spaces called Sobolev spaces. The differential operators under consideration are inverted through integral formulas of the Fourier-Airy type. These have an oscillatory kernel and are under intensive investigation at this time. Currently it is possible to obtain the mapping properties of the integral operators within a small loss of derivative; work is being done to show that one need not assume this extra loss. Other emphasis will be placed on establishing the p-th power regularity of solutions of the wave equation on the complement of strictly convex obstacles. Some good results have been obtained from a model case, by considering the problem with boundary a fixed distance from the obstacle and then letting the distance decrease. If this technique can be validated in the general case it will not only provide an existence proof of regularity but also provide a convergence method for approximating boundary data. Additional work is being carried out on the oblique derivative problem, where boundary data is given in terms of 'flux' passing through the boundary at oblique and even tangential directions. Two methods of solutions are known, each depending on additional assumptions. Efforts will be made to make sharp comparisons between them.

该奖项支持与非椭圆非线性微分方程相关的数学研究。目标是建立属于称为索博列夫空间的各种空间中函数类的解的规律性结果。所考虑的微分算子通过傅里叶-艾里类型的积分公式求逆。它们具有振荡内核，目前正在深入研究中。目前可以在较小的导数损失下获得积分算子的映射性质；正在开展的工作表明人们无需承担这一额外损失。其他重点将放在建立严格凸障碍物补上的波动方程解的 p 次方正则性上。通过考虑边界与障碍物距离固定然后让距离减小的问题，从模型案例中获得了一些良好的结果。如果该技术能够在一般情况下得到验证，那么它不仅可以提供规律性的存在证明，而且可以提供一种逼近边界数据的收敛方法。关于斜导数问题正在进行额外的工作，其中边界数据以在斜向甚至切向方向穿过边界的“通量”给出。已知有两种解决方法，每种方法都依赖于额外的假设。我们将努力对它们进行尖锐的比较。