Mathematical Sciences: LP Regularity for Nonelliptic Differential Equations

数学科学：非椭圆微分方程的 LP 正则性

• 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

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.

该奖项支持与非旋转性非线性微分方程有关的数学研究。 目标是建立有关在称为Sobolev空间的各个空间中属于功能类别的解决方案的规律性结果。 正在考虑的差异操作员通过傅立叶仪类型的整体公式反转。这些具有振荡性核，目前正在进行深入研究。当前，可以在派生型损失少的情况下获得积分运算符的映射属性；正在完成工作以表明人们不必承担额外的损失。 其他重点将放在确定波动方程解决方案解决方案的第p-the Power warrifations在严格凸出障碍的补充上。 通过考虑边界与障碍物固定距离的问题，然后让距离减小，从模型情况下获得了一些良好的结果。 如果在一般情况下可以验证此技术，它将不仅提供规律性证明，而且还提供了近似边界数据的收敛方法。 正在对倾斜导数问题进行其他工作，其中通过“通量”通过斜边界甚至切向方向的边界给出边界数据。 已知两种解决方案方法，每个方法取决于其他假设。 将做出努力以对它们之间进行彻底的比较。