Mathematical Sciences: LP Regularity for Nonelliptic Differential Equations

该奖项支持有关数学研究， 非椭圆型非线性微分方程 的 目标是建立解的正则性结果 属于各种空间中的函数类，称为 Sobolev空间 所考虑的微分算子是反演的 通过积分公式的傅里叶-艾里型。这些有一个 振荡内核，并在深入调查，在这个 时间目前，可以获得映射属性 对积分算子内的导数损失很小;功 是为了表明人们不需要假设这种额外的损失。 其他重点将放在建立p次幂 补上的波动方程解的正则性 严格的凸形障碍物。 取得了较好的效果 从一个模型算例出发，通过考虑边界条件为 与障碍物的固定距离， 减少。 如果这种技术可以在一般情况下得到验证 它不仅提供了正则性存在性证明， 并给出了边界逼近的收敛方法 数据 目前正在对斜向 导数问题，其中边界数据以 “通量”以斜向均匀通过边界 切线方向 已知有两种解决方法，每种方法 取决于其他假设。 将努力 在他们之间进行尖锐的比较。

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.