Mathematical Sciences: LP Regularity for Nonelliptic Differential Equations

该奖项支持与非椭圆型非线性微分方程有关的数学研究。其目的是在称为Sobolev空间的各种空间中建立属于函数类的解的正则性结果。所考虑的微分算子通过傅立叶-艾里型积分公式求逆。它们有一个振荡的核心，目前正在进行深入的研究。目前，在很小的导数损失范围内获得积分算子的映射性质是可能的；人们正在做的工作是证明人们不需要承担这种额外的损失。另一个重点是建立关于严格凸障碍补的波动方程解的p次方正则性。通过一个模型算例，通过考虑距离障碍物一定距离的边界问题，然后使距离变小，得到了较好的结果。如果这种方法在一般情况下能够得到验证，它不仅可以提供正则性的存在性证明，而且可以为逼近边界数据提供一种收敛的方法。关于斜微商问题的其他工作正在进行中，其中边界数据是以斜向甚至切向通过边界的‘通量’的形式给出的。已知有两种求解方法，每一种都取决于附加的假设。将努力在它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.