Mathematical Sciences: LP Regularity for Nonelliptic Differential Equations

该奖项支持有关非椭圆型非线性微分方程的数学研究。目标是建立属于各种空间中称为Sobolev空间的函数类的解的正则性结果。通过傅里叶-艾里型积分公式对所考虑的微分算子进行了反演。它们有一个振荡核，目前正在深入研究中。目前可以在很小的导数损失范围内得到积分算子的映射性质；正在进行的工作表明，人们不必承担这一额外损失。另一个重点将放在建立在严格凸障碍补上的波动方程解的p次幂正则性。在一个模型中，考虑边界与障碍物的距离为固定距离，然后逐渐减小，得到了较好的结果。如果这种方法能在一般情况下得到验证，它不仅提供了正则性的存在性证明，而且提供了一种逼近边界数据的收敛方法。在斜导数问题上正在进行额外的工作，其中边界数据是根据在斜方向和甚至切向上通过边界的“通量”给出的。已知有两种解法，每一种都依赖于额外的假设。我们将努力对它们进行比较。

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.