Fundamental and applicable study of integral operators

积分算子的基础与应用研究

基本信息

批准号：
60540116
负责人：
OKANO Hatsuo
金额：
$ 1.15万
依托单位：
University of Osaka Prefecture
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1985
资助国家：
日本
起止时间：
1985 至 1986
项目状态：
已结题

项目摘要

We investigated properties of Fourier integral operators and studied the singularities of solutions of weakly hyperbolic operators. We clarified the Gevrey wave front sets of solutions of differential equations with coefficients in Gevrey classes. The results are as follows:1. The appearance of wave front sets of fundamental solutions of weakly hyperbolic operators depends on their lower order terms and their fundamental solutions may have the Gevrey wave front sets. In our project we construct exactly the fundamental solutions and investigate their wave front sets and Gevrey wave front sets. Then we can show that the solutions of the Cauchy problem have the branching singularities in a Gevrey class. In order to construct the fundamental solutions we use the Stokes coefficients of associated ordinary differential equations. Then, we can determine exactly whether or not the solution of the Cauchy problem of a degenerate hyperbolic equation has branching singularities.2. In the study of singularities of solutions of differential equations with coefficients in Gevrey classes we give a precise estimate of multi-products of Fourier integral operators and pseudo-differential operators. We construct, as an application, the fundamental solution for a weakly hyperbolic operator and the parametrix of a hypoelliptic operator, and investigate the Gevrey wave front sets of solutions. Throughout this project we use mainly oscillatory integrals.

我们研究了傅里叶积分算子的性质并研究了弱双曲算子解的奇异性。我们阐明了具有 Gevrey 类系数的微分方程解的 Gevrey 波前组。研究结果如下： 1．弱双曲算子的基本解的波前集的出现取决于它们的低阶项，并且它们的基本解可能具有Gevrey波前集。在我们的项目中，我们准确构建了基本解决方案并研究了它们的波前集和 Gevrey 波前集。然后我们可以证明柯西问题的解在 Gevrey 类中具有分支奇点。为了构造基本解，我们使用相关常微分方程的斯托克斯系数。这样就可以准确判断简并双曲方程柯西问题的解是否存在分支奇点。 2．在研究带有 Gevrey 类系数的微分方程解的奇异性时，我们给出了傅里叶积分算子和伪微分算子的多重乘积的精确估计。作为一个应用，我们构造了弱双曲算子的基本解和亚椭圆算子的参数矩阵，并研究了 Gevrey 波前解集。在整个项目中，我们主要使用振荡积分。