Modern analysis of fundamental structures of solutions to partial differential equations

首席研究员Okaji研究了偏微分方程解的两个基本性质，强唯一延拓性质和奇点传播。至于第一个性质，他处理了一阶方程的椭圆系统，这在数学物理中很重要，如狄拉克算子或时间调和麦克斯韦方程。在与De Carli的联合工作中，他得到了保证具有库仑型奇异势的Dirac算子具有强唯一连续性的一个很好的条件。进一步证明了非各向同性非均匀连续介质中的时谐Maxwell方程，如果其系数连续可微，则具有强的延拓性。至于第二个主题，他发明了一种新的方法来研究薛定谔方程解的奇点传播。该方法基于解的维格纳变换所满足的微局部守恒律。它与解的波包变换密切相关。作为应用，他可以阐明具有矢量势的薛定谔方程解的微局部奇点是如何传播的，以及可能在无穷远处增长的电势。它包括平滑效果、奇点重建和从振荡初始数据中创建奇点。

The head investigator Okaji has investigated two fundamental properties, strong unique continuation property and propagation of singularities, of solutions to partial differential equations. As for the first property, he has treated elliptic systems of first order equations which are important in mathematical physics like Dirac operator or time harmonic Maxwell equations. In a joint work with De Carli, he obtained a nice condition which assures the strong unique continuation property for the Dirac operator with singular potential of Coulomb type. Furthermore, he has shown that the time harmonic Maxwell equation in non-isotropic and non-uniform continuous media has the strong continuation property if its coefficients are continuously differentiable.As for the second topic, he has invented a new approach to the study of propagation of singularities of solutions to Schrodinger equations. This approach is based on a microlocal conservation law satisfied by the Wigner transformation of the solution. It is strongly connected to the wave packet transform of the solutions. As applications, he can clarify how propagate microlocal singularities of solutions to Schrodinger equations with vector potential as well as electric potential which may grow at the infinity. It includes smoothing effects, reconstruction of singularities and creation of singularities from oscillatory initial data.