Modern analysis of fundamental structures of solutions to partial differential equations

点击翻译按钮获取中文摘要

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.